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8.1 E: Exercises


Exercise (PageIndex{1})

Find the first six terms of each of the following sequences, starting with (displaystyle n=1).

1) (displaystyle a_n=1+(−1)^n) for (displaystyle n≥1)

Answer

Solution: (displaystyle a_n=0) if (displaystyle n) is odd and (displaystyle a_n=2) if (displaystyle n) is even

2) (displaystyle a_n=n^2−1) for (displaystyle n≥1)

3) (displaystyle a_1=1) and (displaystyle a_n=a_{n−1}+n) for (displaystyle n≥2)

Answer

Solution: (displaystyle {a_n}={1,3,6,10,15,21,…})

4) (displaystyle a_1=1, a_2=1) and (displaystyle a_n+2=a_n+a_{n+1}) for (displaystyle n≥1)

Exercise (PageIndex{2})

1) Find an explicit formula for (displaystyle a_n) where (displaystyle a_1=1) and (displaystyle a_n=a_{n−1}+n) for (displaystyle n≥2).

Answer

(displaystyle a_n=frac{n(n+1)}{2})

2) Find a formula (displaystyle a_n) for the (displaystyle nth) term of the arithmetic sequence whose first term is (displaystyle a_1=1) such that (displaystyle a_{n−1}−a_n=17) for (displaystyle n≥1).

3) Find a formula (displaystyle a_n) for the (displaystyle nth) term of the arithmetic sequence whose first term is (displaystyle a_1=−3) such that (displaystyle a_{n−1}−a_n=4) for (displaystyle n≥1).

Answer

(displaystyle a_n=4n−7)

4) Find a formula (displaystyle a_n) for the (displaystyle nth) term of the geometric sequence whose first term is (displaystyle a_1=1) such that (displaystyle frac{a_{n+1}}{a_n}=10) for (displaystyle n≥1).

5) Find a formula (displaystyle a_n) for the (displaystyle nth) term of the geometric sequence whose first term is (displaystyle a_1=3) such that (displaystyle frac{a_{n+1}}{a_n}=1/10) for (displaystyle n≥1).

Answer

Solution: (displaystyle a_n=3.10^{1−n}=30.10^{−n})

6) Find an explicit formula for the (displaystyle nth) term of the sequence whose first several terms are (displaystyle {0,3,8,15,24,35,48,63,80,99,…}.) (Hint: First add one to each term.)

7) Find an explicit formula for the (displaystyle nth) term of the sequence satisfying (displaystyle a_1=0) and (displaystyle a_n=2a_{n−1}+1) for (displaystyle n≥2).

Answer

Solution: (displaystyle a_n=2^n−1)

Find a formula for the general term (displaystyle a_n) of each of the following sequences.

8) (displaystyle {1,0,−1,0,1,0,−1,0,…}) (Hint: Find where (displaystyle sinx) takes these values)

9) (displaystyle {1,−1/3,1/5,−1/7,…})

Answer

Solution: (displaystyle a_n=frac{(−1)^{n−1}}{2n−1})

Exercise (PageIndex{3})

Find a function (displaystyle f(n)) that identifies the (displaystyle nth) term (displaystyle a_n) of the following recursively defined sequences, as (displaystyle a_n=f(n)).

1) (displaystyle a_1=1) and (displaystyle a_{n+1}=−a_n) for (displaystyle n≥1)

2) (displaystyle a_1=2) and (displaystyle a_{n+1}=2a_n) for (displaystyle n≥1)

Answer

Solution: (displaystyle f(n)=2^n)

3) (displaystyle a_1=1) and (displaystyle a_{n+1}=(n+1)a_n) for (displaystyle n≥1)

4) (displaystyle a_1=2) and (displaystyle a_{n+1}=(n+1)a_n/2) for (displaystyle n≥1)

Answer

Solution: (displaystyle a_1=1) and (displaystyle a_{n+1}=a_n/2^n) for (displaystyle n≥1)

Exercise (PageIndex{4})

Plot the first (displaystyle N) terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges.

1) [T] (displaystyle a_1=1, a_2=2), and for (displaystyle n≥2, a_n=frac{1}{2}(a_{n−1}+a_{n−2})); (displaystyle N=30)

Answer

Solution: Terms oscillate above and below (displaystyle 5/3) and appear to converge to (displaystyle 5/3).

2) [T] (displaystyle a_1=1, a_2=2, a_3=3) and for (displaystyle n≥4, a_n=frac{1}{3}(a_{n−1}+a_{n−2}+a_{n−3}), N=30)

3) [T] (displaystyle a_1=1, a_2=2), and for (displaystyle n≥3, a_n=sqrt{a_{n−1}a_{n−2}}; N=30)

Answer

Solution: Terms oscillate above and below (displaystyle y≈1.57..) and appear to converge to a limit.

4) [T] (displaystyle a_1=1, a_2=2, a_3=3), and for (displaystyle n≥4, a_n=sqrt{a_{n−1}a_{n−2}a_{n−3}}; N=30)

Exercise (PageIndex{5})

Suppose that (displaystyle lim_{n→∞}a_n=1, lim_{n→∞}b_n=−1), and (displaystyle 0<−b_n

1) (displaystyle lim_{n→∞}3a_n−4b_n)

Answer

Solution: (displaystyle 7)

2) (displaystyle lim_{n→∞}frac{1}{2}b_n−frac{1}{2}a_n)

3) (displaystyle lim_{n→∞}frac{a_n+b_n}{a_n−b_n})

Answer

Solution: (displaystyle 0)

4) (displaystyle lim_{n→∞}frac{a_n−b_n}{a_n+b_n})

Exercise (PageIndex{6})

Find the limit of each of the following sequences, using L’Hôpital’s rule when appropriate.

1) (displaystyle frac{n^2}{2^n})

Answer

Solution: (displaystyle 0)

2) (displaystyle frac{(n−1)^2}{(n+1)^2})

3) (displaystyle frac{sqrt{n}}{sqrt{n+1}})

Answer

Solution: (displaystyle 1)

4) (displaystyle n^{1/n}) (Hint: (displaystyle n^{1/n}=e^{frac{1}{n}lnn}))

Exercise (PageIndex{7})

For each of the following sequences, whose (displaystyle nth) terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing.

1) (displaystyle n/2^n, n≥2)

Answer

Solution: bounded, decreasing for (displaystyle n≥1)

2) (displaystyle ln(1+frac{1}{n}))

3) (displaystyle sinn)

Answer

Solution: bounded, not monotone

4) (displaystyle cos(n^2))

5) (displaystyle n^{1/n}, n≥3)

Answer

Solution: bounded, decreasing

6) (displaystyle n^{−1/n}, n≥3)

7) (displaystyle tan(n))

Answer

Solution: not monotone, not bounded

Exercise (PageIndex{8})

Determine whether the sequence defined as follows has a limit. If it does, find the limit.

1) (displaystyle a_1=sqrt{2}, a_2=sqrt{2sqrt{2}}. a_3=sqrt{2sqrt{2sqrt{2}}}) etc.

2) (displaystyle a_1=3, a_n=sqrt{2a_{n−1}}, n=2,3,….)

Answer

Solution: (displaystyle a-n) is decreasing and bounded below by (displaystyle 2). The limit a must satisfy (displaystyle a=sqrt{2a}) so (displaystyle a=2), independent of the initial value.

Exercise (PageIndex{9})

Use the Squeeze Theorem to find the limit of each of the following sequences.

1) (displaystyle nsin(1/n))

2) (displaystyle frac{cos(1/n)−1}{1/n})

Answer

Solution: (displaystyle 0)

3) (displaystyle a_n=frac{n!}{n^n})

4) (displaystyle a_n=sinnsin(1/n))

Answer

Solution: (displaystyle 0:|sinx|≤|x|) and (displaystyle |sinx|≤1) so (displaystyle −frac{1}{n}≤a_n≤frac{1}{n})).

Exercise (PageIndex{10})

For the following sequences, plot the first (displaystyle 25) terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges.

1) [T] (displaystyle a_n=sinn)

2) [T] (displaystyle a_n=cosn)

Answer

Solution: Graph oscillates and suggests no limit.

Exercise (PageIndex{11})

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit.

1) (displaystyle a_n=tan^{−1}(n^2))

2) (displaystyle a_n=(2n)^{1/n}−n^{1/n})

Answer

Solution: (displaystyle n^{1/n}→1) and (displaystyle 2^{1/n}→1,) so (displaystyle a_n→0)

3) (displaystyle a_n=frac{ln(n^2)}{ln(2n)})

4) (displaystyle a_n=(1−frac{2}{n})^n)

Answer

Solution: Since (displaystyle (1+1/n)^n→e), one has (displaystyle (1−2/n)^n≈(1+k)^{−2k}→e^{−2}) as (displaystyle k→∞.)

5) (displaystyle a_n=ln(frac{n+2}{n^2−3}))

6) (displaystyle a_n=frac{2^n+3^n}{4^n})

Answer

Solution: (displaystyle 2^n+3^n≤2⋅3^n) and (displaystyle 3^n/4^n→0) as (displaystyle n→∞), so (displaystyle a_n→0) as (displaystyle n→∞.)

7) (displaystyle a_n=frac{(1000)^n}{n!})

8) (displaystyle a_n=frac{(n!)^2}{(2n)!})

Answer

Solution: (displaystyle frac{a_{n+1}}{a_n}=n!/(n+1)(n+2)⋯(2n) =frac{1⋅2⋅3⋯n}{(n+1)(n+2)⋯(2n)}<1/2^n). In particular, (displaystyle a_{n+1}/a_n≤1/2), so (displaystyle a_n→0) as (displaystyle n→∞)

Exercise (PageIndex{12})

Newton’s method seeks to approximate a solution (displaystyle f(x)=0) that starts with an initial approximation (displaystyle x_0) and successively defines a sequence (displaystyle x_{n+1}=x_n−frac{f(x_n)}{f′(x_n)}). For the given choice of (displaystyle f) and (displaystyle x_0), write out the formula for (displaystyle x_{n+1}). If the sequence appears to converge, give an exact formula for the solution (displaystyle x), then identify the limit (displaystyle x) accurate to four decimal places and the smallest (displaystyle n) such that (displaystyle x_n) agrees with (displaystyle x) up to four decimal places.

1) [T] (displaystyle f(x)=x^2−2, x_0=1)

2) [T] (displaystyle f(x)=(x−1)^2−2, x_0=2)

Answer

Solution: (displaystyle x_{n+1}=x_n−((x_n−1)^2−2)/2(x_n−1); x=1+sqrt{2}, x≈2.4142, n=5)

3) [T] (displaystyle f(x)=e^x−2, x_0=1)

4) [T] (displaystyle f(x)=lnx−1, x_0=2)

Answer

Solution: (displaystyle x_{n+1}=x_n−x_n(ln(x_n)−1); x=e, x≈2.7183, n=5)

Exercise (PageIndex{13})

1) [T] Suppose you start with one liter of vinegar and repeatedly remove (displaystyle 0.1L), replace with water, mix, and repeat.

a. Find a formula for the concentration after (displaystyle n) steps.

b. After how many steps does the mixture contain less than (displaystyle 10%) vinegar?

2) [T] A lake initially contains (displaystyle 2000) fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by (displaystyle 6%) each month. However, factoring in all causes, (displaystyle 150) fish are lost each month.

a. Explain why the fish population after (displaystyle n) months is modeled by (displaystyle P_n=1.06P_{n−1}−150) with (displaystyle P_0=2000).

b. How many fish will be in the pond after one year?

Answer

Solution: a. Without losses, the population would obey (displaystyle P_n=1.06P_{n−1}). The subtraction of (displaystyle 150) accounts for fish losses. b. After (displaystyle 12) months, we have (displaystyle P_{12}≈1494.)

3) [T] A bank account earns (displaystyle 5%) interest compounded monthly. Suppose that (displaystyle $1000) is initially deposited into the account, but that (displaystyle $10) is withdrawn each month.

a. Show that the amount in the account after (displaystyle n) months is (displaystyle A_n=(1+.05/12)A_{n−1}−10; A_0=1000.)

b. How much money will be in the account after (displaystyle 1) year?

c. Is the amount increasing or decreasing?

d. Suppose that instead of (displaystyle $10), a fixed amount (displaystyle d) dollars is withdrawn each month. Find a value of (displaystyle d) such that the amount in the account after each month remains (displaystyle $1000).

e. What happens if (displaystyle d) is greater than this amount?

4) [T] A student takes out a college loan of (displaystyle $10,000) at an annual percentage rate of (displaystyle 6%,) compounded monthly.

a. If the student makes payments of (displaystyle $100) per month, how much does the student owe after (displaystyle 12) months?

b. After how many months will the loan be paid off?

Answer

Solution: a. The student owes (displaystyle $9383) after (displaystyle 12) months. The loan will be paid in full after (displaystyle 139) months or eleven and a half years.

5) [T] Consider a series combining geometric growth and arithmetic decrease. Let (displaystyle a_1=1). Fix (displaystyle a>1) and (displaystyle 0

6) [T] The binary representation (displaystyle x=0.b_1b_2b_3...) of a number (displaystyle x) between (displaystyle 0) and (displaystyle 1) can be defined as follows. Let (displaystyle b_1=0) if (displaystyle x<1/2) and (displaystyle b_1=1) if (displaystyle 1/2≤x<1.) Let (displaystyle x_1=2x−b_1). Let (displaystyle b_2=0) if (displaystyle x_1<1/2) and (displaystyle b_2=1) if (displaystyle 1/2≤x<1). Let (displaystyle x_2=2x_1−b_2) and in general, (displaystyle x_n=2x_{n−1}−b_n) and (displaystyle b_{n−}1=0) if (displaystyle x_n<1/2) and (displaystyle b_{n−1}=1) if (displaystyle 1/2≤x_n<1). Find the binary expansion of (displaystyle 1/3).

Answer

Solution: (displaystyle b_1=0, x_1=2/3, b_2=1, x_2=4/3−1=1/3,) so the pattern repeats, and (displaystyle 1/3=0.010101….)

7) [T] To find an approximation for (displaystyle π), set (displaystyle a_0=sqrt{2+1}, a_1=sqrt{2+a_0}), and, in general, (displaystyle a_{n+1}=sqrt{2+a_n}). Finally, set (displaystyle p_n=3.2^nsqrt{2−a_n}). Find the first ten terms of (displaystyle p_n) and compare the values to (displaystyle π).

Exercise (PageIndex{14})

For the following two exercises, assume that you have access to a computer program or Internet source that can generate a list of zeros and ones of any desired length. Pseudorandom number generators (PRNGs) play an important role in simulating random noise in physical systems by creating sequences of zeros and ones that appear like the result of flipping a coin repeatedly. One of the simplest types of PRNGs recursively defines a random-looking sequence of (displaystyle N) integers (displaystyle a_1,a_2,…,a_N) by fixing two special integers (displaystyle (K) and (displaystyle M) and letting (displaystyle a_{n+1}) be the remainder after dividing (displaystyle K.a_n) into (displaystyle M), then creates a bit sequence of zeros and ones whose (displaystyle nth) term (displaystyle b_n) is equal to one if (displaystyle a_n) is odd and equal to zero if (displaystyle a_n) is even. If the bits (displaystyle b_n) are pseudorandom, then the behavior of their average (displaystyle (b_1+b_2+⋯+b_N)/N) should be similar to behavior of averages of truly randomly generated bits.

1) [T] Starting with (displaystyle K=16,807) and (displaystyle M=2,147,483,647), using ten different starting values of (displaystyle a_1), compute sequences of bits (displaystyle b_n) up to (displaystyle n=1000,) and compare their averages to ten such sequences generated by a random bit generator.

Answer

Solution: For the starting values (displaystyle a_1=1, a_2=2,…, a_1=10,) the corresponding bit averages calculated by the method indicated are (displaystyle 0.5220, 0.5000, 0.4960, 0.4870, 0.4860, 0.4680, 0.5130, 0.5210, 0.5040,) and (displaystyle 0.4840). Here is an example of ten corresponding averages of strings of (displaystyle 1000) bits generated by a random number generator: (displaystyle 0.4880, 0.4870, 0.5150, 0.5490, 0.5130, 0.5180, 0.4860, 0.5030, 0.5050, 0.4980.) There is no real pattern in either type of average. The random-number-generated averages range between (displaystyle 0.4860) and (displaystyle 0.5490), a range of (displaystyle 0.0630), whereas the calculated PRNG bit averages range between (displaystyle 0.4680) and (displaystyle 0.5220), a range of (displaystyle 0.0540.)

2) [T] Find the first (displaystyle 1000) digits of (displaystyle π) using either a computer program or Internet resource. Create a bit sequence (displaystyle b_n) by letting (displaystyle b_n=1) if the (displaystyle nth) digit of (displaystyle π) is odd and (displaystyle b_n=0) if the (displaystyle nth) digit of (displaystyle π) is even. Compute the average value of (displaystyle b_n) and the average value of (displaystyle d_n=|b_{n+1}−b_n|, n=1,...,999.) Does the sequence (displaystyle b_n) appear random? Do the differences between successive elements of (displaystyle b_n) appear random?


8.E: Testing Hypotheses (Exercises)

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number (mu _0) and write (H_0:mu =mu _0) and the appropriate analogous expression for (H_a).)

  1. The average July temperature in a region historically has been (74.5^F). Perhaps it is higher now.
  2. The average weight of a female airline passenger with luggage was (145) pounds ten years ago. The FAA believes it to be higher now.
  3. The average stipend for doctoral students in a particular discipline at a state university is ($14,756). The department chairman believes that the national average is higher.
  4. The average room rate in hotels in a certain region is ($82.53). A travel agent believes that the average in a particular resort area is different.
  5. The average farm size in a predominately rural state was (69.4) acres. The secretary of agriculture of that state asserts that it is less today.

State the null and alternative hypotheses for each of the following situations. (That is, identify the correct number (mu _0) and write (H_0:mu =mu _0) and the appropriate analogous expression for (H_a).)

  1. The average time workers spent commuting to work in Verona five years ago was (38.2) minutes. The Verona Chamber of Commerce asserts that the average is less now.
  2. The mean salary for all men in a certain profession is ($58,291). A special interest group thinks that the mean salary for women in the same profession is different.
  3. The accepted figure for the caffeine content of an (8)-ounce cup of coffee is (133) mg. A dietitian believes that the average for coffee served in a local restaurants is higher.
  4. The average yield per acre for all types of corn in a recent year was (161.9) bushels. An economist believes that the average yield per acre is different this year.
  5. An industry association asserts that the average age of all self-described fly fishermen is (42.8) years. A sociologist suspects that it is higher.

Describe the two types of errors that can be made in a test of hypotheses.

Under what circumstance is a test of hypotheses certain to yield a correct decision?

Answers

    1. (H_0:mu =74.5 vs H_a:mu >74.5)
    2. (H_0:mu =145 vs H_a:mu >145)
    3. (H_0:mu =14756 vs H_a:mu >14756)
    4. (H_0:mu =82.53 vs H_a:mu eq 82.53)
    5. (H_0:mu =69.4 vs H_a:mu <69.4)

    8.2: IONIC BONDING

    Conceptual Problems

    Describe the differences in behavior between NaOH and CH3OH in aqueous solution. Which solution would be a better conductor of electricity? Explain your reasoning.

    What is the relationship between the strength of the electrostatic attraction between oppositely charged ions and the distance between the ions? How does the strength of the electrostatic interactions change as the size of the ions increases?

    Which will result in the release of more energy: the interaction of a gaseous sodium ion with a gaseous oxide ion or the interaction of a gaseous sodium ion with a gaseous bromide ion? Why?

    Which will result in the release of more energy: the interaction of a gaseous chloride ion with a gaseous sodium ion or a gaseous potassium ion? Explain your answer.

    What are the predominant interactions when oppositely charged ions are

    1. far apart?
    2. at internuclear distances close to r0?
    3. very close together (at a distance that is less than the sum of the ionic radii)?

    Several factors contribute to the stability of ionic compounds. Describe one type of interaction that destabilizes ionic compounds. Describe the interactions that stabilize ionic compounds.

    What is the relationship between the electrostatic attractive energy between charged particles and the distance between the particles?

    Conceptual Answer

    The interaction of a sodium ion and an oxide ion. The electrostatic attraction energy between ions of opposite charge is directly proportional to the charge on each ion (Q1 and Q2 in Equation 9.1). Thus, more energy is released as the charge on the ions increases (assuming the internuclear distance does not increase substantially). A sodium ion has a +1 charge an oxide ion, a &minus2 charge and a bromide ion, a &minus1 charge. For the interaction of a sodium ion with an oxide ion, Q1 = +1 and Q2 = &minus2, whereas for the interaction of a sodium ion with a bromide ion, Q1 = +1 and Q2 = &minus1. The larger value of Q1 × Q2 for the sodium ion&ndashoxide ion interaction means it will release more energy.

    Numerical Problems

    How does the energy of the electrostatic interaction between ions with charges +1 and &minus1 compare to the interaction between ions with charges +3 and &minus1 if the distance between the ions is the same in both cases? How does this compare with the magnitude of the interaction between ions with +3 and &minus3 charges?

    How many grams of gaseous MgCl2 are needed to give the same electrostatic attractive energy as 0.5 mol of gaseous LiCl? The ionic radii are Li + = 76 pm, Mg +2 = 72 pm, and Cl &minus = 181 pm.

    Sketch a diagram showing the relationship between potential energy and internuclear distance (from r = &infin to r = 0) for the interaction of a bromide ion and a potassium ion to form gaseous KBr. Explain why the energy of the system increases as the distance between the ions decreases from r = r0 to r = 0.

    Calculate the magnitude of the electrostatic attractive energy (E, in kilojoules) for 85.0 g of gaseous SrS ion pairs. The observed internuclear distance in the gas phase is 244.05 pm.

    What is the electrostatic attractive energy (E, in kilojoules) for 130 g of gaseous HgI2? The internuclear distance is 255.3 pm.

    Numerical Answers

    According to Equation 9.1, in the first case Q1Q2 = (+1)(&minus1) = &minus1 in the second case, Q1Q2 = (+3)(&minus1) = &minus3. Thus, E will be three times larger for the +3/&minus1 ions. For +3/&minus3 ions, Q1Q2 = (+3)(&minus3) = &minus9, so E will be nine times larger than for the +1/&minus1 ions.

    At r < r0, the energy of the system increases due to electron&ndashelectron repulsions between the overlapping electron distributions on adjacent ions. At very short internuclear distances, electrostatic repulsions between adjacent nuclei also become important.


    NCERT Solutions for Class 10 Maths Chapter 8 Exercise 8.1

    NCERT Solutions for class 10 Maths Chapter 8 Exercise 8.1 Introduction to Trigonometry in PDF format free download Hindi Medium and English Medium or View in Video Format updated for new academic session 2021-22.

    NCERT Solutions and Offline Apps 2021-22 are re-modified removing errors and making more simplified for the students. Solutions are appropriate for CBSE board students, MP Board, UP Board students and all the other state boards, who are following NCERT Books 2021-2022. Download (Exercise 8.1) in PDF file format or use online given below.

    NCERT Solutions for class 10 Maths Chapter 8 Exercise 8.1

    Class 10 Maths Exercise 8.1 Solutions

    10 Maths Chapter 8 Exercise 8.1 Solutions

    NCERT Solutions for class 10 Maths Chapter 8 Exercise 8.1 Introduction to Trigonometry in ENGLISH & HINDI MEDIUM to free download in PDF. These Solutions are updated and modified according to the new CBSE Curriculum 2021-22 for CBSE board as well as UP Board students who are using NCERT Books 2021-22. Visit to Class X Mathematics Solution Chapter 8 main page for other exercises whether download or online study.


    8 Strength Exercises All Beginners Should Learn How to Do

    If you’re just starting an exercise routine for the first time, you’re probably feeling a mix of emotions. It’s always exciting to try something new, but it can also be equal parts confusing and daunting. But the thing is, when it comes to working out, the best place to start really is at the beginning, with simple and effective exercises that’ll let you build a sturdy base you can use as a jumping off point as you get stronger and stronger.

    Trust me, I know it can be tempting to try and tackle a workout that you found online that seems challenging, or a circuit that your favorite trainer posted on Instagram. But if you're new to this whole exercise thing (welcome!), it really is absolutely essential that you start with the basics. And by the basics I mean classic exercises that let you practice the foundational movements upon which hundreds of other exercises are created. Most of these movement patterns are also functional, meaning they’re movements you do in everyday life, not just in the gym.

    For example, the hip-hinge movement is one important movement pattern. It’s the motion of bending forward from your hips (not your back), and pushing your butt behind you. You do this movement in a squat (and almost every squat variation) and any type of deadlift. Learning how to properly do the basic versions of these exercises is key if you want to safely build on them as you get stronger. If you skip over mastering basic exercises that teach you to do foundational movements properly, you’ll be doing yourself (and your fitness goals) a disservice long term.

    Below are eight basic exercises that are great for many beginners to start with. Of course, exercise is not one size fits all, and you should absolutely speak with your doctor or another health-care professional you trust before starting a new exercise regimen, especially if you’re unsure whether it’s safe for you. And as you’re working on these exercises, if you’re having trouble maintaining proper form or feel any sort of pain (other than a little post-workout soreness a day or two after), stop and check in with a doctor or physical therapist. A base level of body control, stability, and mobility is needed for these exercises, so you may need to start by taking a closer look at those things.

    When you're first learning the following moves, use just your body weight. (There are two you'll need resistance bands for—more on that below.) Adding resistance in the form of free weights, like dumbbells or kettlebells, will make them more challenging and it's best to wait to do that until you've fully mastered each movement. You should be able to do 10 to 15 reps comfortably with great form before even thinking about adding weights, says Jacque Crockford, M.S., C.S.C.S., certified personal trainer and exercise physiology content manager at American Council on Exercise (ACE).

    A squat is a classic exercise that shows up in tons of workouts. Learning a basic bodyweight squat will help you master the hip-hinge movement. It’s a compound exercise, meaning it works more than one muscle group at once, including the glutes, quads, and core.

    • Stand with your feet slightly wider than hip-width apart, toes slightly turned out, arms at your sides, palms in.
    • Engage your core and keep your chest lifted and back flat as you shift your weight into your heels, push your hips back, and bend your knees to lower into a squat. Bend your elbows and bring your palms together in front of your chest. (You can also just hold your hands in front of your chest the entire time.)
    • Drive through your heels to stand and squeeze your glutes at the top for 1 rep.

    The deadlift also trains the hip-hinge motion, but targets your hamstrings more than a squat does. It also works the glutes and core. You probably have usually seen deadlifts done with weights, but they can absolutely be done without them, Crockford says.

    • Stand with your feet hip-width apart, knees slightly bent, arms relaxed by the front of your quads. This is the starting position.
    • Hinge forward at your hips and bend your knees slightly as you push your butt way back. Keep your back flat and shoulders engaged as you slowly lower your arms along your shins toward the floor until you feel a stretch in your hamstrings.
    • Keeping your core tight, push through your heels to stand up straight and return to the starting position. Keep your arms close to your shins as you pull. Pause at the top and squeeze your butt. That's 1 rep.

    If you’re just starting to exercise, the Romanian deadlift (pictured here) is a great deadlift to start with. A traditional deadlift is done by fully bending the knees to lift the weight off the floor. The Romanian deadlift, which involves a slight bend in the knees but not a full knee bend, helps keep the focus on the hip-hinge movement. (A stiff-leg deadlift, where you don’t bend your knees at all, requires a lot more flexibility to do properly so isn’t the best to start with.)

    When you lunge, you're training your body’s ability to do single-leg movements. Any lunge that has you transitioning from two feet to one foot and back again—like a forward lunge, reverse lunge, transverse lunge, or lateral lunge—fits the bill, says Crockford. By changing your base of support with each rep, you’ll train your balance and stability more than doing exercises where your base of support stays firmly on both feet. You’re also working your glutes, quads, and core.

    I chose a reverse lunge here because they are typically easier on the knees and easier to control than forward lunges. But if you feel more comfortable lunging forward and don’t have any knee pain when you do, feel free to do that instead.

    • Stand with your feet together with your arms by your sides (or pictured) or hands on your hips. This is the starting position.
    • Step back (about 2 feet) with your right foot, landing on the ball of your foot and keeping your heel off the floor.
    • Bend both knees until your left quad and right shin are parallel to the floor, your torso leaning slightly forward so your back is flat. Your left knee should be above your left foot and your butt and core should be engaged.
    • Push through the heel of your left foot to return to the starting position. This is 1 rep.
    • You can either alternate legs each time, or do all your reps on one side before switching to the other side.

    A row works the “pulling” movement pattern and specifically targets the muscles in the upper back. Unlike the other exercises here, you can’t really do a pull exercise without some sort of equipment, whether it’s dumbbells or a resistance band. Crockford recommends starting with a very light resistance band (you can simply stand on the other end) and thinking about keeping your shoulder blades back and down as you perform the rowing movement—your shoulders shouldn’t be rounded forward or hunched up tensely by your ears.

    • Stand with your feet hip-width apart, holding a weight in each hand with your arms at your sides.
    • With your core engaged, hinge forward at the hips, push your butt back, and bend your knees slightly, so that your back is no lower than parallel to the floor. (Depending on your hamstring flexibility, you may not be able to bend so far over.) Gaze at the ground a few inches in front of your feet to keep your neck in a comfortable position.
    • Do a row by pulling the weights up toward your chest, keeping your elbows hugged close to your body, and squeezing your shoulder blades for two seconds at the top of the movement. Your elbows should go past your back as you bring the weight toward your chest
    • Slowly lower the weights by extending your arms toward the floor. This is 1 rep.

    Crockford says that the pull motion can be challenging to learn because it’s hard for many people to know what correctly stabilizing the scapula (shoulder blade) feels like. “What I always recommend for people to do first is lie on their backs and extend their arms above them like they’re reaching for the ceiling. Then, squeeze the shoulder blades together and actually feel the shoulder blades press into the ground.” Do a few reps of this, keeping your arms straight and only squeezing and releasing your shoulder blades. You can also do it with your back against the wall, Crockford says. The goal is to just get familiar with that motion of locking the shoulder blades in that position so that when you do the rowing movement, you will just bend your elbows and won’t be tempted to round forward and overextend your shoulders.

    A plank is a great exercise for working on total-body stability as it engages your entire core, plus your shoulders and upper back. Crockford notes that it also helps you get in the right position for a push-up (more on that next). She recommends doing a high plank, with your arms straight and palms flat on the floor, as this will help you get used to engaging your upper back and pulling your shoulder blades back and in a stable position.


    8.1 E: Exercises

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    8.5: DRAWING LEWIS STRUCTURES

    Conceptual Problems

    Compare and contrast covalent and ionic compounds with regard to

    1. volatility.
    2. melting point.
    3. electrical conductivity.
    4. physical appearance.

    What are the similarities between plots of the overall energy versus internuclear distance for an ionic compound and a covalent compound? Why are the plots so similar?

    Which atom do you expect to be the central atom in each of the following species?

    Which atom is the central atom in each of the following species?

    What is the relationship between the number of bonds typically formed by the period 2 elements in groups 14, 15, and 16 and their Lewis electron structures?

    Although formal charges do not represent actual charges on atoms in molecules or ions, they are still useful. Why?

    Numerical Problems

    Give the electron configuration and the Lewis dot symbol for the following. How many more electrons can each atom accommodate?

    Give the electron configuration and the Lewis dot symbol for the following. How many more electrons can each atom accommodate?

    Based on Lewis dot symbols, predict the preferred oxidation state of Be, F, B, and Cs.

    Based on Lewis dot symbols, predict the preferred oxidation state of Br, Rb, O, Si, and Sr.

    Based on Lewis dot symbols, predict how many bonds gallium, silicon, and selenium will form in their neutral compounds.

    Determine the total number of valence electrons in the following.

    1. Cr
    2. Cu +
    3. NO +
    4. XeF2
    5. Br2
    6. CH2Cl2
    7. NO3 &minus
    8. H3O +

    Determine the total number of valence electrons in the following.

    1. Ag
    2. Pt 2+
    3. H2S
    4. OH &minus
    5. I2
    6. CH4
    7. SO4 2&minus
    8. NH4 + .

    Draw Lewis electron structures for the following.

    1. F2
    2. SO2
    3. AlCl4 &minus
    4. SO3 2&minus
    5. BrCl
    6. XeF4
    7. NO +
    8. PCl3

    Draw Lewis electron structures for the following.

    Draw Lewis electron structures for CO2, NO2 &minus , SO2, and NO2 + . From your diagram, predict which pair(s) of compounds have similar electronic structures.

    Write Lewis dot symbols for each pair of elements. For a reaction between each pair of elements, predict which element is the oxidant, which element is the reductant, and the final stoichiometry of the compound formed.

    Write Lewis dot symbols for each pair of elements. For a reaction between each pair of elements, predict which element is the oxidant, which element is the reductant, and the final stoichiometry of the compound formed.

    Use Lewis dot symbols to predict whether ICl and NO4 &minus are chemically reasonable formulas.

    Draw a plausible Lewis electron structure for a compound with the molecular formula Cl3PO.

    Draw a plausible Lewis electron structure for a compound with the molecular formula CH4O.

    While reviewing her notes, a student noticed that she had drawn the following structure in her notebook for acetic acid:

    Why is this structure not feasible? Draw an acceptable Lewis structure for acetic acid. Show the formal charges of all nonhydrogen atoms in both the correct and incorrect structures.

    A student proposed the following Lewis structure shown for acetaldehyde.

    Why is this structure not feasible? Draw an acceptable Lewis structure for acetaldehyde. Show the formal charges of all nonhydrogen atoms in both the correct and incorrect structures.

    Draw the most likely structure for HCN based on formal charges, showing the formal charge on each atom in your structure. Does this compound have any plausible resonance structures? If so, draw one.

    Draw the most plausible Lewis structure for NO3 &minus . Does this ion have any other resonance structures? Draw at least one other Lewis structure for the nitrate ion that is not plausible based on formal charges.

    At least two Lewis structures can be drawn for BCl3. Using arguments based on formal charges, explain why the most likely structure is the one with three B&ndashCl single bonds.

    Using arguments based on formal charges, explain why the most feasible Lewis structure for SO4 2&minus has two sulfur&ndashoxygen double bonds.

    At least two distinct Lewis structures can be drawn for N3 &minus . Use arguments based on formal charges to explain why the most likely structure contains a nitrogen&ndashnitrogen double bond.

    Is H&ndashO&ndashN=O a reasonable structure for the compound HNO2? Justify your answer using Lewis electron dot structures.

    Is H&ndashO=C&ndashH a reasonable structure for a compound with the formula CH2O? Use Lewis electron dot structures to justify your answer.

    Explain why the following Lewis structure for SO3 2&minus is or is not reasonable.

    Numerical Answers

    Selenium can accommodate two more electrons, giving the Se 2&minus ion.

    Krypton has a closed shell electron configuration, so it cannot accommodate any additional electrons.

    Lithium can accommodate one additional electron in its 2s orbital, giving the Li &minus ion.

    Strontium has a filled 5s subshell, and additional electrons would have to be placed in an orbital with a higher energy. Thus strontium has no tendency to accept an additional electron.

    Hydrogen can accommodate one additional electron in its 1s orbital, giving the H &minus ion.

    K is the reductant S is the oxidant. The final stoichiometry is K2S.

    Sr is the reductant Br is the oxidant. The final stoichiometry is SrBr2.

    Al is the reductant O is the oxidant. The final stoichiometry is Al2O3.

    Mg is the reductant Cl is the oxidant. The final stoichiometry is MgCl2.

    The only structure that gives both oxygen and carbon an octet of electrons is the following:

    The student&rsquos proposed structure has two flaws: the hydrogen atom with the double bond has four valence electrons (H can only accommodate two electrons), and the carbon bound to oxygen only has six valence electrons (it should have an octet). An acceptable Lewis structure is

    The formal charges on the correct and incorrect structures are as follows:

    The most plausible Lewis structure for NO3 &minus is:

    There are three equivalent resonance structures for nitrate (only one is shown), in which nitrogen is doubly bonded to one of the three oxygens. In each resonance structure, the formal charge of N is +1 for each singly bonded O, it is &minus1 and for the doubly bonded oxygen, it is 0.

    The following is an example of a Lewis structure that is not plausible:

    This structure nitrogen has six bonds (nitrogen can form only four bonds) and a formal charge of &ndash1.

    With four S&ndashO single bonds, each oxygen in SO4 2&minus has a formal charge of &minus1, and the central sulfur has a formal charge of +2. With two S=O double bonds, only two oxygens have a formal charge of &ndash1, and sulfur has a formal charge of zero. Lewis structures that minimize formal charges tend to be lowest in energy, making the Lewis structure with two S=O double bonds the most probable.

    Yes. This is a reasonable Lewis structure, because the formal charge on all atoms is zero, and each atom (except H) has an octet of electrons.


    Introduction to Logic

    So far we’ve see truth tables used to define the operators. They have other uses as well: they make it possible to classify and to compare statements to appreciate their logical properties, to test arguments for validity, and to define rules of deduction and replacement. A truth table exhibits all the truth-values that it is possible for a given statement or set of statements to have. But let’s back up just a bit.

    In the previous section we introduced the truth-functional definitions of the operators. With that information (exhibited on truth-tables, which showed all the possible values “p” and “q” could have), we have enough information that we can “calculate” or figure out the truth-value of compound statements as long as we know the truth-values of the simple statements that make them up.

    For instance, since we know that

    “Bananas are fruit” is true

    and “Apples are fruit” is true

    and “Pears are fruit” is true,

    we can figure out that this statement:

    How? List the truth values under the letters, and then combine the values according to the definitions of the five operators, starting at the smallest unit and working up to the largest.

    This table shows us the values of these three statements. Each is true, so we have a “T” under each statement and since the negation of “Pears are fruit” occurs (“Pears are not fruit”), we have an “F” under the tilde.

    The simplest or smallest level at which any “calculation” can be done is that negation of a simple statement. The next level is in the conjoining of the negated statement with “Apples are fruit.” The claim that “Apples are fruit but Pears are not” is false, so an “F” goes under the dot.

    That dot statement is the consequent of the conditional, and the antecedent of the conditional is true, so the conditional itself is false an “F” goes under the horseshoe. I’ve colored it red to make it more noticeable.

    Now, in this one-line format, the only way to get these symbols to line up straight is to present them in a table. But the table showing us that B ⊃ (A ∙

    P) is false is not what we’ll call a “Truth Table.” A truth table shows all the possible truth values that the simple statements in a compound or set of compounds can have, and it shows us a result of those values it is always at least two lines long. The example we are looking at here is simply calculating the value of a single compound statement, not exhibiting all the possibilities that the form of this statement allows for.

    The tables we used to define the operators, repeated below, are truth tables. There are no combinations of truth values for these statements that have not been shown.

    p q
    T T T
    T F F
    F F T
    F F F

    p v q
    T T T
    T T F
    F T T
    F F F

    p q
    T T T
    T F F
    F T T
    F T F

    p q
    T T T
    T F F
    F F T
    F T F

    Truth tables provide the truth-functional definition of the five operators. With those definitions, we can calculate the truth value of compound statements once we know the truth values of the simple ones that make them up. Here are some examples, and some exercises you can practice at.

    James Farmer taught The History of the Civil Rights Movement at Mary Washington.

    Martin Luther King taught theology at Mary Washington.

    Since the first of these is true and the second is false, the conjunction made up of them (F . K) is false.

    If King taught theology or Farmer taught Civil Rights at Mary Washington, then a major figure of the Civil Rights Movement was a member of the UMW faculty.

    This conditional has a disjunction for its antecedent:

    M = a major figure of the CRM was on the faculty.

    (K v F) M
    F T T T T

    I’ve marked the truth values: the first one we can enter is the bolded one, because K v F is the smallest unit the second one is the slanted one, which combines the value from K v F with the value of M. By the way, don’t infer from this example that the first value you can calculate will always be the left-most one. The last value you can do is the one for what’s called the main operator: this statement is a conditional, its main operator is the horseshoe.

    Here are some more you can practice with:

    1. Obama and Clinton are Democrats if Christie is a Republican.

    2. Either Clinton will run or Christie is a Democrat.

    3. If Clinton runs then she is at least 35 years old.

    4. Obama is commander in chief if and only if he is President.

    5. Being born in America is a necessary condition for being president.

    6. If being born here is a necessary condition for running, then Schwarzenegger cannot run.

    7. Being sound is a sufficient condition for being valid and being valid is a necessary condition for being sound in addition, being deductive is a necessary condition for being sound and for being valid.

    8. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege but Kant denied that “existence” was a predicate only if Aristotelian logic dominated for two thousand years.

    9. Either Hume did not invent truth tables or else Wittgenstein wrote the Tractatus, and Russell’s paradox was bad news to Frege only if Kant denied that “existence” was a predicate, given that Aristotelian logic dominated for two thousand years.

    10. If it is false both that Hume invented truth tables and that Kant denied “existence” was a predicate, then given that Aristotelian logic dominated for two thousand years, Wittgenstein’s writing the Tractatus implies that Russell’s paradox was bad news to Frege.

    1. Obama and Clinton are Democrats if Christie is a Republican. C > (O . C)

    2. Either Clinton will run or Christie is a Democrat. C v D

    3. If Clinton runs then she is at least 35 years old. C > T

    4. Obama is commander in chief if and only if he is President. O ≡ C

    5. Being born in America is a necessary condition for being president. P > A

    6. If being born here is a necessary condition for running, then Schwarzenegger cannot run. (P > R) >

    7. Being sound is a sufficient condition for being valid and being valid is a necessary condition for being sound in addition, being deductive is a necessary condition for being sound and for being valid. [(S > V) . (S > V)] . [(S v V) > D]

    8. If either Hume did not invent truth tables or Wittgenstein wrote the Tractatus, then Russell’s paradox was bad news to Frege but Kant denied that “existence” was a predicate only if Aristotelian logic dominated for two thousand years.

    9. Either Hume did not invent truth tables or else Wittgenstein wrote the Tractatus, and Russell’s paradox was bad news to Frege only if Kant denied that “existence” was a predicate, given that Aristotelian logic dominated for two thousand years.

    10. If it is false both that Hume invented truth tables and that Kant denied “existence” was a predicate, then given that Aristotelian logic dominated for two thousand years, Wittgenstein’s writing the Tractatus implies that Russell’s paradox was bad news to Frege.


    Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1

    Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1

    Answer 1E.



    Answer 2E.




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    Steps:

    (i) False, because sides of a right-angled triangle may have any length. So ( ext,A) may have any value.

    As hypotenuse is largest side, the ratio on RHS will be greater than 1. Hence (egin ext ext>1.end) Thus, the given statement is true.

    (iii) Abbreviation used for cosecant of ( angle A) is ( ext,A) and ( ext,A) is the abbreviation used for cosine of ( m angle A). Hence the given statement is false.

    (iv) ( ext,A) is not the product of ( m) and () It is the cotangent of ( m angle A) . Hence, the given statement is false.

    We know that in a right-angled triangle,

    In a right-angled triangle, hypotenuse is always greater than the remaining two sides. Also, the value of Sine should be less than (1.) Therefore, such value of ( m) is not possible. Hence the given statement is false.


    Watch the video: 8th Class Math, Ch 8 - Practical Geometry Exercise Q 1. 8th Class Maths (November 2021).