Q: A function $f(x)$ has a local maximum at $x = c$ if:

$f'(c) = 0$ \,\text{and}\, f''(c) < 0$

Q: A company wants to minimize the cost $C(x)$ of producing $x$ units. If $C(x)$ is differentiable, what must be true at a local minimum?

$C'(x) = 0$ \,\text{and}\, C''(x) > 0$

Question 1

Which of the following is the correct derivative of $y = 	an^{-1}(x)$?

Accepted Answer

$\frac{1}{1 + x^2}$

Question 2

Suppose $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. According to the Mean Value Theorem, which statement must be true?

Accepted Answer

There exists $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$

Question 3

Given $y = \ln(x^2 + 1)$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{2x}{x^2 + 1}$

Question 4

A function $f(x)$ has a local maximum at $x = c$ if:

Accepted Answer

$f'(c) = 0$ \,	ext{and}\, f''(c) < 0$

Question 5

If $y = \sec^{-1}(x)$, what is $\frac{dy}{dx}$ for $|x| > 1$?

Accepted Answer

$\frac{1}{|x| \sqrt{x^2 - 1}}$

Question 6

Which of the following is the correct derivative of $y = \ln|x|$?

Accepted Answer

$\frac{1}{x}$

Question 7

A pharmaceutical company wants to maximize the profit $P(x)$ from selling $x$ units of a drug. If $P(x)$ is continuous and differentiable, which method should they use to find the maximum profit on $[a, b]$?

Accepted Answer

Evaluate $P(x)$ at the endpoints and at any critical points in $(a, b)$, then choose the largest value.

Question 8

If $y = 	an^{-1}(x)$, what is $\frac{d^2y}{dx^2}$?

Accepted Answer

$\frac{-2x}{(1 + x^2)^2}$

Question 9

Which statement is true about the function $f(x)$ if $f'(x) > 0$ for all $x$ in $(a, b)$?

Accepted Answer

$f(x)$ is increasing on $(a, b)$

Question 10

A biologist models the population of a species by $P(t) = 1000e^{0.05t}$. What is the instantaneous rate of change of the population at $t = 10$?

Accepted Answer

$P'(10) = 50e^{0.5}$

Question 11

Which of the following is the correct derivative of $y = \ln(\sin x)$?

Accepted Answer

$\cot x$

Question 12

If $f(x)$ is continuous on $[a, b]$, which theorem guarantees that $f(x)$ attains both an absolute maximum and minimum on $[a, b]$?

Accepted Answer

Extreme Value Theorem

Question 13

A company wants to minimize the cost $C(x)$ of producing $x$ units. If $C(x)$ is differentiable, what must be true at a local minimum?

Accepted Answer

$C'(x) = 0$ \,	ext{and}\, C''(x) > 0$

Question 14

Which of the following is the correct derivative of $y = \ln(	an x)$?

Accepted Answer

$\frac{1}{\sin x \cos x}$

Question 15

If $y = \sin^{-1}(x)$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{1}{\sqrt{1 - x^2}}$

Question 16

A medical researcher models the concentration of a drug in the bloodstream by $C(t) = 5\ln(t + 1)$. What is the rate of change of concentration at $t = 4$?

Accepted Answer

$C'(4) = 1$

Question 17

Which of the following is the correct derivative of $y = 	an^{-1}(2x)$?

Accepted Answer

$\frac{2}{1 + 4x^2}$

Question 18

Which of the following is the correct derivative of $y = \ln(x^2)$?

Accepted Answer

$\frac{2}{x}$

Question 19

If $f(x)$ is continuous on $[a, b]$ and $f(a) = f(b)$, which theorem guarantees there is $c$ in $(a, b)$ such that $f'(c) = 0$?

Accepted Answer

Rolle's Theorem

Calculus I: Differentiation, Applications, and Extrema – Exam Review Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards