Question 1

Given the function $f(x) = 4x^2 - 12x + 9$ on the interval $[0,2]$, what is the absolute minimum value of $f$ on this interval?

Accepted Answer

$f(\frac{3}{2}) = 0$

Question 2

For the function $g(x) = x \ln(x)$ defined on $(0, \infty)$, at which $x$ does $g$ have a critical point?

Question 3

Suppose $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. According to the Mean Value Theorem, which equation must have a solution $c$ in $(a, b)$?

Accepted Answer

$f'(c) = \frac{f(b) - f(a)}{b - a}$

Question 4

Given the graph of $f'(x)$, at which $x$-values does $f(x)$ have inflection points?

Accepted Answer

Where $f'(x)$ has a local maximum or minimum

Question 5

Evaluate $\lim_{x 	o 1} \frac{x^3 - 1}{x - 1}$.

Accepted Answer

$3$

Question 6

For the function $f(x) = 4x^2 - 4x + 1$ on $(-\infty, \infty)$, what is the concavity of $f$?

Accepted Answer

Concave up everywhere

Question 7

Find the critical point of $f(x) = 4x^2 - 4x + 1$.

Accepted Answer

$x = \frac{1}{2}$

Question 8

Suppose $xy = 24$ and you want to minimize $3x + y$ for positive $x$ and $y$. What is the minimum value of $x$?

Accepted Answer

$x = \sqrt{8}$

Question 9

Given $f(x) = \frac{2}{2 + x}$, what is the linear approximation $L(x)$ at $x = 1$?

Accepted Answer

$L(x) = \frac{2}{3} - \frac{2}{9}(x - 1)$

Question 10

Evaluate $\lim_{x 	o 0^+} \sqrt{x} \ln(x)$.

Accepted Answer

$0$

Question 11

If $f(x) = x + \frac{1}{x}$ on $[1,3]$, which $c$ in $(1,3)$ is guaranteed by the Mean Value Theorem?

Accepted Answer

$c = \sqrt{3}$

Question 12

Evaluate $\lim_{x 	o 0^+} x^7 e^{7x}$.

Accepted Answer

$0$

Question 13

When using the linear approximation of $\ln(x)$ near $x = e$, are the approximations overestimates or underestimates of the true values?

Accepted Answer

Overestimates

Question 14

For $g(x) = x \ln(x)$, on which interval is $g$ increasing?

Accepted Answer

$(e^{-1}, \infty)$

Question 15

Given $f(x) = 4x^2 - 12x + 9$ on $[0,2]$, what is the absolute maximum value of $f$?

Accepted Answer

$f(0) = 9$

Question 16

Evaluate $\lim_{x 	o \frac{\pi}{2}} \frac{\sin(x)}{x}$.

Accepted Answer

$\frac{2}{\pi}$

Question 17

Suppose $f'(x)$ is positive on $(-\infty, -2)$. What can you say about $f(x)$ on this interval?

Accepted Answer

$f(x)$ is increasing

Question 18

If $f''(x) > 0$ for all $x$, what can you conclude about the graph of $f$?

Accepted Answer

It is concave up everywhere

Question 19

For the function $f(x) = 4x^2 - 4x + 1$, use the second derivative test to classify the critical point at $x = \frac{1}{2}$.

Accepted Answer

Local minimum

Calculus Exam Study Guide: Critical Points, Extrema, Concavity, Limits, and Approximations

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