diff --git a/source/calculus/source/09-PS/05.ptx b/source/calculus/source/09-PS/05.ptx new file mode 100644 index 000000000..fa502417e --- /dev/null +++ b/source/calculus/source/09-PS/05.ptx @@ -0,0 +1,361 @@ + + +
+ Taylor's Theorem (PS5) + + + + + Activities + + + +

+ Recall that we can use a Taylor series for a function + to approximate that function by using an kth degree Taylor + polynomial. +

+ + + +

+ Which of the following is the 3rd degree Taylor polynomial for f(x)=\sin x + centered at 0. +

+
    +
  1. 1-\dfrac{x^2}{2}
    2. +
  2. x-\dfrac{x^3}{3!}
    3. +
  3. x+\dfrac{x^3}{3!}
    4. +
  4. x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}
    5. +
+ + + + + +

+ Use the 3rd degree Taylor polynomial for f(x)=\sin x + to approximate \sin(1). +

+ + + + + +

+ Use technology to approximate \sin(1). +

+ + + + + + +

+ Given a infinitely differentiable function f(x)=\displaystyle\sum_{n=0}^\infty \dfrac{f^{(n)}(c)}{n!}(x-c)^n, we define + the remainder, denoted R_k(x), to be the difference between the function f(x) + and its kth degree Taylor polynomial T_k(x). That is, + + R_k(x)=f(x)-T_k(x). + +

+

+ The error in the approximation f(x)\approx T_k(x) is given by |R_k(x)|. +

+ + + + + +

+ We saw in , the Maclaurin series for + f(x)=e^x is + + e^x=\displaystyle\sum_{n=0}^\infty \dfrac{1}{n!}x^n. + + +

+ + + +

+ Compute R_2(4) using technology. +

+ + + + +

+ Compute R_3(4) using technology. +

+ + + + +

+ What do you expect from R_4(4)? +

+
    +
  1. +

    + There is not enough information. +

    +
    2. +
  2. +

    + It will be greater than both R_2(4) and R_3(4). +

    +
    3. +
  3. +

    + It will be between R_2(4) and R_3(4). +

    +
    4. +
  4. +

    + It will be less than both R_2(4) and R_3(4). +

    +
    5. +
+ + + + + + +

+ Let f(x) be a function represented by a power series centered at x=c + + f(x)=\displaystyle\sum_{n=0}^\infty a_n(x-c)^n + + with an interval of convergence I. Then for all x in I, + + \lim_{k\rightarrow\infty} R_k(x)=0. + + +

+ + + + + Taylor's Theorem + + +

+ Let f(x) be an (k+1) times differentiable function on an interval I of c, and let T_k(x) + be its kth degree Taylor polynomial centered at x=c. Then for any x in the interval I, + there exists p between c and x such that + + R_k(x)=\dfrac{f^{(k+1)}(p)}{(k+1)!}(x-c)^{k+1}. + + If there exists M_k such that |f^{(k+1)}(x)|\leq M_k for all x in I, then the error in + the approximation f(x)\approx T_k(x) has an upper bound: + + |R_k(x)|\leq \dfrac{M_k}{(k+1)!}|x-c|^{k+1}. + + +

+ + + + + Using Taylor's Theorem +

+ The trickiest part to using Taylor's Theorem is calculating M_k to get a bound for the error + |R_k(x)| for the approximation f(x)\approx T_k(x). +

+ + + + +

+ Consider the function f(x)=1/x defined on the interval I=[1,2]. +

+ + + +

+ Calculate the derivatives f'(x), f''(x), f'''(x), and f^{(4)}(x). +

+ +

+ f'(x)=-1/x^2, f''(x)=2/x^3, f'''(x)=-6/x^4, f^{(4)}(x)=24/x^5 +

+ + + + + +

+ Which of the following can we say above the values of |f^{(k)}(x)| on I for k=1,2,3,4? +

+
    +
  1. +

    + |f'(x)| and |f'''(x)| are increasing, while |f''(x)| and |f^{(4)}(x)| are decreasing. +

    +
    2. +
  2. +

    + All are decreasing. +

    +
    3. +
  3. +

    + All are increasing. +

    +
    4. +
  4. +

    + |f'(x)| and |f'''(x)| are decreasing, while |f''(x)| and |f^{(4)}(x)| are decreasing. +

    +
    5. +
+ +

+ B. +

+ + + + + + +

+ Calculate M_k for each k=1,2,3,4 using your results from part (b). +

+ +

+ M_1=1, M_2=2, M_3=6, M_4=24 +

+ + + + + + +

+ Use Taylor's Theorem to calculate |R_k(1.5)| for each k=1,2,3,4 + to 3 decimal places. Use a=1 as the center of the approximation. +

+ +

+ 0.125, 0.042, 0.016, 0.006 +

+ + + + + + +

+ Are the errors decreasing? Explain why or why not. +

+ + + + + + + + +

+ Let f(x)=e^x. Your goal is to approximate f(1)=e. +

+ + + +

+ Explain and demonstrate how to determine the upper bound M_k from Taylor's + Theorem for f(x)=e^x. +

+ + + + +

+ Use your value for M_k in part (a) to find an upper bound for the error + |R_4(1)|. +

+ + + + +

+ Use your value for M_k in part (a) to find an upper bound for the error + |R_8(1)|. +

+ + + + + + + + + + + + Sample Problem + +

+ Here you are tasked with approximating the value of \cos(1). +

+ + + +

+ Calculate the 4th degree Taylor polynomial for f(x)=\cos x centered at \pi, + then use it to approximate the value of \cos(1) to three decimal places. +

+ + + + + +

+ Apply Taylor's Theorem to find an upper bound for the error in this approximation. +

+ + + + + +

+ Use technology to calculate |R_4(1)|. Is the error within the upper bound found + in part (b)? +

+ + + + + +

+ Explain whether the approximation error |R_{k}(1)| increases or decreases as + k\rightarrow\infty. +

+ + + + + + + + +
\ No newline at end of file diff --git a/source/calculus/source/09-PS/main.ptx b/source/calculus/source/09-PS/main.ptx index d6699c3a5..5e4170cfa 100644 --- a/source/calculus/source/09-PS/main.ptx +++ b/source/calculus/source/09-PS/main.ptx @@ -9,5 +9,6 @@ + diff --git a/source/calculus/source/09-PS/outcomes/05.ptx b/source/calculus/source/09-PS/outcomes/05.ptx new file mode 100644 index 000000000..1380e80c7 --- /dev/null +++ b/source/calculus/source/09-PS/outcomes/05.ptx @@ -0,0 +1,4 @@ + +

+Determine an upper bound for the error in an approximation of a function via a Taylor polynomial. +

\ No newline at end of file diff --git a/source/calculus/source/09-PS/outcomes/main.ptx b/source/calculus/source/09-PS/outcomes/main.ptx index d1f2ecb64..6ba275383 100644 --- a/source/calculus/source/09-PS/outcomes/main.ptx +++ b/source/calculus/source/09-PS/outcomes/main.ptx @@ -21,6 +21,9 @@ By the end of this chapter, you should be able to...
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