Linear and Quadratic Approximation Assignment.sagews

Linear and Quadratic Approximation Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Use linear approximation to estimate the following and give the percent error:

Part a

$67^{4/3}$

[Note: $64^{4/3}=\left(\sqrt[3]{64}\right)^4=4^4=256$, so use $x=64$ for your point of tangency.]

Part b

$66^{4/3}$ [Use the same tangent line as part a.]

Is the percent error bigger or smaller than Part a? Why?

Part c

$\displaystyle\cos\left(\frac{\pi}{7}\right)$

[Note: $\frac{\pi}{6}$ is close to $\frac{\pi}{7}$, and $\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$, so use $x=\frac{\pi}{6}$ for your point of tangency.]

Question 2

Consider a function $f$ such that $f(5)=10$ and $f'(5)=-3$. Estimate $f(6)$ using a tangent line.

[Hint: Since I have not given you a formula for $f(x)$, you can't copy and paste the linear approximation code from the notes. Instead, use the tangent line formula and plug in the given numbers.]

Question 3

Use quadratic approximation to estimate $67^{4/3}$ and find the percent error (use the same point of tangency as Question 1). Compare with your result for Question 1, Part a.

Question 4

Use quadratic approximation to estimate $\displaystyle\cos\left(\frac{\pi}{7}\right)$ and find the percent error (use the same point of tangency as Question 1). Compare with your result for Question 1, Part c.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.