Problems

01

Inverse of exponential function

Find a formula for the inverse of $f(x)=e^{8x-3}$. What is the domain of your $f^{-1}$?

02

Falling ball

Assume that a ball dropped from a tower travels a distance $d$ that is a function of time as follows: $$d(t)=4.9t^2$$ What is the average speed of the ball during the time range $t\in [0,20]$?

What is the instantaneous speed of the ball at $t=10$? (For this question, please do not use calculus rules and formulas, you should find an approximate value, valid to three decimal places, by using a calculator to approximate the relevant limit. Your solution should include a table of values you computed.)

03

Asymptotes

Find the vertical and horizontal asymptotes of this function: $$f(x)=\frac{x^2+1}{3x-2x^2}$$

04

Confusing-but-not-hard limits

Find the following limits:

• (a) ${\lim }_{x\to {\pi }^{-}}x\cot x$
• (b) ${\lim }_{x\to 0^{+}}\left(\frac{1}{x}-\ln x\right)$
• (c) ${\lim }_{x\to 4}\frac{\ln x-\ln 4}{x-4}$ (for this one, first approximate with a calculator, then guess the exact answer)