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10: Polar Coordinates, Parametric Equations (Exercises)


These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

10.1: Polar Coordinates

10.1.1Plot these polar coordinate points on one graph: $(2,pi/3)), $(-3,pi/2)), $(-2,-pi/4)), $(1/2,pi)), $(1,4pi/3)), $(0,3pi/2)$.

Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates.

10.1.2 ( y=3x) (answer)

10.1.3 ( y=-4) (answer)

10.1.4 ( xy^2=1) (answer)

10.1.5 ( x^2+y^2=5) (answer)

10.1.6 ( y=x^3) (answer)

10.1.7 ( y=sin x) (answer)

10.1.8 ( y=5x+2) (answer)

10.1.9 ( x=2) (answer)

10.1.10 ( y=x^2+1) (answer)

10.1.11 ( y=3x^2-2x) (answer)

10.1.12 ( y=x^2+y^2) (answer)

Sketch the curve.

10.1.13 ( r=cos heta$

10.1.14 ( r=sin( heta+pi/4)$

10.1.15 ( r=-sec heta$

10.1.16 ( r= heta/2), $ hetage0)

10.1.17 ( r=1+ heta^1/pi^2)

10.1.18 ( r=cot hetacsc heta)

10.1.19 ( r={1oversin heta+cos heta})

10.1.20 ( r^2=-2sec hetacsc heta)

Find an equation in rectangular coordinates that has the same graph as the given equation in polar coordinates.

10.1.21 ( r=sin(3 heta)) (answer)

10.1.22 ( r=sin^2 heta) (answer)

10.1.23 ( r=sec hetacsc heta) (answer)

10.1.24 ( r= an heta) (answer)

10.2: Slopes in polar coordinates

Compute (y'=dy/dx) and ( y''=d^2y/dx^2)

10.2.1 (r= heta) (answer)

10.2.2 (r=1+sin heta) (answer)

10.2.3 (r=cos heta) (answer)

10.2.4 (r=sin heta) (answer)

10.2.5 (r=sec heta) (answer)

10.2.6 (r=sin(2 heta)) (answer)

Sketch the curves over the interval ([0,2pi]) unless otherwise stated.

10.2.7 (r=sin heta+cos heta)

10.2.8 (r=2+2sin heta)

10.2.9 ( r={3over2}+sin heta)

10.2.10 (r= 2+cos heta)

10.2.11 ( r={1over2}+cos heta)

10.2.12 ( r=cos( heta/2), 0le hetale4pi)

10.2.13 (r=sin( heta/3), 0le hetale6pi)

10.2.14 ( r=sin^2 heta)

10.2.15 ( r=1+cos^2(2 heta))

10.2.16 ( r=sin^2(3 heta))

10.2.17 ( r= an heta)

10.2.18 ( r=sec( heta/2), 0le hetale4pi)

10.2.19 ( r=1+sec heta)

10.2.20 ( r={1over 1-cos heta})

10.2.21 ( r={1over 1+sin heta})

10.2.22 ( r=cot(2 heta))

10.2.23 ( r=pi/ heta, 0le hetaleinfty)

10.2.24 ( r=1+pi/ heta, 0le hetaleinfty)

10.2.25 ( r=sqrt{pi/ heta}, 0le hetaleinfty)

10.3: Areas in polar coordinates

Find the area enclosed by the curve.

10.3.1 ( r=sqrt{sin heta}) (answer)

10.3.2 ( r=2+cos heta) (answer)

10.3.3 ( r=sec heta, pi/6le hetalepi/3) (answer)

10.3.4 ( r=cos heta, 0le hetalepi/3) (answer)

10.3.5 ( r=2acos heta, a>0) (answer)

10.3.6 ( r=4+3sin heta) (answer)

10.3.7 Find the area inside the loop formed by ( r= an( heta/2)$. (answer)

10.3.8 Find the area inside one loop of ( r=cos(3 heta)$. (answer)

10.3.9 Find the area inside one loop of ( r=sin^2 heta$. (answer)

10.3.10 Find the area inside the small loop of ( r=(1/2)+cos heta$. (answer)

10.3.11 Find the area inside ( r=(1/2)+cos heta), including the area inside the small loop. (answer)

10.3.12 Find the area inside one loop of ( r^2=cos(2 heta)$. (answer)

10.3.13 Find the area enclosed by $r= an heta$ and ( r={csc hetaoversqrt2}$. (answer)

10.3.14 Find the area inside $r=2cos heta$ and outside $r=1$. (answer)

10.3.15 Find the area inside $r=2sin heta$ and above the line $r=(3/2)csc heta$. (answer)

10.3.16 Find the area inside $r= heta), $0le hetale2pi$. (answer)

10.3.17 Find the area inside ( r=sqrt{ heta}), $0le hetale2pi$. (answer)

10.3.18 Find the area inside both ( r=sqrt3cos heta$ and $r=sin heta$. (answer)

10.3.19 Find the area inside both $r=1-cos heta$ and $r=cos heta$. (answer)

10.3.20 The center of a circle of radius 1 is on the circumference of a circle of radius 2. Find the area of the region inside both circles. (answer)

10.3.21 Find the shaded area in figure 10.3.4. The curve is $r= heta), $0le hetale3pi$. (answer)

Figure 10.3.4. An area bounded by the spiral of Archimedes.

10.4: Parametric Equations

10.4.1 What curve is described by ( x=t^2),( y=t^4)? If (t) is interpreted as time, describe how the object moves on the curve.

10.4.2 What curve is described by (x=3cos t),(y=3sin t)? If (t) is interpreted as time, describe how the object moves on the curve.

10.4.3 What curve is described by (x=3cos t),(y=2sin t)? If (t) is interpreted as time, describe how the object moves on the curve.

10.4.4 What curve is described by (x=3sin t),(y=3cos t)? If (t) is interpreted as time, describe how the object moves on the curve.

10.4.5 Sketch the curve described by ( x=t^3-t),( y=t^2). If (t) is interpreted as time, describe how the object moves on the curve.

10.4.6 A wheel of radius 1 rolls along a straight line, say the(x)-axis. A point(P)is located halfway between the center of the wheel and the rim; assume(P)starts at the point ((0,1/2)). As the wheel rolls, (P) traces a curve; find parametric equations for the curve.(answer)

10.4.7 A wheel of radius 1 rolls around the outside of a circle of radius 3. A point(P)on the rim of the wheel traces out a curve called a hypercycloid, as indicated in figure 10.4.3. Assuming(P)starts at the point ((3,0)), find parametric equations for the curve. (answer)

Figure 10.4.3. A hypercycloid and a hypocycloid.

10.4.8 A wheel of radius 1 rolls around the inside of a circle of radius 3. A point(P)on the rim of the wheel traces out a curve called a hypocycloid, as indicated in figure 10.4.3. Assuming(P)starts at the point((3,0)), find parametric equations for the curve. (answer)

10.4.9 An involute of a circle is formed as follows: Imagine that a long (that is, infinite) string is wound tightly around a circle, and that you grasp the end of the string and begin to unwind it, keeping the string taut. The end of the string traces out the involute. Find parametric equations for this curve, using a circle of radius 1, and assuming that the string unwinds counter-clockwise and the end of the string is initially at((1,0)). Figure 10.4.4 shows part of the curve; the dotted lines represent the string at a few different times. (answer)

Figure 10.4.4. An involute of a circle.

10.5: Calculus with Parametric Equations

10.5.1 Consider the curve of exercise 6 in section 10.4. Find all values of $t$ for which the curve has a horizontal tangent line. (answer)

10.5.2 Consider the curve of exercise 6 in section 10.4. Find the area under one arch of the curve. (answer)

10.5.3 Consider the curve of exercise 6 in section 10.4. Set up an integral for the length of one arch of the curve. (answer)

10.5.4 Consider the hypercycloid of exercise 7 in section 10.4. Find all points at which the curve has a horizontal tangent line. (answer)

10.5.5 Consider the hypercycloid of exercise 7 in section 10.4. Find the area between the large circle and one arch of the curve. (answer)

10.5.6 Consider the hypercycloid of exercise 7 in section 10.4. Find the length of one arch of the curve. (answer)

10.5.7 Consider the hypocycloid of exercise 8 in section 10.4. Find the area inside the curve. (answer)

10.5.8 Consider the hypocycloid of exercise 8 in section 10.4. (answer)

10.5.9 Recall the involute of a circle from exercise 9 in section 10.4. Find the point in the first quadrant in figure 10.4.4 at which the tangent line is vertical. (answer)

10.5.10 Recall the involute of a circle from exercise 9 in section 10.4. Instead of an infinite string, suppose we have a string of length $pi$ attached to the unit circle at $(-1,0)), and initially laid around the top of the circle with its end at $(1,0)$. If we grasp the end of the string and begin to unwind it, we get a piece of the involute, until the string is vertical. If we then keep the string taut and continue to rotate it counter-clockwise, the end traces out a semi-circle with center at $(-1,0)), until the string is vertical again. Continuing, the end of the string traces out the mirror image of the initial portion of the curve; see figure 10.5.1. Find the area of the region inside this curve and outside the unit circle. (answer)

10.5.11 Find the length of the curve from the previous exercise, shown in figure 10.5.1. (answer)

10.5.12 Find the length of the spiral of Archimedes (figure 10.3.4) for $0le hetale2pi$. (answer)

Figure 10.5.1. A region formed by the end of a string.


Rectangular coordinates are the ordinary ((x, y)) coordinates that you are used to.

Polar coordinates represent the same point, but describe the point by its distance from the origin ((r)) and its angle on the unit circle (( heta)). To translate back and forth between polar and rectangular coordinates you should use the basic trig relationships:

(sin heta=frac ightarrow r cdot sin heta=y)

(cos heta=frac ightarrow r cdot cos heta=x)

( an heta=frac ightarrow heta= an ^ <-1>frac)

You can also express the relationship between (x, y) and (r) using the Pythagorean Theorem.

Note that coordinates in polar form are not unique. This is because there are an infinite number of coterminal angles that point towards any given ((x, y)) coordinate.

For example, the point (3,4) can be written in polar coordinates in at least three different ways. To find ( heta), use the third equation from above and to find (r) use the pythagorean theorem.

Three equivalent polar coordinates for the point (3,4) are:

Notice how the third coordinate points in the opposite direction and has a seemingly negative radius. This means go in the opposite direction of the angle.

Once you can translate back and forth between points, use the same substitutions to change equations too. A polar equation is written with the radius as a function of the angle. This means an equation in polar form should be written in the form (r=) ___.

To write an equation in polar form, use the conversion equations to substitute. For example, to convert (y=-x+1) to polar form make substitutions for (y) and (x). Then, solve for (r).

(egin r cdot sin heta &=-r cdot cos heta+1 r cdot sin heta+r cdot cos heta &=1 r(sin heta+cos heta) &=1 r &=frac<1> end)


10 Polar Coordinates, Parametric Equations

1 Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle In polar coordinates a point in the plane is identified by a pair of numbers (r,&theta) The number &theta measures the angle between the positive x-axis and a ray that goes through the point, as shown in figure the number r measures the distance from the origin to the point Figure shows the point with rectangular coordinates (, 3) and polar coordinates (2,&pi/3), 2 units from the origin and &pi/3 radians from the positive x-axis 3 (2,&pi/3) Figure Polar coordinates of the point (, 3) 239

2 24 Chapter Polar Coordinates, Parametric Equations Just as we describe curves in the plane using equations involving x and y, so can we describe curves using equations involving r and &theta Most common are equations of the form r = f(&theta) EXAMPLE Graph the curve given by r = 2 All points with r = 2 are at distance 2 from the origin, so r = 2 describes the circle of radius 2 with center at the origin EXAMPLE 2 Graph the curve given by r = + cos&theta We first consider y = +cosx, as in figure 2 As &theta goes through the values in [,2&pi], the value of r tracks the value of y, forming the cardioid shape of figure 2 For example, when &theta = &pi/2, r = + cos(&pi/2) =, so we graph the point at distance from the origin along the positive y-axis, which is at an angle of &pi/2 from the positive x-axis When &theta = 7&pi/4, r = +cos(7&pi/4) = + 2/2 7, and the corresponding point appears in the fourth quadrant This illustrates one of the potential benefits of using polar coordinates: the equation for this curve in rectangular coordinates would be quite complicated 2 &pi/2 &pi 3&pi/2 2&pi Figure 2 A cardioid: y = +cosx on the left, r = +cos&theta on the right Each point in the plane is associated with exactly one pair of numbers in the rectangular coordinate system each point is associated with an infinite number of pairs in polar coordinates In the cardioid example, we considered only the range &theta 2&pi, and already there was a duplicate: (2,) and (2,2&pi) are the same point Indeed, every value of &theta outside the interval [,2&pi) duplicates a point on the curve r = +cos&theta when &theta < 2&pi We can even make sense of polar coordinates like ( 2,&pi/4): go to the direction &pi/4 and then move a distance 2 in the opposite direction see figure 3 As usual, a negative angle &theta means an angle measured clockwise from the positive x-axis The point in figure 3 also has coordinates (2, 5&pi/4) and (2, 3&pi/4) The relationship between rectangular and polar coordinates is quite easy to understand The point with polar coordinates (r,&theta) has rectangular coordinates x = rcos&theta and y = rsin&theta this follows immediately from the definition of the sine and cosine functions Using figure 3 as an example, the point shown has rectangular coordinates

3 Polar Coordinates Figure 3 The point ( 2, &pi/4) = (2, 5&pi/4) = (2, 3&pi/4) in polar coordinates x = ( 2)cos(&pi/4) = and y = ( 2)sin(&pi/4) = 2 This makes it very easy to convert equations from rectangular to polar coordinates EXAMPLE 3 Find the equation of the line y = 3x+2 in polar coordinates We 2 merely substitute: rsin&theta = 3rcos&theta+2, or r = sin&theta 3cos&theta EXAMPLE 4 Find the equation of the circle (x /2) 2 + y 2 = /4 in polar coordinates Again substituting: (rcos&theta /2) 2 +r 2 sin 2 &theta = /4 A bit of algebra turns this into r = cos(t) You should try plotting a few (r,&theta) values to convince yourself that this makes sense EXAMPLE 5 Graph the polar equation r = &theta Here the distance from the origin exactly matches the angle, so a bit of thought makes it clear that when &theta we get the spiral of Archimedes in figure 4 When &theta <, r is also negative, and so the full graph is the right hand picture in the figure &pi/4 (&pi/2, &pi/2) (,) (&pi,&pi) (2&pi, 2&pi) ( &pi/2, &pi/2) ( 2&pi, 2&pi) (, ) ( &pi, &pi) Figure 4 The spiral of Archimedes and the full graph of r = &theta Converting polar equations to rectangular equations can be somewhat trickier, and graphing polar equations directly is also not always easy

4 242 Chapter Polar Coordinates, Parametric Equations EXAMPLE 6 Graph r = 2sin&theta Because the sine is periodic, we know that we will get the entire curve for values of &theta in [,2&pi) As &theta runs from to &pi/2, r increases from to 2 Then as &theta continues to &pi, r decreases again to When &theta runs from &pi to 2&pi, r is negative, and it is not hard to see that the first part of the curve is simply traced out again, so in fact we get the whole curve for values of &theta in [,&pi) Thus, the curve looks something like figure 5 Now, this suggests that the curve could possibly be a circle, and if it is, it would have to be the circle x 2 +(y ) 2 = Having made this guess, we can easily check it First we substitute for x and y to get (rcos&theta) 2 + (rsin&theta ) 2 = expanding and simplifying does indeed turn this into r = 2sin&theta Figure 5 Graph of r = 2sin&theta Exercises Plot these polar coordinate points on one graph: (2,&pi/3), ( 3,&pi/2), ( 2, &pi/4), (/2,&pi), (,4&pi/3), (,3&pi/2) Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates 2 y = 3x 3 y = 4 4 xy 2 = 5 x 2 +y 2 = 5 6 y = x 3 7 y = sinx 8 y = 5x+2 9 x = 2 y = x 2 + y = 3x 2 2x 2 y = x 2 +y 2 Sketch the curve 3 r = cos&theta

6 244 Chapter Polar Coordinates, Parametric Equations This fraction is zero when the numerator is zero (and the denominator is not zero) The numerator is 2cos 2 &theta +cos&theta so by the quadratic formula cos&theta = ± = or This means &theta is &pi or ±&pi/3 However, when &theta = &pi, the denominator is also, so we cannot conclude that the tangent line is horizontal Setting the denominator to zero we get &theta 2sin&thetacos&theta = sin&theta(+2cos&theta) =, so either sin&theta = or cos&theta = /2 The first is true when &theta is or &pi, the second when &theta is 2&pi/3 or 4&pi/3 However, as above, when &theta = &pi, the numerator is also, so we cannot conclude that the tangent line is vertical Figure 2 shows points corresponding to &theta equal to, ±&pi/3, 2&pi/3 and 4&pi/3 on the graph of the function Note that when &theta = &pi the curve hits the origin and does not have a tangent line 2 Figure 2 Points of vertical and horizontal tangency for r = +cos&theta We know that the second derivative f (x) is useful in describing functions, namely, in describing concavity We can compute f (x) in terms of polar coordinates as well We already know how to write dy/dx = y in terms of &theta, then d dy dx dx = dy dx = dy d&theta d&theta dx = dy /d&theta dx/d&theta EXAMPLE 22 We find the second derivative for the cardioid r = +cos&theta: d cos&theta +cos 2 &theta sin 2 &theta d&theta sin&theta 2sin&thetacos&theta dx/d&theta = = = 3(+cos&theta) (sin&theta+2sin&thetacos&theta) 2 3(+cos&theta) (sin&theta+2sin&thetacos&theta) 3 (sin&theta +2sin&thetacos&theta) The ellipsis here represents rather a substantial amount of algebra We know from above that the cardioid has horizontal tangents at ±&pi/3 substituting these values into the second

7 3 Areas in polar coordinates 245 derivative we get y (&pi/3) = 3/2 and y ( &pi/3) = 3/2, indicating concave down and concave up respectively This agrees with the graph of the function Exercises 2 Compute y = dy/dx and y = d 2 y/dx 2 r = &theta 2 r = +sin&theta 3 r = cos&theta 4 r = sin&theta 5 r = sec&theta 6 r = sin(2&theta) Sketch the curves over the interval [, 2&pi] unless otherwise stated 7 r = sin&theta +cos&theta 8 r = 2+2sin&theta 9 r = 3 +sin&theta r = 2+cos&theta 2 r = +cos&theta 2 r = cos(&theta/2), &theta 4&pi 2 3 r = sin(&theta/3), &theta 6&pi 4 r = sin 2 &theta 5 r = +cos 2 (2&theta) 6 r = sin 2 (3&theta) 7 r = tan&theta 8 r = sec(&theta/2), &theta 4&pi 9 r = +sec&theta 2 r = cos&theta 2 r = +sin&theta 22 r = cot(2&theta) 23 r = &pi/&theta, &theta 24 r = +&pi/&theta, &theta 25 r = &pi/&theta, &theta ½¼º Ö Ò ÔÓÐ Ö ÓÓÖ Ò Ø We can use the equation of a curve in polar coordinates to compute some areas bounded by such curves The basic approach is the same as with any application of integration: find an approximation that approaches the true value For areas in rectangular coordinates, we approximated the region using rectangles in polar coordinates, we use sectors of circles, as depicted in figure 3 Recall that the area of a sector of a circle is &alphar 2 /2, where &alpha is the angle subtended by the sector If the curve is given by r = f(&theta), and the angle subtended by a small sector is &theta, the area is ( &theta)(f(&theta)) 2 /2 Thus we approximate the total area as n i= 2 f(&theta i) 2 &theta In the limit this becomes b a 2 f(&theta)2 d&theta

8 246 Chapter Polar Coordinates, Parametric Equations EXAMPLE 3 We find the area inside the cardioid r = +cos&theta 2&pi 2 (+cos&theta)2 d&theta = 2&pi 2 +2cos&theta+cos 2 &theta d&theta = 2 (&theta +2sin&theta+ &theta 2 + sin2&theta 4 ) 2&pi = 3&pi 2 Figure 3 Approximating area by sectors of circles EXAMPLE 32 We find the area between the circles r = 2 and r = 4sin&theta, as shown in figure 32 The two curves intersect where 2 = 4sin&theta, or sin&theta = /2, so &theta = &pi/6 or 5&pi/6 The area we want is then 2 5&pi/6 &pi/6 6sin 2 &theta 4 d&theta = 4 3 &pi +2 3 Figure 32 An area between curves This example makes the process appear more straightforward than it is Because points have many different representations in polar coordinates, it is not always so easy to identify points of intersection

9 3 Areas in polar coordinates 247 EXAMPLE 33 We find the shaded area in the first graph of figure 33 as the difference of the other two shaded areas The cardioid is r = + sin&theta and the circle is r = 3sin&theta We attempt to find points of intersection: +sin&theta = 3sin&theta = 2sin&theta /2 = sin&theta This has solutions &theta = &pi/6 and 5&pi/6 &pi/6 corresponds to the intersection in the first quadrant that we need Note that no solution of this equation corresponds to the intersection point at the origin, but fortunately that one is obvious The cardioid goes through the origin when &theta = &pi/2 the circle goes through the origin at multiples of &pi, starting with Now the larger region has area and the smaller has area so the area we seek is &pi/8 &pi/6 (+sin&theta) 2 d&theta = &pi 2 &pi/ &pi/6 (3sin&theta) 2 d&theta = 3&pi Figure 33 An area between curves

10 248 Chapter Polar Coordinates, Parametric Equations Exercises 3 Find the area enclosed by the curve r = sin&theta 2 r = 2+cos&theta 3 r = sec&theta,&pi/6 &theta &pi/3 4 r = cos&theta, &theta &pi/3 5 r = 2acos&theta,a > 6 r = 4+3sin&theta 7 Find the area inside the loop formed by r = tan(&theta/2) 8 Find the area inside one loop of r = cos(3&theta) 9 Find the area inside one loop of r = sin 2 &theta Find the area inside the small loop of r = (/2)+cos&theta Find the area inside r = (/2)+cos&theta, including the area inside the small loop 2 Find the area inside one loop of r 2 = cos(2&theta) 3 Find the area enclosed by r = tan&theta and r = csc&theta 2 4 Find the area inside r = 2cos&theta and outside r = 5 Find the area inside r = 2sin&theta and above the line r = (3/2)csc&theta 6 Find the area inside r = &theta, &theta 2&pi 7 Find the area inside r = &theta, &theta 2&pi 8 Find the area inside both r = 3cos&theta and r = sin&theta 9 Find the area inside both r = cos&theta and r = cos&theta 2 The center of a circle of radius is on the circumference of a circle of radius 2 Find the area of the region inside both circles 2 Find the shaded area in figure 34 The curve is r = &theta, &theta 3&pi Figure 34 An area bounded by the spiral of Archimedes

11 4 Parametric Equations 249 ½¼º È Ö Ñ ØÖ ÕÙ Ø ÓÒ When we computed the derivative dy/dx using polar coordinates, we used the expressions x = f(&theta) cos &theta and y = f(&theta) sin &theta These two equations completely specify the curve, though the form r = f(&theta) is simpler The expanded form has the virtue that it can easily be generalized to describe a wider range of curves than can be specified in rectangular or polar coordinates Suppose f(t) and g(t) are functions Then the equations x = f(t) and y = g(t) describe a curve in the plane In the case of the polar coordinates equations, the variable t is replaced by &theta which has a natural geometric interpretation But t in general is simply an arbitrary variable, often called in this case a parameter, and this method of specifying a curve is known as parametric equations One important interpretation of t is time In this interpretation, the equations x = f(t) and y = g(t) give the position of an object at time t EXAMPLE 4 Describe the path of an object that moves so that its position at time t is given by x = cost, y = cos 2 t We see immediately that y = x 2, so the path lies on this parabola The path is not the entire parabola, however, since x = cost is always between and It is now easy to see that the object oscillates back and forth on the parabola between the endpoints (,) and (,), and is at point (,) at time t = It is sometimes quite easy to describe a complicated path in parametric equations when rectangular and polar coordinate expressions are difficult or impossible to devise EXAMPLE 42 A wheel of radius rolls along a straight line, say the x-axis A point on the rim of the wheel will trace out a curve, called a cycloid Assume the point starts at the origin find parametric equations for the curve Figure 4 illustrates the generation of the curve (click on the AP link to see an animation) The wheel is shown at its starting point, and again after it has rolled through about 49 degrees We take as our parameter t the angle through which the wheel has turned, measured as shown clockwise from the line connecting the center of the wheel to the ground Because the radius is, the center of the wheel has coordinates (t,) We seek to write the coordinates of the point on the rim as (t + x, + y), where x and y are as shown in figure 42 These values are nearly the sine and cosine of the angle t, from the unit circle definition of sine and cosine However, some care is required because we are measuring t from a nonstandard starting line and in a clockwise direction, as opposed to the usual counterclockwise direction A bit of thought reveals that x = sint and y = cost Thus the parametric equations for the cycloid are x = t sint, y = cost

12 25 Chapter Polar Coordinates, Parametric Equations t Figure 4 A cycloid (AP) y x Figure 42 The wheel Exercises 4 What curve is described by x = t 2, y = t 4? If t is interpreted as time, describe how the object moves on the curve 2 What curve is described by x = 3cost, y = 3sint? If t is interpreted as time, describe how the object moves on the curve 3 What curve is described by x = 3cost, y = 2sint? If t is interpreted as time, describe how the object moves on the curve 4 What curve is described by x = 3sint, y = 3cost? If t is interpreted as time, describe how the object moves on the curve 5 Sketch the curve described by x = t 3 t, y = t 2 If t is interpreted as time, describe how the object moves on the curve 6 A wheel of radius rolls along a straight line, say the x-axis A point P is located halfway between the center of the wheel and the rim assume P starts at the point (,/2) As the wheel rolls, P traces a curve find parametric equations for the curve 7 A wheel of radius rolls around the outside of a circle of radius 3 A point P on the rim of the wheel traces out a curve called a hypercycloid, as indicated in figure 43 Assuming P starts at the point (3,), find parametric equations for the curve 8 A wheel of radius rolls around the inside of a circle of radius 3 A point P on the rim of the wheel traces out a curve called a hypocycloid, as indicated in figure 43 Assuming P starts at the point (3,), find parametric equations for the curve 9 An involute of a circle is formed as follows: Imagine that a long (that is, infinite) string is wound tightly around a circle, and that you grasp the end of the string and begin to unwind it, keeping the string taut The end of the string traces out the involute Find parametric equations for this curve, using a circle of radius, and assuming that the string unwinds counter-clockwise and the end of the string is initially at (, ) Figure 44 shows part of the curve the dotted lines represent the string at a few different times

13 5 Calculus with Parametric Equations 25 Figure 43 A hypercycloid and a hypocycloid Figure 44 An involute of a circle ½¼º ÐÙÐÙ Û Ø È Ö Ñ ØÖ ÕÙ Ø ÓÒ We have already seen how to compute slopes of curves given by parametric equations it is how we computed slopes in polar coordinates EXAMPLE 5 Find the slope of the cycloid x = t sint, y = cost We compute x = cost, y = sint, so dy dx = sint cost Note that when t is an odd multiple of &pi, like &pi or 3&pi, this is (/2) =, so there is a horizontal tangent line, in agreement with figure 4 At even multiples of &pi, the fraction is /, which is undefined The figure shows that there is no tangent line at such points

14 252 Chapter Polar Coordinates, Parametric Equations Areas can be a bit trickier with parametric equations, depending on the curve and the area desired We can potentially compute areas between the curve and the x-axis quite easily EXAMPLE 52 Findtheareaunder onearchofthecycloidx = t sint, y = cost We would like to compute 2&pi y dx, but we do not know y in terms of x However, the parametric equations allow us to make a substitution: use y = cost to replace y, and compute dx = ( cost) dt Then the integral becomes 2&pi ( cost)( cost) dt = 3&pi Note that we need to convert the original x limits to t limits using x = t sint When x =, t = sint, which happens only when t = Likewise, when x = 2&pi, t 2&pi = sint and t = 2&pi Alternately, because we understand how the cycloid is produced, we can see directly that one arch is generated by t 2&pi In general, of course, the t limits will be different than the x limits This technique will allow us to compute some quite interesting areas, as illustrated by the exercises As a final example, we see how to compute the length of a curve given by parametric equations Section 99 investigates arc length for functions given as y in terms of x, and develops the formula for length: b a + ( ) 2 dy dx dx Using some properties of derivatives, including the chain rule, we can convert this to use parametric equations x = f(t), y = g(t): b a + ( ) 2 dy b dx = dx a = = v u v u (dx ) 2 + dt (dx ) 2 + dt ( dx dt ) 2 ( ) 2 dy dt dx dx dx ( ) 2 dy dt dt (f (t)) 2 +(g (t)) 2 dt Here u and v are the t limits corresponding to the x limits a and b

15 5 Calculus with Parametric Equations 253 EXAMPLE 53 Find the length of one arch of the cycloid From x = t sint, y = cost, we get the derivatives f = cost and g = sint, so the length is 2&pi 2&pi ( cost) 2 +sin 2 t dt = 2 2cost dt Now we use the formula sin 2 (t/2) = ( cos(t))/2 or 4sin 2 (t/2) = 2 2cost to get 2&pi 4sin 2 (t/2) dt Since t 2&pi, sin(t/2), so we can rewrite this as 2&pi 2sin(t/2) dt = 8 Exercises 5 Consider the curve of exercise 6 in section 4 Find all values of t for which the curve has a horizontal tangent line 2 Consider the curve of exercise 6 in section 4 Find the area under one arch of the curve 3 Consider the curve of exercise 6 in section 4 Set up an integral for the length of one arch of the curve 4 Consider the hypercycloid of exercise 7 in section 4 Find all points at which the curve has a horizontal tangent line 5 Consider the hypercycloid of exercise 7 in section 4 Find the area between the large circle and one arch of the curve 6 Consider the hypercycloid of exercise 7 in section 4 Find the length of one arch of the curve 7 Consider the hypocycloid of exercise 8 in section 4 Find the area inside the curve 8 Consider the hypocycloid of exercise 8 in section 4 Find the length of one arch of the curve 9 Recall the involute of a circle from exercise 9 in section 4 Find the point in the first quadrant in figure 44 at which the tangent line is vertical Recall the involute of a circle from exercise 9 in section 4 Instead of an infinite string, suppose we have a string of length &pi attached to the unit circle at (,), and initially laid around the top of the circle with its end at (,) If we grasp the end of the string and begin to unwind it, we get a piece of the involute, until the string is vertical If we then keep the string taut and continue to rotate it counter-clockwise, the end traces out a semi-circle with center at (,), until the string is vertical again Continuing, the end of the string traces out the mirror image of the initial portion of the curve see figure 5 Find the area of the region inside this curve and outside the unit circle

16 254 Chapter Polar Coordinates, Parametric Equations Figure 5 A region formed by the end of a string Find the length of the curve from the previous exercise, shown in figure 5 2 Find the length of the spiral of Archimedes (figure 34) for &theta 2&pi


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Solutions for Chapter 10: PARAMETRIC EQUATIONS AND POLAR COORDINATES

Solutions for Chapter 10: PARAMETRIC EQUATIONS AND POLAR COORDINATES

  • 10.10.1: Sketch and identify the curve defined by the parametric equations x.
  • 10.10.2: What curve is represented by the following parametric equations?x c.
  • 10.10.3: What curve is represented by the given parametric equations?x sin 2.
  • 10.10.4: Find parametric equations for the circle with center and radius r
  • 10.10.5: Sketch the curve with parametric equations x sin t ,y sin2 t
  • 10.10.6: Use a graphing device to graph the curve x y 4 3y 2
  • 10.10.7: The curve traced out by a point on the circumference of a circle as.
  • 10.10.8: Investigate the family of curves with parametric equations What do .
  • 10.10.9: Sketch the curve by using the parametric equations to plot x 1 3t p.
  • 10.10.10: Sketch the curve by using the parametric equations to plot x 1 3t p.
  • 10.10.11: Sketch the curve by using the parametric equations to plot x 1 3t p.
  • 10.10.12: Sketch the curve by using the parametric equations to plot x 1 3t p.
  • 10.10.13: Sketch the curve by using the parametric equations to plot points. .
  • 10.10.14: Sketch the curve by using the parametric equations to plot points. .
  • 10.10.15: Sketch the curve by using the parametric equations to plot points. .
  • 10.10.16: Sketch the curve by using the parametric equations to plot points. .
  • 10.10.17: Sketch the curve by using the parametric equations to plot points. .
  • 10.10.18: Sketch the curve by using the parametric equations to plot points. .
  • 10.10.19: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.20: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.21: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.22: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.23: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.24: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.25: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.26: a) Eliminate the parameter to find a Cartesian equation of the curv.
  • 10.10.27: Describe the motion of a particle with position as varies in the gi.
  • 10.10.28: Describe the motion of a particle with position as varies in the gi.
  • 10.10.29: Describe the motion of a particle with position as varies in the gi.
  • 10.10.30: Describe the motion of a particle with position as varies in the gi.
  • 10.10.31: Suppose a curve is given by the parametric equations , , where the .
  • 10.10.32: Match the graphs of the parametric equations and in (a)(d) with the.
  • 10.10.33: Use the graphs of and to sketch the parametric curve , . Indicate w.
  • 10.10.34: Use the graphs of and to sketch the parametric curve , . Indicate w.
  • 10.10.35: Use the graphs of and to sketch the parametric curve , . Indicate w.
  • 10.10.36: Match the parametric equations with the graphs labeled I-VI. Give r.
  • 10.10.37: Graph the curve x y 3y 3 y 5 x
  • 10.10.38: Graph the curves and and find their points of intersection correct .
  • 10.10.39: (a) Show that the parametric equations 41. ba where , describe the .
  • 10.10.40: Use a graphing device and the result of Exercise 31(a) to draw the .
  • 10.10.41: Find parametric equations for the path of a particle that moves alo.
  • 10.10.42: (a) Find parametric equations for the ellipse . [Hint: Modify the e.
  • 10.10.43: Use a graphing calculator or computer to reproduce the picture.
  • 10.10.44: Compare the curves represented by the parametric equations. How do .
  • 10.10.45: Compare the curves represented by the parametric equations. How do .
  • 10.10.46: Derive Equations 1 for the case 2
  • 10.10.47: Let be a point at a distance from the center of a circle of radius .
  • 10.10.48: If and are fixed numbers, find parametric equations for the curve t.
  • 10.10.49: If and are fixed numbers, find parametric equations for the curve t.
  • 10.10.50: A curve, called a witch of Maria Agnesi, consists of all possible p.
  • 10.10.51: a) Find parametric equations for the set of all points as shown in .
  • 10.10.52: Suppose that the position of one particle at time is given by and t.
  • 10.10.53: If a projectile is fired with an initial velocity of meters per sec.
  • 10.10.54: Investigate the family of curves defined by the parametric equation.
  • 10.10.55: The swallowtail catastrophe curves are defined by the parametric eq.
  • 10.10.56: The curves with equations , are called Lissajous figures. Investiga.
  • 10.10.57: Investigate the family of curves defined by the parametric equation.
  • 10.10.58: A curve is defined by the parametric equations , (a) Show that has .
  • 10.10.59: (a) Find the tangent to the cycloid , at the point where . (See Exa.
  • 10.10.60: Find the area under one arch of the cycloid x rsin y r1 cos V or
  • 10.10.61: If we use the representation of the unit circle given in Example 2 .
  • 10.10.62: Find the length of one arch of the cycloid
  • 10.10.63: Show that the surface area of a sphere of radius is 4 r 2
  • 10.10.64: Find an equation of the tangent to the curve at the point correspon.
  • 10.10.65: Find an equation of the tangent to the curve at the point correspon.
  • 10.10.66: Find an equation of the tangent to the curve at the point correspon.
  • 10.10.67: Find an equation of the tangent to the curve at the point correspon.
  • 10.10.68: Find an equation of the tangent to the curve at the given point by .
  • 10.10.69: Find an equation of the tangent to the curve at the given point by .
  • 10.10.70: Find an equation of the tangent(s) to the curve at the given point.
  • 10.10.71: Find an equation of the tangent(s) to the curve at the given point.
  • 10.10.72: Find and . For which values of is the curve concave upward?
  • 10.10.73: Find and . For which values of is the curve concave upward?
  • 10.10.74: Find and . For which values of is the curve concave upward?
  • 10.10.75: Find and . For which values of is the curve concave upward?
  • 10.10.76: Find and . For which values of is the curve concave upward?
  • 10.10.77: Find and . For which values of is the curve concave upward?
  • 10.10.78: Find the points on the curve where the tangent is horizontal or ver.
  • 10.10.79: Find the points on the curve where the tangent is horizontal or ver.
  • 10.10.80: Find the points on the curve where the tangent is horizontal or ver.
  • 10.10.81: Find the points on the curve where the tangent is horizontal or ver.
  • 10.10.82: Use a graph to estimate the coordinates of the rightmost point on t.
  • 10.10.83: Use a graph to estimate the coordinates of the lowest point and the.
  • 10.10.84: Graph the curve in a viewing rectangle that displays all the import.
  • 10.10.85: Graph the curve in a viewing rectangle that displays all the import.
  • 10.10.86: Show that the curve , has two tangents at and find their equations.
  • 10.10.87: Graph the curve , to discover where it crosses itself. Then find eq.
  • 10.10.88: a) Find the slope of the tangent line to the trochoid , in terms of.
  • 10.10.89: a) Find the slope of the tangent to the astroid , in terms of . (As.
  • 10.10.90: At what points on the curve , does the tangent line have slope ?
  • 10.10.91: Find equations of the tangents to the curve , that pass through the.
  • 10.10.92: Use the parametric equations of an ellipse, , y b sin , 0 2, to fin.
  • 10.10.93: Find the area enclosed by the curve , and n 6 the y-axis
  • 10.10.94: Find the area enclosed by the and the curve
  • 10.10.95: Find the area of the region enclosed by the astroid , . (Astroids a.
  • 10.10.96: Find the area under one arch of the trochoid of Exercise 40 in Sect.
  • 10.10.97: Let be the region enclosed by the loop of the curve in Example 1. (.
  • 10.10.98: Set up an integral that represents the length of the curve. Then us.
  • 10.10.99: Set up an integral that represents the length of the curve. Then us.
  • 10.10.100: Set up an integral that represents the length of the curve. Then us.
  • 10.10.101: Set up an integral that represents the length of the curve. Then us.
  • 10.10.102: Find the exact length of the curve
  • 10.10.103: Find the exact length of the curve
  • 10.10.104: Find the exact length of the curve
  • 10.10.105: Find the exact length of the curve
  • 10.10.106: Graph the curve and find its length.
  • 10.10.107: Graph the curve and find its length.
  • 10.10.108: Graph the curve and find its length.
  • 10.10.109: Find the length of the loop of the curve , y 3t .
  • 10.10.110: Use Simpsons Rule with to estimate the length of the curve y t e 6 .
  • 10.10.111: In Exercise 43 in Section 10.1 you were asked to derive the paramet.
  • 10.10.112: Find the distance traveled by a particle with position as varies in.
  • 10.10.113: Find the distance traveled by a particle with position as varies in.
  • 10.10.114: Show that the total length of the ellipse (e ca , is where c sa2 b2.
  • 10.10.115: Find the total length of the astroid , , where a>0
  • 10.10.116: (a) Graph the epitrochoid with equations What parameter interval gi.
  • 10.10.117: A curve called Cornus spiral is defined by the parametric equations.
  • 10.10.118: Set up an integral that represents the area of the surface obtained.
  • 10.10.119: Set up an integral that represents the area of the surface obtained.
  • 10.10.120: Find the exact area of the surface obtained by rotating the y fx gi.
  • 10.10.121: Find the exact area of the surface obtained by rotating the y fx gi.
  • 10.10.122: Find the exact area of the surface obtained by rotating the y fx gi.
  • 10.10.123: Graph the curve If this curve is rotated about the -axis, find the .
  • 10.10.124: If the curve is rotated about the -axis, use your calculator to est.
  • 10.10.125: If the arc of the curve in Exercise 50 is rotated about the -axis, .
  • 10.10.126: Find the surface area generated by rotating the given curve about t.
  • 10.10.127: Find the surface area generated by rotating the given curve about t.
  • 10.10.128: If is continuous and for , show that the parametric curve , , , can.
  • 10.10.129: Use Formula 2 to derive Formula 7 from Formula 8.2.5 for the case i.
  • 10.10.130: The curvature at a point of a curve is defined as where is the angl.
  • 10.10.131: a) Use the formula in Exercise 69(b) to find the curvature of the p.
  • 10.10.132: Use the formula in Exercise 69(a) to find the curvature of the cycl.
  • 10.10.133: a) Show that the curvature at each point of a straight line is . (b.
  • 10.10.134: A string is wound around a circle and then unwound while being held.
  • 10.10.135: A cow is tied to a silo with radius by a rope just long enough to r.
  • 10.10.136: Plot the points whose polar coordinates are given. (a) (b) (c) (d)
  • 10.10.137: Convert the point from polar to Cartesian coordinates.
  • 10.10.138: Represent the point with Cartesian coordinates in terms of polar co.
  • 10.10.139: What curve is represented by the polar equation r=2?
  • 10.10.140: Sketch the polar curve 0=1
  • 10.10.141: (a) Sketch the curve with polar equation . (b) Find a Cartesian equ.
  • 10.10.142: Sketch the curve r 1 sin
  • 10.10.143: Sketch the curve r cos 2
  • 10.10.144: a) For the cardioid of Example 7, find the slope of the tangent lin.
  • 10.10.145: Graph the curve r sin85 t
  • 10.10.146: Investigate the family of polar curves given by . How does the shap.
  • 10.10.147: Plot the point whose polar coordinates are given. Then find (s2 , 5.
  • 10.10.148: Plot the point whose polar coordinates are given. Then find (s2 , 5.
  • 10.10.149: Plot the point whose polar coordinates are given. Then find the Car.
  • 10.10.150: Plot the point whose polar coordinates are given. Then find the Car.
  • 10.10.151: The Cartesian coordinates of a point are given. (i) Find polar coor.
  • 10.10.152: The Cartesian coordinates of a point are given. (i) Find polar coor.
  • 10.10.153: Sketch the region in the plane consisting of points whose polar coo.
  • 10.10.154: Sketch the region in the plane consisting of points whose polar coo.
  • 10.10.155: Sketch the region in the plane consisting of points whose polar coo.
  • 10.10.156: Sketch the region in the plane consisting of points whose polar coo.
  • 10.10.157: Sketch the region in the plane consisting of points whose polar coo.
  • 10.10.158: Sketch the region in the plane consisting of points whose polar coo.
  • 10.10.159: Find the distance between the points with polar coordinates and
  • 10.10.160: Find a formula for the distance between the points with polar coord.
  • 10.10.161: Identify the curve by finding a Cartesian equation for the curver 2
  • 10.10.162: Identify the curve by finding a Cartesian equation for the curver c.
  • 10.10.163: Identify the curve by finding a Cartesian equation for the curver 3.
  • 10.10.164: Identify the curve by finding a Cartesian equation for the curver 2.
  • 10.10.165: Identify the curve by finding a Cartesian equation for the curver c.
  • 10.10.166: Identify the curve by finding a Cartesian equation for the curver t.
  • 10.10.167: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.168: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.169: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.170: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.171: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.172: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.173: For each of the described curves, decide if the curve would be more.
  • 10.10.174: For each of the described curves, decide if the curve would be more.
  • 10.10.175: Sketch the curve with the given polar equation6 2
  • 10.10.176: Sketch the curve with the given polar equationr 6 2 3r 2 0 2,
  • 10.10.177: Sketch the curve with the given polar equationr sin r
  • 10.10.178: Sketch the curve with the given polar equationr 3 cos
  • 10.10.179: Sketch the curve with the given polar equationr 21 sin 0 r 1
  • 10.10.180: Sketch the curve with the given polar equationr 1 3 cos
  • 10.10.181: Sketch the curve with the given polar equationr 0 r
  • 10.10.182: Sketch the curve with the given polar equationr ln 1 r
  • 10.10.183: Sketch the curve with the given polar equationr 4 sin 3r
  • 10.10.184: Sketch the curve with the given polar equationr cos 5
  • 10.10.185: Sketch the curve with the given polar equationr 2 cos 4r
  • 10.10.186: Sketch the curve with the given polar equationr 3 cos 6
  • 10.10.187: Sketch the curve with the given polar equationr 1 2 sin 4
  • 10.10.188: Sketch the curve with the given polar equationr 2 sin
  • 10.10.189: Sketch the curve with the given polar equation
  • 10.10.190: Sketch the curve with the given polar equation
  • 10.10.191: Sketch the curve with the given polar equation
  • 10.10.192: Sketch the curve with the given polar equation
  • 10.10.193: Sketch the curve with the given polar equation
  • 10.10.194: Sketch the curve with the given polar equation
  • 10.10.195: The figure shows the graph of as a function of in Cartesian coordin.
  • 10.10.196: The figure shows the graph of as a function of in Cartesian coordin.
  • 10.10.197: Show that the polar curve (called a conchoid) has the line as a ver.
  • 10.10.198: Show that the curve (also a conchoid) has the line as a horizontal .
  • 10.10.199: Show that the curve (called a cissoid of Diocles) has the line as a.
  • 10.10.200: Sketch the curve x 2 y 2 3 4x 2 y 2
  • 10.10.201: a) In Example 11 the graphs suggest that the limaon has an inner lo.
  • 10.10.202: Match the polar equations with the graphs labeled IVI. Give reasons.
  • 10.10.203: Find the slope of the tangent line to the given polar curve at the .
  • 10.10.204: Find the slope of the tangent line to the given polar curve at the .
  • 10.10.205: Find the slope of the tangent line to the given polar curve at the .
  • 10.10.206: Find the slope of the tangent line to the given polar curve at the .
  • 10.10.207: Find the slope of the tangent line to the given polar curve at the .
  • 10.10.208: Find the slope of the tangent line to the given polar curve at the .
  • 10.10.209: Find the points on the given curve where the tangent line is horizo.
  • 10.10.210: Find the points on the given curve where the tangent line is horizo.
  • 10.10.211: Find the points on the given curve where the tangent line is horizo.
  • 10.10.212: Find the points on the given curve where the tangent line is horizo.
  • 10.10.213: Find the points on the given curve where the tangent line is horizo.
  • 10.10.214: Find the points on the given curve where the tangent line is horizo.
  • 10.10.215: Show that the polar equation , where , represents a circle, and fin.
  • 10.10.216: Show that the curves and intersect at right angles.
  • 10.10.217: Use a graphing device to graph the polar curve. Choose the paramete.
  • 10.10.218: Use a graphing device to graph the polar curve. Choose the paramete.
  • 10.10.219: Use a graphing device to graph the polar curve. Choose the paramete.
  • 10.10.220: Use a graphing device to graph the polar curve. Choose the paramete.
  • 10.10.221: Use a graphing device to graph the polar curve. Choose the paramete.
  • 10.10.222: Use a graphing device to graph the polar curve. Choose the paramete.
  • 10.10.223: How are the graphs of and related to the graph of ? In general, how.
  • 10.10.224: Use a graph to estimate the -coordinate of the highest points on th.
  • 10.10.225: (a) Investigate the family of curves defined by the polar equations.
  • 10.10.226: A family of curves is given by the equations , where is a real numb.
  • 10.10.227: A family of curves has polar equations Investigate how the graph ch.
  • 10.10.228: The astronomer Giovanni Cassini (16251712) studied the family of cu.
  • 10.10.229: Let be any point (except the origin) on the curve . If is the angle.
  • 10.10.230: a) Use Exercise 83 to show that the angle between the tangent line .
  • 10.10.231: Find the area of the region that is bounded by the given curve r 3 .
  • 10.10.232: Find the area of the region that is bounded by the given curve r 3 .
  • 10.10.233: Find the area of the region that is bounded by the given curve r 3 .
  • 10.10.234: Find the area of the region that is bounded by the given curve r 3 .
  • 10.10.235: Find the area of the shaded region
  • 10.10.236: Find the area of the shaded region
  • 10.10.237: Find the area of the shaded region
  • 10.10.238: Find the area of the shaded region
  • 10.10.239: Sketch the curve and find the area that it encloses.r 3 cos r
  • 10.10.240: Sketch the curve and find the area that it encloses.r 3 cos r
  • 10.10.241: Sketch the curve and find the area that it encloses.r r 2 sin 11. 2.
  • 10.10.242: Sketch the curve and find the area that it encloses.r r 2 sin 11. 2.
  • 10.10.243: Sketch the curve and find the area that it encloses.r 2 cos 3r
  • 10.10.244: Sketch the curve and find the area that it encloses.r 2 cos 2
  • 10.10.245: Graph the curve and find the area that it encloses
  • 10.10.246: Graph the curve and find the area that it encloses
  • 10.10.247: Find the area of the region enclosed by one loop of the curve r sin.
  • 10.10.248: Find the area of the region enclosed by one loop of the curve r 4 s.
  • 10.10.249: Find the area of the region enclosed by one loop of the curve r 3 c.
  • 10.10.250: Find the area of the region enclosed by one loop of the curve r 2 s.
  • 10.10.251: Find the area of the region enclosed by one loop of the curve r 1 2.
  • 10.10.252: Find the area enclosed by the loop of the strophoid r 2 cos sec 2
  • 10.10.253: Find the area of the region that lies inside the first curve and ou.
  • 10.10.254: Find the area of the region that lies inside the first curve and ou.
  • 10.10.255: Find the area of the region that lies inside the first curve and ou.
  • 10.10.256: Find the area of the region that lies inside the first curve and ou.
  • 10.10.257: Find the area of the region that lies inside the first curve and ou.
  • 10.10.258: Find the area of the region that lies inside the first curve and ou.
  • 10.10.259: Find the area of the region that lies inside both curves.
  • 10.10.260: Find the area of the region that lies inside both curves.
  • 10.10.261: Find the area of the region that lies inside both curves.
  • 10.10.262: Find the area of the region that lies inside both curves.
  • 10.10.263: Find the area of the region that lies inside both curves.
  • 10.10.264: Find the area of the region that lies inside both curves.r a sin r .
  • 10.10.265: Find the area inside the larger loop and outside the smaller loop o.
  • 10.10.266: Find the area between a large loop and the enclosed small loop of t.
  • 10.10.267: Find all points of intersection of the given curves.
  • 10.10.268: Find all points of intersection of the given curves.
  • 10.10.269: Find all points of intersection of the given curves.
  • 10.10.270: Find all points of intersection of the given curves.
  • 10.10.271: Find all points of intersection of the given curves.
  • 10.10.272: Find all points of intersection of the given curves.
  • 10.10.273: The points of intersection of the cardioid and the spiral loop , , .
  • 10.10.274: When recording live performances, sound engineers often use a micro.
  • 10.10.275: Find the exact length of the polar curve.
  • 10.10.276: Find the exact length of the polar curve.
  • 10.10.277: Find the exact length of the polar curve.
  • 10.10.278: Find the exact length of the polar curve.
  • 10.10.279: Use a calculator to find the length of the curve correct to four de.
  • 10.10.280: Use a calculator to find the length of the curve correct to four de.
  • 10.10.281: Use a calculator to find the length of the curve correct to four de.
  • 10.10.282: Use a calculator to find the length of the curve correct to four de.
  • 10.10.283: Graph the curve and find its length
  • 10.10.284: Graph the curve and find its length
  • 10.10.285: a) Use Formula 10.2.7 to show that the area of the surface generate.
  • 10.10.286: (a) Find a formula for the area of the surface generated by rotatin.
  • 10.10.287: Find the focus and directrix of the parabola and sketch the graph
  • 10.10.288: Sketch the graph of and locate the foci.
  • 10.10.289: Find an equation of the ellipse with foci and vertices 0, 3
  • 10.10.290: Find the foci and asymptotes of the hyperbola and sketch its graph
  • 10.10.291: Find the foci and equation of the hyperbola with vertices and asymp.
  • 10.10.292: Find an equation of the ellipse with foci , and vertices 1, 25, 2 2,
  • 10.10.293: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.294: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.295: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.296: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.297: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.298: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.299: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.300: Find the vertex, focus, and directrix of the parabola and sketch it.
  • 10.10.301: Find an equation of the parabola. Then find the focus and directrix
  • 10.10.302: Find an equation of the parabola. Then find the focus and directrix
  • 10.10.303: Find the vertices and foci of the ellipse and sketch its graphx 2 9.
  • 10.10.304: Find the vertices and foci of the ellipse and sketch its graphx 2 6.
  • 10.10.305: Find the vertices and foci of the ellipse and sketch its graph4x 2 .
  • 10.10.306: Find the vertices and foci of the ellipse and sketch its graph4x 2 .
  • 10.10.307: Find the vertices and foci of the ellipse and sketch its graph9x 2 .
  • 10.10.308: Find the vertices and foci of the ellipse and sketch its graphx 2 3.
  • 10.10.309: Find an equation of the ellipse. Then find its foci
  • 10.10.310: Find an equation of the ellipse. Then find its foci
  • 10.10.311: Find the vertices, foci, and asymptotes of the hyperbola and sketch.
  • 10.10.312: Find the vertices, foci, and asymptotes of the hyperbola and sketch.
  • 10.10.313: Find the vertices, foci, and asymptotes of the hyperbola and sketch.
  • 10.10.314: Find the vertices, foci, and asymptotes of the hyperbola and sketch.
  • 10.10.315: Find the vertices, foci, and asymptotes of the hyperbola and sketch.
  • 10.10.316: Find the vertices, foci, and asymptotes of the hyperbola and sketch.
  • 10.10.317: Identify the type of conic section whose equation is given and find.
  • 10.10.318: Identify the type of conic section whose equation is given and find.
  • 10.10.319: Identify the type of conic section whose equation is given and find.
  • 10.10.320: Identify the type of conic section whose equation is given and find.
  • 10.10.321: Identify the type of conic section whose equation is given and find.
  • 10.10.322: Identify the type of conic section whose equation is given and find.
  • 10.10.323: Find an equation for the conic that satisfies the given conditions.
  • 10.10.324: Find an equation for the conic that satisfies the given conditions.
  • 10.10.325: Find an equation for the conic that satisfies the given conditions.
  • 10.10.326: Find an equation for the conic that satisfies the given conditions.
  • 10.10.327: Find an equation for the conic that satisfies the given conditions.
  • 10.10.328: Find an equation for the conic that satisfies the given conditions.
  • 10.10.329: Find an equation for the conic that satisfies the given conditions.
  • 10.10.330: Find an equation for the conic that satisfies the given conditions.
  • 10.10.331: Find an equation for the conic that satisfies the given conditions.
  • 10.10.332: Find an equation for the conic that satisfies the given conditions.
  • 10.10.333: Find an equation for the conic that satisfies the given conditions.
  • 10.10.334: Find an equation for the conic that satisfies the given conditions.
  • 10.10.335: Find an equation for the conic that satisfies the given conditions.
  • 10.10.336: Find an equation for the conic that satisfies the given conditions.
  • 10.10.337: Find an equation for the conic that satisfies the given conditions.
  • 10.10.338: Find an equation for the conic that satisfies the given conditions.
  • 10.10.339: Find an equation for the conic that satisfies the given conditions.
  • 10.10.340: Find an equation for the conic that satisfies the given conditions.
  • 10.10.341: The point in a lunar orbit nearest the surface of the moon is calle.
  • 10.10.342: A cross-section of a parabolic reflector is shown in the figure. Th.
  • 10.10.343: In the LORAN (LOng RAnge Navigation) radio navigation system, two r.
  • 10.10.344: Use the definition of a hyperbola to derive Equation 6 for a hyperb.
  • 10.10.345: Show that the function defined by the upper branch of the hyperbola.
  • 10.10.346: Find an equation for the ellipse with foci and and major axis of le.
  • 10.10.347: Determine the type of curve represented by the equation in each of .
  • 10.10.348: (a) Show that the equation of the tangent line to the parabola at t.
  • 10.10.349: Show that the tangent lines to the parabola drawn from any point on.
  • 10.10.350: Show that if an ellipse and a hyperbola have the same foci, then th.
  • 10.10.351: Use Simpsons Rule with to estimate the length of the ellipse x 2 4y.
  • 10.10.352: The planet Pluto travels in an elliptical orbit around the sun (at .
  • 10.10.353: Find the area of the region enclosed by the hyperbola and the verti.
  • 10.10.354: (a) If an ellipse is rotated about its major axis, find the volume .
  • 10.10.355: Let be a point on the ellipse with foci and and let and be the angl.
  • 10.10.356: Let be a point on the hyperbola with foci and and let and be the an.
  • 10.10.357: Find a polar equation for a parabola that has its focus at the orig.
  • 10.10.358: A conic is given by the polar equation Find the eccentricity, ident.
  • 10.10.359: Sketch the conic r 12 2 4 sin
  • 10.10.360: If the ellipse of Example 2 is rotated through an angle about the o.
  • 10.10.361: a) Find an approximate polar equation for the elliptical orbit of t.
  • 10.10.362: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.363: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.364: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.365: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.366: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.367: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.368: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.369: Write a polar equation of a conic with the focus at the origin and .
  • 10.10.370: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.371: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.372: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.373: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.374: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.375: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.376: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.377: a) Find the eccentricity, (b) identify the conic, (c) give an equat.
  • 10.10.378: (a) Find the eccentricity and directrix of the conic and graph the .
  • 10.10.379: Graph the conic and its directrix. Also graph the conic obtained by.
  • 10.10.380: Graph the conics with , , , and on a common screen. How does the va.
  • 10.10.381: (a) Graph the conics for and various values of . How does the value.
  • 10.10.382: Show that a conic with focus at the origin, eccentricity , and dire.
  • 10.10.383: Show that a conic with focus at the origin, eccentricity , and dire.
  • 10.10.384: Show that a conic with focus at the origin, eccentricity , and dire.
  • 10.10.385: Show that the parabolas and intersect at right angles.
  • 10.10.386: The orbit of Mars around the sun is an ellipse with eccentricity an.
  • 10.10.387: Jupiters orbit has eccentricity and the length of the major axis is.
  • 10.10.388: The orbit of Halleys comet, last seen in 1986 and due to return in .
  • 10.10.389: The Hale-Bopp comet, discovered in 1995, has an elliptical orbit wi.
  • 10.10.390: The planet Mercury travels in an elliptical orbit with eccentricity.
  • 10.10.391: The distance from the planet Pluto to the sun is km at perihelion a.
  • 10.10.392: Using the data from Exercise 29, find the distance traveled by the .
  • 10.10.393: a) What is a parametric curve? (b) How do you sketch a parametric c.
  • 10.10.394: a) How do you find the slope of a tangent to a parametric curve? (b.
  • 10.10.395: Write an expression for each of the following: (a) The length of a .
  • 10.10.396: a) Use a diagram to explain the meaning of the polar coordinates of.
  • 10.10.397: (a) How do you find the slope of a tangent line to a polar curve? (.
  • 10.10.398: (a) Give a geometric definition of a parabola. (b) Write an equatio.
  • 10.10.399: (a) Give a definition of an ellipse in terms of foci. (b) Write an .
  • 10.10.400: a) Give a definition of a hyperbola in terms of foci. (b) Write an .
  • 10.10.401: a) What is the eccentricity of a conic section? (b) What can you sa.
  • 10.10.402: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.403: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.404: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.405: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.406: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.407: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.408: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.409: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.410: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.411: Determine whether the statement is true or false. If it is true, ex.
  • 10.10.412: Sketch the parametric curve and eliminate the parameter to find the.
  • 10.10.413: Sketch the parametric curve and eliminate the parameter to find the.
  • 10.10.414: Sketch the parametric curve and eliminate the parameter to find the.
  • 10.10.415: Sketch the parametric curve and eliminate the parameter to find the.
  • 10.10.416: Write three different sets of parametric equations for the curve y sx
  • 10.10.417: Use the graphs of and to sketch the parametric curve , . Indicate w.
  • 10.10.418: a) Plot the point with polar coordinates . Then find its Cartesian .
  • 10.10.419: Sketch the region consisting of points whose polar coordinates sati.
  • 10.10.420: Sketch the polar curver 1 cos r
  • 10.10.421: Sketch the polar curve]r sin 4
  • 10.10.422: Sketch the polar curver cos 3 r
  • 10.10.423: Sketch the polar curver 3 cos 3
  • 10.10.424: Sketch the polar curver 1 cos 2 r
  • 10.10.425: Sketch the polar curver 2 cos 2 r
  • 10.10.426: Sketch the polar curver 3 1 2 sin
  • 10.10.427: Sketch the polar curver 3 2 2 cos r
  • 10.10.428: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.429: Find a polar equation for the curve represented by the given Cartes.
  • 10.10.430: The curve with polar equation is called a cochleoid. Use a graph of.
  • 10.10.431: Graph the ellipse and its directrix. Also graph the ellipse obtaine.
  • 10.10.432: Find the slope of the tangent line to the given curve at the point .
  • 10.10.433: Find the slope of the tangent line to the given curve at the point .
  • 10.10.434: Find the slope of the tangent line to the given curve at the point .
  • 10.10.435: Find the slope of the tangent line to the given curve at the point .
  • 10.10.436: Use a graph to estimate the coordinates of the lowest point on the .
  • 10.10.437: Find the area enclosed by the loop of the curve in Exercise 27.
  • 10.10.438: At what points does the curve have vertical or horizontal tangents.
  • 10.10.439: Find the area enclosed by the curve in Exercise 29
  • 10.10.440: Find the area enclosed by the curve r 2 9 cos 5
  • 10.10.441: Find the area enclosed by the inner loop of the curve r 1 3 sin
  • 10.10.442: Find the points of intersection of the curves and r 4 cos
  • 10.10.443: Find the points of intersection of the curves r 2 cos r
  • 10.10.444: Find the area of the region that lies inside both of the circles r .
  • 10.10.445: Find the area of the region that lies inside the curve but outside .
  • 10.10.446: Find the length of the curve.y 2t 0 t 2 3 x
  • 10.10.447: Find the length of the curve. x 2 3t y cosh 3t 0 t 1y
  • 10.10.448: Find the length of the curve. r 12
  • 10.10.449: Find the length of the curve. r sin , 0 3
  • 10.10.450: Find the area of the surface obtained by rotating the given 3, 2cur.
  • 10.10.451: Find the area of the surface obtained by rotating the given 3, 2cur.
  • 10.10.452: The curves defined by the parametric equations are called strophoid.
  • 10.10.453: The curves defined by the parametric equations are called strophoid.
  • 10.10.454: A family of curves has polar equations where is a positive number. .
  • 10.10.455: Find the foci and vertices and sketch the graph.2 9 y 2 8 1
  • 10.10.456: Find the foci and vertices and sketch the graph.4x 2 y 2 16 x
  • 10.10.457: Find the foci and vertices and sketch the graph.6y 2 x 36y 55 0 4x
  • 10.10.458: Find the foci and vertices and sketch the graph.25x 2 4y 2 50x 16y .
  • 10.10.459: Find an equation of the ellipse with foci and vertices
  • 10.10.460: Find an equation of the parabola with focus and directrix .
  • 10.10.461: Find an equation of the parabola with focus and directrix .
  • 10.10.462: Find an equation of the ellipse with foci and major axis with lengt.
  • 10.10.463: Find an equation for the ellipse that shares a vertex and a focus w.
  • 10.10.464: Show that if is any real number, then there are exactly two lines o.
  • 10.10.465: Find a pplar equation for the ellipse with focus at the origin,ecce.
  • 10.10.466: Show that the angles between the polar axis and the asymptotes of t.
  • 10.10.467: In the figure the circle of radius is stationary, and for every , t.
  • 10.10.468: A curve is defined by the parametric equations Find the length of t.
  • 10.10.469: (a) Find the highest and lowest points on the curve . (b) Sketch th.
  • 10.10.470: What is the smallest viewing rectangle that contains every member o.
  • 10.10.471: Four bugs are placed at the four corners of a square with side leng.
  • 10.10.472: A curve called the folium of Descartes is defined by the parametric.
  • 10.10.473: A circle of radius has its center at the origin. A circle of radius.
Textbook: Calculus: Early Transcendentals
Edition: 6
Author: James Stewart
ISBN: 9780495011668

This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780495011668. Since 473 problems in chapter 10: PARAMETRIC EQUATIONS AND POLAR COORDINATES have been answered, more than 70595 students have viewed full step-by-step solutions from this chapter. Chapter 10: PARAMETRIC EQUATIONS AND POLAR COORDINATES includes 473 full step-by-step solutions.

An expression a + bi, where a (the real part) and b (the imaginary part) are real numbers

The probability of an event A given that an event B has already occurred

A sequence or series diverges if it does not converge

The set of all points in the coordinate plane corresponding to the pairs (x, ƒ(x)) for x in the domain of ƒ.

The amount of time required for half of a radioactive substance to decay.

A model of population growth: ƒ1x2 = c 1 + a # bx or ƒ1x2 = c1 + ae-kx, where a, b, c, and k are positive with b < 1. c is the limit to growth

Any of the real numbers in a matrix

The eight regions of space determined by the coordinate planes.

The order of an m x n matrix is m x n.

Data items more than 1.5 times the IQR below the first quartile or above the third quartile.

The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

A function ƒ for which there is a positive number c such that for every value t in the domain of ƒ. The smallest such number c is the period of the function.

A degree 4 polynomial function.

See Division algorithm for polynomials.

See Elementary row operations.

One-half of the sum of the lengths of the sides of a triangle.

The function that associates points on the unit circle with points on the real number line


Watch the video: Parametric Equations (December 2021).