Linear and Quadratic Approximation Assignment

Question 1

[2 points] We are going to use linear approximation to estimate values of the 4/3 power function.

[Hint: Don't forget parentheses in x^(4/3)]

Part a

Use linear approximation to estimate $67^{4/3}$ and give the percent error (Hint: follow Example 2).

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

Part b

Use linear approximation to estimate $66^{4/3}$ and give the percent error.

[Use the same tangent line as part a.]

Part c

Since our approximation is based on $x=64$, we expect the error to be larger at $x=67$ than it is at $x=66$. Check that your error is larger for part a (Type Yes or No).

Question 2

[2 points] Use linear approximation to estimate $\displaystyle\cos\left(\frac{\pi}{7}\right)$ and give the percent error.

[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 3

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

Hint: Since you do not know $f(x)$, you can't copy and paste from Example 2. However, you have enough numbers to define one tangent line.

In general, a tangent line looks like $TL(x)=f(x_0)+f'(x_0)\cdot(x-x_0)$. Put the given numbers in the appropriate spots to define a line at $x_0=5$.

Then see what happens at $x=6$ by computing $TL(6)$.

Question 4

[2 points] Use quadratic approximation to estimate $67^{4/3}$ and find the percent error (Hint: follow Example 4).

[Use the same point of tangency as Question 1.]

The percent error should be much smaller than the percent error from Question 1, Part a.

Question 5

[2 points] 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 2.]

The percent error should be much smaller than the percent error from Question 2.