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10.3E: Exercises for Taylor Polynomials and Taylor Series


Taylor Polynomials

In exercises 1 - 8, find the Taylor polynomials of degree two approximating the given function centered at the given point.

1) ( f(x)=1+x+x^2) at ( a=1)

2) ( f(x)=1+x+x^2) at ( a=−1)

Answer:
( f(−1)=1;;f′(−1)=−1;;f''(−1)=2;quad p_2(x)=1−(x+1)+(x+1)^2)

3) ( f(x)=cos(2x)) at ( a=π)

4) ( f(x)=sin(2x)) at ( a=frac{π}{2})

Answer:
( f′(x)=2cos(2x);;f''(x)=−4sin(2x);quad p_2(x)=−2(x−frac{π}{2}))

5) ( f(x)=sqrt{x}) at ( a=4)

6) ( f(x)=ln x) at ( a=1)

Answer:
( f′(x)=dfrac{1}{x};; f''(x)=−dfrac{1}{x^2};quad p_2(x)=0+(x−1)−frac{1}{2}(x−1)^2)

7) ( f(x)=dfrac{1}{x}) at ( a=1)

8) ( f(x)=e^x) at ( a=1)

Answer:
( p_2(x)=e+e(x−1)+dfrac{e}{2}(x−1)^2)

Taylor Remainder Theorem

In exercises 9 - 14, verify that the given choice of (n) in the remainder estimate ( |R_n|≤dfrac{M}{(n+1)!}(x−a)^{n+1}), where (M) is the maximum value of ( ∣f^{(n+1)}(z)∣) on the interval between (a) and the indicated point, yields ( |R_n|≤frac{1}{1000}). Find the value of the Taylor polynomial ( p_n) of ( f) at the indicated point.

9) [T] ( sqrt{10};; a=9,; n=3)

10) [T] ( (28)^{1/3};; a=27,; n=1)

Answer:
( dfrac{d^2}{dx^2}x^{1/3}=−dfrac{2}{9x^{5/3}}≥−0.00092…) when ( x≥28) so the remainder estimate applies to the linear approximation ( x^{1/3}≈p_1(27)=3+dfrac{x−27}{27}), which gives ( (28)^{1/3}≈3+frac{1}{27}=3.ar{037}), while ( (28)^{1/3}≈3.03658.)

11) [T] ( sin(6);; a=2π,; n=5)

12) [T] ( e^2; ; a=0,; n=9)

Answer:
Using the estimate ( dfrac{2^{10}}{10!}<0.000283) we can use the Taylor expansion of order 9 to estimate ( e^x) at ( x=2). as ( e^2≈p_9(2)=1+2+frac{2^2}{2}+frac{2^3}{6}+⋯+frac{2^9}{9!}=7.3887)… whereas ( e^2≈7.3891.)

13) [T] ( cos(frac{π}{5});; a=0,; n=4)

14) [T] ( ln(2);; a=1,; n=1000)

Answer:
Since ( dfrac{d^n}{dx^n}(ln x)=(−1)^{n−1}dfrac{(n−1)!}{x^n},R_{1000}≈frac{1}{1001}). One has (displaystyle p_{1000}(1)=sum_{n=1}^{1000}dfrac{(−1)^{n−1}}{n}≈0.6936) whereas ( ln(2)≈0.6931⋯.)

Approximating Definite Integrals Using Taylor Series

15) Integrate the approximation (sin t≈t−dfrac{t^3}{6}+dfrac{t^5}{120}−dfrac{t^7}{5040}) evaluated at ( π)t to approximate (displaystyle ∫^1_0frac{sin πt}{πt},dt).

16) Integrate the approximation ( e^x≈1+x+dfrac{x^2}{2}+⋯+dfrac{x^6}{720}) evaluated at ( −x^2) to approximate (displaystyle ∫^1_0e^{−x^2},dx.)

Answer:
(displaystyle ∫^1_0left(1−x^2+frac{x^4}{2}−frac{x^6}{6}+frac{x^8}{24}−frac{x^{10}}{120}+frac{x^{12}}{720} ight),dx =1−frac{1^3}{3}+frac{1^5}{10}−frac{1^7}{42}+frac{1^9}{9⋅24}−frac{1^{11}}{120⋅11}+frac{1^{13}}{720⋅13}≈0.74683) whereas (displaystyle ∫^1_0e^{−x^2}dx≈0.74682.)

More Taylor Remainder Theorem Problems

In exercises 17 - 20, find the smallest value of (n) such that the remainder estimate ( |R_n|≤dfrac{M}{(n+1)!}(x−a)^{n+1}), where (M) is the maximum value of ( ∣f^{(n+1)}(z)∣) on the interval between (a) and the indicated point, yields ( |R_n|≤frac{1}{1000}) on the indicated interval.

17) ( f(x)=sin x) on ( [−π,π],; a=0)

18) ( f(x)=cos x) on ( [−frac{π}{2},frac{π}{2}],; a=0)

Answer:
Since ( f^{(n+1)}(z)) is (sin z) or (cos z), we have ( M=1). Since ( |x−0|≤frac{π}{2}), we seek the smallest (n) such that ( dfrac{π^{n+1}}{2^{n+1}(n+1)!}≤0.001). The smallest such value is ( n=7). The remainder estimate is ( R_7≤0.00092.)

19) ( f(x)=e^{−2x}) on ( [−1,1],a=0)

20) ( f(x)=e^{−x}) on ( [−3,3],a=0)

Answer:
Since ( f^{(n+1)}(z)=±e^{−z}) one has ( M=e^3). Since ( |x−0|≤3), one seeks the smallest (n) such that ( dfrac{3^{n+1}e^3}{(n+1)!}≤0.001). The smallest such value is ( n=14). The remainder estimate is ( R_{14}≤0.000220.)

In exercises 21 - 24, the maximum of the right-hand side of the remainder estimate ( |R_1|≤dfrac{max|f''(z)|}{2}R^2) on ( [a−R,a+R]) occurs at (a) or ( a±R). Estimate the maximum value of (R) such that ( dfrac{max|f''(z)|}{2}R^2≤0.1) on ( [a−R,a+R]) by plotting this maximum as a function of (R).

21) [T] ( e^x) approximated by ( 1+x,; a=0)

22) [T] ( sin x) approximated by ( x,; a=0)

Answer:

Since ( sin x) is increasing for small ( x) and since ( frac{d^2}{dx^2}left(sin x ight)=−sin x), the estimate applies whenever ( R^2sin(R)≤0.2), which applies up to ( R=0.596.)

23) [T] ( ln x) approximated by ( x−1,; a=1)

24) [T] ( cos x) approximated by ( 1,; a=0)

Answer:

Since the second derivative of ( cos x) is ( −cos x) and since ( cos x) is decreasing away from ( x=0), the estimate applies when ( R^2cos R≤0.2) or ( R≤0.447).

Taylor Series

In exercises 25 - 35, find the Taylor series of the given function centered at the indicated point.

25) (f(x) = x^4) at ( a=−1)

26) (f(x) = 1+x+x^2+x^3) at ( a=−1)

Answer:
( (x+1)^3−2(x+1)^2+2(x+1))

27) (f(x) = sin x) at ( a=π)

28) (f(x) = cos x) at ( a=2π)

Answer:
Values of derivatives are the same as for ( x=0) so (displaystyle cos x=sum_{n=0}^∞(−1)^nfrac{(x−2π)^{2n}}{(2n)!})

29) (f(x) = sin x) at ( x=frac{π}{2})

30) (f(x) = cos x) at ( x=frac{π}{2})

Answer:
( cos(frac{π}{2})=0,;−sin(frac{π}{2})=−1) so (displaystyle cos x=sum_{n=0}^∞(−1)^{n+1}frac{(x−frac{π}{2})^{2n+1}}{(2n+1)!}), which is also ( −cos(x−frac{π}{2})).

31) (f(x) = e^x) at ( a=−1)

32) (f(x) = e^x) at ( a=1)

Answer:
The derivatives are ( f^{(n)}(1)=e,) so (displaystyle e^x=esum_{n=0}^∞frac{(x−1)^n}{n!}.)

33) (f(x) = dfrac{1}{(x−1)^2}) at ( a=0) (Hint: Differentiate the Taylor Series for( dfrac{1}{1−x}).)

34) (f(x) = dfrac{1}{(x−1)^3}) at ( a=0)

Answer:
(displaystyle frac{1}{(x−1)^3}=−frac{1}{2}frac{d^2}{dx^2}left(frac{1}{1−x} ight)=−sum_{n=0}^∞left(frac{(n+2)(n+1)x^n}{2} ight))

35) (displaystyle F(x)=∫^x_0cos(sqrt{t}),dt;quad ext{where}; f(t)=sum_{n=0}^∞(−1)^nfrac{t^n}{(2n)!}) at a=0 (Note: ( f) is the Taylor series of (cos(sqrt{t}).))

In exercises 36 - 44, compute the Taylor series of each function around ( x=1).

36) ( f(x)=2−x)

Answer:
( 2−x=1−(x−1))

37) ( f(x)=x^3)

38) ( f(x)=(x−2)^2)

Answer:
( ((x−1)−1)^2=(x−1)^2−2(x−1)+1)

39) ( f(x)=ln x)

40) ( f(x)=dfrac{1}{x})

Answer:
(displaystyle frac{1}{1−(1−x)}=sum_{n=0}^∞(−1)^n(x−1)^n)

41) ( f(x)=dfrac{1}{2x−x^2})

42) ( f(x)=dfrac{x}{4x−2x^2−1})

Answer:
(displaystyle xsum_{n=0}^∞2^n(1−x)^{2n}=sum_{n=0}^∞2^n(x−1)^{2n+1}+sum_{n=0}^∞2^n(x−1)^{2n})

43) ( f(x)=e^{−x})

44) ( f(x)=e^{2x})

Answer:
(displaystyle e^{2x}=e^{2(x−1)+2}=e^2sum_{n=0}^∞frac{2^n(x−1)^n}{n!})

Maclaurin Series

[T] In exercises 45 - 48, identify the value of (x) such that the given series (displaystyle sum_{n=0}^∞a_n) is the value of the Maclaurin series of ( f(x)) at ( x). Approximate the value of ( f(x)) using (displaystyle S_{10}=sum_{n=0}^{10}a_n).

45) (displaystyle sum_{n=0}^∞frac{1}{n!})

46) (displaystyle sum_{n=0}^∞frac{2^n}{n!})

Answer:
( x=e^2;quad S_{10}=dfrac{34,913}{4725}≈7.3889947)

47) (displaystyle sum_{n=0}^∞frac{(−1)^n(2π)^{2n}}{(2n)!})

48) (displaystyle sum_{n=0}^∞frac{(−1)^n(2π)^{2n+1}}{(2n+1)!})

Answer:
(sin(2π)=0;quad S_{10}=8.27×10^{−5})

In exercises 49 - 52 use the functions ( S_5(x)=x−dfrac{x^3}{6}+dfrac{x^5}{120}) and ( C_4(x)=1−dfrac{x^2}{2}+dfrac{x^4}{24}) on ( [−π,π]).

49) [T] Plot (sin^2x−(S_5(x))^2) on ( [−π,π]). Compare the maximum difference with the square of the Taylor remainder estimate for ( sin x.)

50) [T] Plot (cos^2x−(C_4(x))^2) on ( [−π,π]). Compare the maximum difference with the square of the Taylor remainder estimate for ( cos x).

Answer:

The difference is small on the interior of the interval but approaches ( 1) near the endpoints. The remainder estimate is ( |R_4|=frac{π^5}{120}≈2.552.)

51) [T] Plot ( |2S_5(x)C_4(x)−sin(2x)|) on ( [−π,π]).

52) [T] Compare ( dfrac{S_5(x)}{C_4(x)}) on ( [−1,1]) to ( an x). Compare this with the Taylor remainder estimate for the approximation of ( an x) by ( x+dfrac{x^3}{3}+dfrac{2x^5}{15}).

Answer:

The difference is on the order of ( 10^{−4}) on ( [−1,1]) while the Taylor approximation error is around ( 0.1) near ( ±1). The top curve is a plot of ( an^2x−left(dfrac{S_5(x)}{C_4(x)} ight)^2) and the lower dashed plot shows ( t^2−left(dfrac{S_5}{C_4} ight)^2).

53) [T] Plot ( e^x−e_4(x)) where ( e_4(x)=1+x+dfrac{x^2}{2}+dfrac{x^3}{6}+dfrac{x^4}{24}) on ( [0,2]). Compare the maximum error with the Taylor remainder estimate.

54) (Taylor approximations and root finding.) Recall that Newton’s method ( x_{n+1}=x_n−dfrac{f(x_n)}{f'(x_n)}) approximates solutions of ( f(x)=0) near the input ( x_0).

a. If ( f) and ( g) are inverse functions, explain why a solution of ( g(x)=a) is the value ( f(a)) of ( f).

b. Let ( p_N(x)) be the ( N^{ ext{th}}) degree Maclaurin polynomial of ( e^x). Use Newton’s method to approximate solutions of ( p_N(x)−2=0) for ( N=4,5,6.)

c. Explain why the approximate roots of ( p_N(x)−2=0) are approximate values of (ln(2).)

Answer:
a. Answers will vary.
b. The following are the ( x_n) values after ( 10) iterations of Newton’s method to approximation a root of ( p_N(x)−2=0): for ( N=4,x=0.6939...;) for ( N=5,x=0.6932...;) for ( N=6,x=0.69315...;.) (Note: ( ln(2)=0.69314...))
c. Answers will vary.

Evaluating Limits using Taylor Series

In exercises 55 - 58, use the fact that if (displaystyle q(x)=sum_{n=1}^∞a_n(x−c)^n) converges in an interval containing ( c), then (displaystyle lim_{x→c}q(x)=a_0) to evaluate each limit using Taylor series.

55) (displaystyle lim_{x→0}frac{cos x−1}{x^2})

56) (displaystyle lim_{x→0}frac{ln(1−x^2)}{x^2})

Answer:
( dfrac{ln(1−x^2)}{x^2}→−1)

57) (displaystyle lim_{x→0}frac{e^{x^2}−x^2−1}{x^4})

58) (displaystyle lim_{x→0^+}frac{cos(sqrt{x})−1}{2x})

Answer:
(displaystyle frac{cos(sqrt{x})−1}{2x}≈frac{(1−frac{x}{2}+frac{x^2}{4!}−⋯)−1}{2x}→−frac{1}{4})

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


One sequence is the sequence of terms < a n >. The other is the sequence of n th partial sums, < S n >= < ∑ i = 1 n a i >.

Converges because it is a geometric series with r = 1 5 .

lim n → ∞ ⁡ a n = 1 by Theorem 9.2.4 the series diverges.

lim n → ∞ ⁡ a n = e by Theorem 9.2.4 the series diverges.

Using partial fractions, we can show that a n = 1 4 ⁢ ( 1 2 ⁢ n - 1 + 1 2 ⁢ n + 1 ) . The series is effectively twice the sum of the odd terms of the Harmonic Series which was shown to diverge in Example 9.2.5 . Thus this series diverges.


7 Answers 7

Yes. There is a geometric explanation. For simplicity, let me take $x=0$ and $h=1$ . By the Fundamental Theorem of Calculus (FTC), $ f(1)=f(0)+int_<0>^<1>dt_1 f'(t_1) . $ Now use the FTC for the $f'(t_1)$ inside the integral, which gives $ f'(t_1)=f'(0)+int_<0>^dt_2 f''(t_2) , $ and insert this in the previous equation. We then get $ f(1)=f(0)+f'(0)+int_<0>^<1>dt_1int_<0>^dt_2 f''(t_2) . $ Keep iterating this, using the FTC to rewrite the last integrand, each time invoking a new variable $t_k$ . At the end of the day, one obtains $ f(1)=sum_^int_ dt_1cdots dt_k f^<(k)>(0) + < m remainder>$ where $Delta_k$ is the simplex $ <(t_1,ldots,t_k)inmathbb^k | 1>t_1>cdots>t_k>0> . $ For example $Delta_<2>$ is a triangle in the plane, and $Delta_3$ is a tetrahedron in 3D, etc. The $frac<1>$ is just the volume of $Delta_k$ . Indeed, by a simple change of variables (renaming), the volume is the same for all $k!$ simplices of the form $ <(t_1,ldots,t_k)inmathbb^k | 1>t_>cdots>t_>0> $ where $sigma$ is a permutation of $<1,2,ldots,k>$ . Putting all these simplices together essentially reproduces the cube $[0,1]^k$ which of course has volume $1$ .

Exercise: Recover the usual formula for the integral remainder using the above method.

Remark 1: As Sangchul said in the comment, the method is related to the notion of ordered exponential. In a basic course on ODEs, one usually sees the notion of fundamental solution $Phi(t)$ of a linear system of differential equations $X'(t)=A(t)X(t)$ . One can rewrite the equation for $Phi(t)$ in integral form and do the same iteration as in the above method with the result $ Phi(s)=sum_^int_ dt_1cdots dt_k A(t_1)cdots A(t_k) . $ It is only when the matrices $A(t)$ for different times commute, that one can use the above permutation and cube reconstruction, in order to write the above series as an exponential. This happens in one dimension and also when $A(t)$ is time independent, i.e., for the two textbook examples where one has explicit formulas.

Remark 2: The method I used for the Taylor expansion is related to how Newton approached the question using divided differences. The relation between Newton's iterated divided differences and the iterated integrals I used is provide by the Hermite-Genocchi formula.

Remark 3: These iterated integrals are also useful in proving some combinatorial identities, see this MO answer:


Video:

Taylor Polynomial Approximations. A function f (x) can be approxiamted near the point x = a by using polynomials. To get more accuracy, higher-degree polynomials are taken . Taylorin polynomi If you're seeing this message, it means we're having trouble loading external resources on our website.Solution: We will compute this as if we didn't already know the Taylor series for sine. We look at $T_n(x)$ above, and see we need to find evaluate $f(x)=sin(x)$ and its derivatives at $a=0$. The Taylor polynomial p of degree n of a function f expanded about the point x = a is Taylor also gave a formula for the polynomial. (See it below in the series.) If you don't..

Taylor Polynomial on eMathHel

  1. Let Δ(k)j be its kth-order derivative. To prove that f is Lipschitz α at v we shall approximate f with a polynomial that generalizes the Taylor polynomial
  2. We introduce Taylor polynomials for functions of several variables. Quadric surfaces. Computing the Taylor polynomial is not so bad, you just need to get the hang of it
  3. ..taylor_series_expansion th time_converter trace transpose_matrix trig_calculator valuation variance vector_calculator vector_coordinates vector_difference vector_norm..

Mathwords: Taylor Polynomia

  1. And at this point it is not at all clear what exactly can be done with such formulas. For one thing, there are choices.
  2. From the differentiability of g and (α + x)n+ 1 it follows that G is differentiable on any subinterval of (a, b) and we may apply Rolle's theorem 2.4 to G on the interval [x0, α]. Thus there exists ξα ∈ (x0, α) such that
  3. The general idea here is to approximate &lsquofancy&rsquo functions by polynomials, especially if we restrict ourselves to a fairly small interval around some given point. (That &lsquoapproximation by differentials&rsquo circus was a very crude version of this idea).

We begin by splitting the map ℳ into its linear part M and its purely nonlinear part N, so that we have Taylor Polynomial? Find the Taylor Polynomial of degree 3 for the function f(x) = (sqaure root of x) centered at 4 . Each value of p ⩾ 1 gives a norm, although the only two finite values of p which are used in practice are 1 and 2. It is easy to verify that (5.3) satisfies axioms (i) and (ii) of Definition 5.1. Axiom (iii) is easily verified for p = 1 and does not hold for p < 1, which explains our restriction p ⩾ 1 above. For p > 1 we can approximate to the integral by sums and show that the verification of axiom (iii) is equivalent to carrying out the same exercise for the discrete ip-norm defined by (5.5) below. EVERYDAY IM TAYLORIN` Hey guys Taylor-Chan here. If your wanting to ask me for requests. Hecky to the Noey, i dont do requests. i might do my friends sometime but that doesnt include EVERYONE

This guarantees a fast decay of |Δj(k) (v)| when 2j goes to zero, because α is not an integer so α > ⌊α⌋. At large scales 2j, since |Wf(u, s)| ≤ ‖f‖ ‖ψ‖ with the change of variable u′ = (t − u)/s in (6.23), we have Taylor polynomials are certain polynomials that can approximate functions. Read this article to find out what you need to know for the AP Calculus BC exam Polynomi who? finish it. No teams 1 team 2 teams 3 teams 4 teams 5 teams 6 teams 7 teams 8 teams 9 teams 10 teams Custom where αk(x) = pk(x)/pk+ 1, (x) and pk+ 1(x) divides exactly into pk(x). The expression on the right of (6.67) is called a continued fraction. Since it looks so cumbersome, we usually write it in the more concise form


Basic Mathematics and Physics to be a Great Amateur Theoretical Physicist

You will want to study basic mathematics before you get into mechanics.

For what follows these are good resources:

  • US Navy Training Manual, Mathematics, Basic Math and Algebra , search for NAVEDTRA 14139 in your search engine.
  • US Navy Training Manual. Mathematics, Trigonometry , search for NAVEDTRA 14140 in your search engine.
  • US Navy Training Manual. Mathematics, Pre-Calculus and Introduction to Probability , search for NAVEDTRA 14141 in your search engine.
  • US Navy Training Manual. Mathematics, Introduction to Statistics, Number Systems and Boolean Algebra , search for NAVEDTRA 14142 in your search engine.
  • David A. Santos, (2008), Andragogic Propaedeutic Mathematics . This is a free download for the website: http://www.freemathtexts.org/Santos/Pdf.php. This book does not cover complex numbers, but it does have an introduction to sets.
  • David A. Santos, (2008), Ossifrage and Algebra . This is a free download for the website: http://www.freemathtexts.org/Santos/Pdf.php. This book does not cover complex numbers.
  • David A. Santos, (2008), Precalculus . This is a free download for the website: http://www.freemathtexts.org/Santos/Pdf.php.
  • Silvanus Thompson, (1912), The Project Gutenberg EBook of Calculus Made Easy . Note that this has been redone with modern notation throughout. This is a free download from the website: http://www.gutenberg.org/files/33283/33283-pdf.pdf
  • David Santos, (2008), The Elements of Infinitesimal Calculus . This is a free download for the website: http://www.freemathtexts.org/Santos/Pdf.php
  • S. K. Chung, (2007), Understanding Basic Calculus . This is a free download from the website: https://booktree.ng/wp-content/uploads/2019/01/Understanding-Basic-Calculus-By-S.K.-CHUNG.pdf
  • David Guichard, (2012), Calculus Early Transcendentals . This is a free download from the website: http://www.whitman.edu/mathematics/multivariable/
  • Wilfrid Kaplan and Donal J. Lewis, (), Calculus and Linear Algebra, vol.s 1 and 2 . These are a free downloads from the website: http://quod.lib.umich.edu/s/spobooks/. I strongly recommend these.
  • Dan Sloughter, (2000), Difference Equations to Differential Equations , as a free download located here: http://synechism.org/drupal/de2de/.
  • Kenneth Kuttle, (2010), Calculus, Applications and Theory . A free download from the website: http://www.math.byu.edu/

If you wish to buy a book, I recommend these:

  • Biman Das, (2005), Mathematics for Physics with Calculus , Pearson Prentice Hall. The only problem with this book is that it doesn't cover any linear algebra.
  • George E. Owen, (1964), Fundamentals of Scientific Mathematics , Johns Hopkins University Press, reprinted by Dover Publications in 2003. This is a very nice book to read, covering the use of matices in geometry, vector algebra, analytic geometry, functions and approximations, and calculus.
  • Feynman, Leighton, Sands, The Feynman Lecture on Physics , Basic Books. I also recommand Feynman, Gottlied, Lighton, Feynman's Tips on Physics , Pearson Addison Wesley. And Feynman, Exercises for the Feynman Lectures on Physics , Basic Books.
  • R. Shankar, (2013), Fundamentals of Physics: Mechanics, Relativity, and Thermodynamics . Yale University Press This is well written.
  • Frank Blume, Calvin Piston, (2014), Applied Calculus for Scientists and Engineers, Volume 1 . Available at Amazon.com.

You should master these topics/skills before you move on to mechanics:

This requires either Mathematica 8 or later, or the free Mathematica CDF Viewer, though the viewer cannot run the programs, (you can find that here). You will also need to download the MAST Writing Style into the folder SystemFiles/Front End/Stylesheets. You can download that here. Once you load this file into the folder rename it MAST Writing Style 3. Reload Mathematica and it will be there.

The topics below form several logical units.

Sets of Numbers and Their Generalization

This constitutes a modern introductory course in algebra. The key reason to study this, if you altready have studied elementary algebra, is that the language of modern algebraic structures are introdude early and used throughout.

  1. Numbers and Sets
  2. Logic:
  3. Relations
  4. Binary Operations, Addition, Summation, Multiplication, Products, and Exponentiation
  5. Semigroups and Solving Equations by Subtraction
  6. Integers
  7. Monoids and Groups
  8. Rings and Integral Domains
  9. Solving Equations by Division
  10. Primes, Factoring, and Division
  11. The Fundametnal Theorem of Arithmetic
  12. Rational Numbers
  13. Fields
  14. Decimal Expansions
  15. Scientific Notation
  16. Other Simple Algebraic Structures
  17. Modular Arithmetic
  18. Finite Arithmetic
  19. Diophantine Equations
  20. Rules for Exponents
  21. Roots and Logarithms
  22. Real Numbers
  23. Dedekind Cuts and Ordered Fields
  24. Imaginary Numbers
  25. Complex Numbers
  26. Quaternions and Gaussian Integers
  27. Scientific and Mathematical Writing
  28. Algebraic Expressions
  29. Polynomials
  30. Partial Fractions
  31. The Binomial Theorem
  32. Combinatorics
  33. Probability
  34. Measurement and Error
  35. Equations
  36. Functions, Maps, and Morphisms
  37. Infinite Sets
  38. Algebraic Functions
  39. The Theory of Equations
  40. Exponential and Logarithmic Functions

Sets of Points

This is a modern course in introductory geometry where the concepts of geometric symmetry using the language of algebraic structures is used extensively. This merges the concepts of traditional geometry, algebraic structures, trigonometry, analytic geometry, and linear algebra.


Gateway exam

Friday, February 26 Areas between curves Section 7.1 (through top of page 417) * Monday, March 1 Finding volumes by parallel cross section
Introducing the goblet project Sect 7.1: 13, 14, 16, 17, 19, 23. Wednesday, March 3 Center of Mass of a solid of revolution---work on the goblet project Sect 7.1: 39, 40, 42, 43.
Sect 7.2: 25, 26, 27, 28, 29, 31. Friday, March 5 Work on the goblet project.

Spring Break

Section 7.4 Wednesday, March 24 Solving the logistic equation Sect 7.4: 1, 2, 3, 4, 9, 12, 13, 18, 19, 21, 22. Friday, March 26 Group quiz #2 ---techniques and applications of integration Goblet project due * Monday, March 29 Approximations by polynomials: Taylor polynomials. Section 9.1 Wednesday, March 31 Taylor Polynomials and Taylor's theorem Section 9.2 Sect. 9.1: 3, 5, 7-10, 13, 16, 18, 21, 22, 25, 26, 27, 30, 31, and 32. Friday, April 2

Geometric series---a first look.

* Monday, April 5 L'Hopital's Rule
Introducing Improper integrals L'Hopital's Rule handout Sect. 9.2: 5, 7, 8, 10. Wednesday, April 7 Improper integrals Section 10.1 L'Hopital's Rule exercises (handout) Friday, April 9 Sequences of real numbers: definitions and convergence. Section 11.1 Sect. 10.1, part I: 1, 3, 4, 5, 7, 8, 12. * Monday, April 12

Sequences of real numbers---getting more precise.

Geometric series---ideas to consider and write about.

Sect. 10.1, part II: 9-17 (odds only) and 33, 35, 39, 45.
Sect. 11.1:
1-17 (odds only), 18.
Wednesday, April 14 Convergence and divergence of series Section 11.2 Sect. 11.1: 19-24, 27, 31, 33, 34, 50, 51, 52, 54. Friday, April 16

Sect. 11.2:
4, 7, 8, 11, 12, 13, 17, 20, 23, 24.
(Hint: on the last two. Compute a few partial sums for the series. Find a pattern so you can get a limit.)


10.3E: Exercises for Taylor Polynomials and Taylor Series

Is it possible to express transcendental function such as , , , using polynomials? The next theorem answers such a question. transcendental function Let be polynomials in . Then a function satisfies a polynomial equation

is called algebraic function . A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves roots of polynomials.

NOTE Note that for , we have , where . Thus for , Taylor's theorem is the same as Mean Value Theorem. For , we have , where is the difference of the value of line and function at . In other words, is an error caused by approximation of the value of by the line. Similarly, is an approximation error of by a quadratic polynomial.

Proof Let be an expression satisfying

Then . Thus satisfies the condition of Rolle's Theorem . Thus

Check
.

In Taylor's theorem with is called Maclaurin's Theorem . Set . Then we have

Now error estimate is given by , where

Understanding 1. Taylor polynomial around
P(x) = .
2. Lagrange's remainder

Note that . Now we show , where . Note that can represents all numbers form 0 to . Since , we have .

Error Estimate Error estimate is to find the bound for the absolute value of remainder term.

SOLUTION Since , we have . Find a Taylor polynomial around , we have

We next find error estimate. Since

SOLUTION Since , we have . Thus Taylor polynomial around is Now we divide this into two cases.
case 1. is even. Let . Then
case 2. is odd. Let . Then . Thus

MacLaurin Series Expansion Suppose that is infinitely many times differentiable function on an interval containing . The by MacLaurin's theorem, we have

If , then we can express as

In this case, the right-hand side is called MacLaurin Series Expansion of .

MacLaurin Series Expansion of Basic Functions

Note that takes values from to . Suppose . Then for implies and for implies .

Understanding In MacLaurin's theorem, a function is expressed by Taylor polynomial and error term. If the size of error estimate is getting close to 0, then we should be able to write as infinite series.

To show MacLaurin series expansion, we need to show . But it is not easy to show by using Lagrange's Remainder . So, we use different method. Suppose we express a MacLaurin series expansion of as . Then showing approachs 0 is the same as showing converges. To show converges, it is useful Limit Ration Test. Limit Ration Test

Radius of Convergence Replace by . Then . Now . Thus implies and for , converges.

SOLUTION 1. Let . Then since , we have . Thus Taylor polynomial for is

Let . Then apply the limit ration test.

Thus for all , converges. Therefore and,

2. Let . Then since , we have . Now for is even, and for , we have . Thus

Let . Then by the limit ration test,

Check
Limit Ratio Test handles consecutive term. But for , we need to consider .

SOLUTION 1. Let . Then since , we have . Thus Taylor polynomial is Let . Then by Limit Ratio Test, we have

Thus for , converges. Therefore, and

Radius of Convergence If , then converges. Thus we say a radius of convergence.

Taylor polynomial is given by . Let and apply Limit Ratio Test. Then

Thus for , converges. Therefore, and



Thus . Then a sequence is decreasing and bouded blow by 0. Thus, . On the other hand, since , . Therefore, converges.

SOLUTION Since the denominator is , we find Taylor polynomial of 2nd degree of .

SOLUTION Since the denominator is , we find Taylor polynomial of 3rd degree of . implies ,

1. Find a MacLaurin series expansion of the following functions (a)

1. Show the following MacLaurin series expansion holds

(d) , 2. Find the limit of the following functions using Landau o

(a) By the exercise 1(d)we can obtain Now using this fact, calculate

(b) is called Machin's formula Using this formula, calculate 100 digts after the decimal point


MATH 230 Spring 2021

PREREQUISITE: MATH 229 with a grade of C or better.

  • To understand and connect concepts of the calculus with real world problems and other scientific disciplines.
  • To value mathematics and develop an ability to communicate mathematics, both in writing and orally.
  • To develop mathematical reasoning, and an ability to solve problems.
  • To attain computational facility in integral calculus, and sequences and series.

TEXT: Calculus (eighth edition) by James Stewart, published by Cengage Learning.

Some additional references:
Thomas and Finney, Calculus and Analytic Geometry.
Edwards and Penney, Calculus and Analytic Geometry.
Swokowski, Calculus with Analytic Geometry.
Leithold, The Calculus with Analytic Geometry.

SYLLABUS: Click here for suggested lecture pace.

HOMEWORK: Click here for the list of suggested homework exercises.

GRADING: Grades will be assigned on the basis of 650 points, as follows:

  • 3 hour exams worth 100 points each
  • Quizzes and/or homework, 150 points total
  • Final exam, 200 points

GRADING SCALE: The grading scale for this class will at least guarantee the following:

PEER ASSISTED LEARNING: The Calculus Tutoring Center is currently closed. Online tutoring is available via the Peer Assisted Learning scheme, see here.

EXAMS: There will be two exams given on a section by section basis and one Mid-term exam (the second exam) which will be a mass exam.

MIDTERM and FINAL EXAMS: Because of concerns about COVID-19, the Midterm and Final exam times and locations will be announced at a later date. The dates/times in the syllabus are tentative. The final exam will be a comprehensive, departmental examination. All sections of this course will take the same final exam at the same time.

PREVIOUS MID-TERM EXAMS:Note that the course changes and so do the exams. Our goal is to help you learn the material in Calculus 2, not specifically to prepare you for the midterm exam.
Mid-term Exam (Fall 2019)

PREVIOUS FINAL EXAMS: Note that the course changes and so do the exams. Our goal is to help you learn the material in Calculus 2, not specifically to prepare you for the final exam.
Final Exam (Spring 2019)
Final Exam (Spring 2012)

CALCULATORS: Students may consider having a graphing calculator with roughly the capabilities of the TI-83. You will find this useful for investigating the concepts of the class, so you can experiment with additional examples. You may also want to verify parts of your homework calculations.

RESOURCES ON THE WEB:
Understanding Mathematics: a study guide, from the University of Utah.
Calculus resource list from the Math Archives, from the University of Tennessee at Knoxville.
Symbolic calculators which will compute derivatives and integrals.

ACADEMIC CONDUCT: Good academic work must be based on honesty. The attempt of students to present as their own work that which they have not produced is regarded by the faculty and administration as a serious offense. Students are considered to have cheated if they copy the work of another during an examination or turn in a paper or an assignment written, in whole or in part, by someone else. Students are guilty of plagiarism, intentional or not, if they copy material from books, magazines, or other sources without identifying and acknowledging those sources or if they paraphrase ideas from such sources without acknowledging them. Students guilty of, or assisting others in, either cheating or plagiarism on an assignment, quiz, or examination may receive a grade of F for the course involved and may be suspended or dismissed from the university.

DRC STATEMENT: Northern Illinois University is committed to providing an accessible educational environment in collaboration with the Disability Resource Center (DRC). Any student requiring an academic accommodation due to a disability should let his or her faculty member know as soon as possible. Students who need academic accommodations based on the impact of a disability will be encouraged to contact the DRC if they have not done so already. The DRC is located in Suite 180 of the Campus Life Building, and can be reached at 815-753-1303 or [email protected]

ADVICE: Perhaps the single most important factor in your success in this course is your study habits. This is a fast paced course, with little room for catching up if you fall behind. Successful students have good time management skills. Set aside at least three nights a week to study the topics and work the homework problems. Do not wait until exam time to try to learn new material.

Learn mathematics like you would learn a language. Work on the concepts until they make sense. Don't just memorize facts and then forget them a few weeks later. You will need to know this stuff for Calculus III and other courses.

Master each homework problem---beyond just getting a correct answer. Be aware of your mistakes in algebra and trigonometry.


Engineering mathematics: a foundation for electronic, electrical, communications and systems engineers

Engineering Mathematics is the leading undergraduate textbook for Level 1 and 2 mathematics courses for electrical and electronic engineering, systems and communications engineering students. It includes a basic mathematics review, along with all the relevant maths topics required for these engineering degrees. Features Students see the application of the maths they are learning to their engineering degree through the book&rsquos applications-focussed introduction to engineering mathematics, that integrates the two disciplines Provides the foundation and advanced mathematical techniques most appropriate to students of electrical, electronic, systems and communications engineering, including: algebra, trigonometry and calculus, as well as set theory, sequences and series, Boolean algebra, logic and difference equations Integral transform methods, including the Laplace, z and Fourier transforms are fully covered Students learn and test their understanding of mathematical theory and the application to engineering with a huge number of examples and exercises with solutions New to this edition New Engineering Example showcase feature, covering an extensive range of modern applications, including music technology, electric vehicles, offshore wind power and PWM solar chargers New mathematical sections on number bases, logs and indices, summation notation, the sinc x function, waves, polar curves and the discrete cosine transform New exercises and answers


Answers and Replies

Start with google, select BOOKS, search for ' Machine Learning '
There are plenty of books on the subject.

You may find some of those in a library near you, or as an ebook on the web.
When you find a good book, go to bookfinder.com and have a look for a new or used copy.

I know there are plenty of books on the subject. That's why I am looking for advice for books which do not require a strong mathematical background and would still be good to get knowledge and intuitions on the required maths for ML.

If I search "calculus" on google books, I get Spivak's. That's not really the kind of books that I am looking for.

I am already a gold member and yes, I would have used amazon link if available where I can order.

In the preface to All of statistics the author writes that the book is suitable for graduate students in computer science or honors undergraduates in math, statistics or computer science. I checked it out from the library once and do not believe that the OP is prepared.

Likewise, i own a copy of the book by Jaynes. It is not easy reading, and has as much philosophy as probability. According to the preface, it is "addressed to readers who are already familiar with applied mathematics, at the advanced undergraduate level or preferably higher,"

I do not know what book, if any, would be appropriate for the OP, but I do not think either one of those would be helpful.

I am a software developer (bachelor's degree in Europe, different than a bachelor's degree in the US I believe) and I don't have a strong math/physics background but I am willing to learn.

For a few years now, I have been really interested in machine learning but until now, I only read blog posts and examples in python. Now, I would really like to learn the maths and get the intuition behind ML so I need resources adapted to my maths background.

I haven't started working on Multivariable calculus, linear algebra and differential equations on Khan Academy yet so I have a lot of work to do. I also worked on paul's online notes (http://tutorial.math.lamar.edu/) except Calculus III and diffeq.

I read online that the following maths are required to properly learn machine learning concepts

1) probability and statistics
.

3) Calculus / multivariable calculus ?
.

If you want something structured for multi-variable calc, you may want to work through the assignments here:
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/

key things to learn are how to think about derivatives in higher dimensions, do nested integrals (basically nested for loops), lagrange multipliers, chain rule, taylor polynomials. Taylor polynomials (not series but polynomials) believe it or not are basically the all access pass for most optimization techniques in ML. A lot of the stuff in multi-variable calc will not be needed -- e.g. spherical coordinate systems, line integrals, surface integrals, etc. I don't think really ever come up in machine learning -- you're basically working in rectangular coordinates and need to be able to interpret nested integrals. (Nested integrals also will come up in the below probability courses.)

probability, one of the courses recommended here:
https://www.physicsforums.com/threa. swer-sheet-in-statistics.921188/#post-5810893
You need to do multi-variable calc first though. I am biased towards the stuff from MIT like 6.041 (and 6.041x on edx). This can be mind-bending stuff so doing lots of exercises (with solutions available) is key.

If there's one topic you were going to beat with a hammer on your list, it probably should be linear algebra, which comes up again and again in various forms for ML. My advice would be to pick one of the sources you mentioned and work through it and then ignore the other linear algebra sources on your list. Step two would be to work through the first 7 chapters of Linear Algebra Done Wrong, freely available by the author here: https://www.math.brown.edu/

treil/papers/LADW/LADW_2017-09-04.pdf . It still has my favorite walkthrough of spectral theory that I've seen. I wouldn't recommend it as a first book in linear algebra, but that's probably the only negative I'd say about it.

I woudn't worry too much about optimization for the time being -- you should pick it up as you go along in ML. I do trust that you are familiar with Dynamic Programming, though, as part of your CS background.


Watch the video: Workshop and Qu0026A from TMiL 29th October Geogebra (December 2021).