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Section 9.5 Taylor’s Theorem (PS5)
Learning Outcomes
Subsection 9.5.1 Activities
Activity 9.5.1 .
Recall that we can use a Taylor series for a function to approximate that function by using an \(n\) th-degree Taylor polynomial.
(a) Which of the following is the \(3\) rd degree Taylor polynomial for \(f(x)=\sin x\) centered at 0.
\(\displaystyle 1-\dfrac{x^2}{2}\)
\(\displaystyle x-\dfrac{x^3}{3!}\)
\(\displaystyle x+\dfrac{x^3}{3!}\)
\(\displaystyle x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}\)
(b) Use the 3rd-degree Taylor polynomial for \(f(x)=\sin x\) to approximate \(\sin(1)\text{.}\)
(c) Use technology to approximate \(\sin(1)\text{.}\)
Subsection 9.5.2 Sample Problem
Here you are tasked with approximating the value of \(\cos(1)\text{.}\)
(a) Calculate the 4th-degree Taylor polynomial for \(f(x)=\cos x\) centered at \(\pi\text{,}\) then use it to approximate the value of \(\cos(1)\) to three decimal places.
(b) Apply Taylor’s Theorem to find an upper bound for the error in this approximation.
(c) Use technology to calculate \(|R_4(1)|\text{.}\) Is the error within the upper bound found in part (b)?
(d) Explain whether the approximation error \(|R_{n}(1)|\) increases or decreases as \(n\rightarrow\infty\text{.}\)
