$f'(2)g(2) + f(2)g'(2) = 1 \times 2 + 3 \times 4 = 2 + 12 = 14$

Question 1

Given the curve defined by $x^2 + xy + y^2 = 27$, what is the expression for the slope of the curve at any point $(x, y)$?

Accepted Answer

$\frac{-2x - y}{x + 2y}$

Question 2

For the curve $x^2 + xy + y^2 = 27$, at which points are the tangent lines to the curve vertical?

Accepted Answer

$(3, 6)$ and $(-3, -6)$

Question 3

Consider the function $y = 3x^3 + 3x^2$. What is $\frac{dy}{dx}$ at $x = 0$?

Accepted Answer

$0$

Question 4

Find the equation of the tangent line to $y = \frac{1}{5}(x-2)$ at the point $(2, 0)$.

Accepted Answer

$y = \frac{1}{5}(x-2)$

Question 5

Given $f(x) = \sin(e^x)$, what is $f'(x)$?

Accepted Answer

$\cos(e^x) \cdot e^x$

Question 6

If $g(x) = \ln(x^2 + 2x)$, what is $g'(x)$?

Accepted Answer

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

Question 7

Find $\frac{d}{dx} \arctan(6x)$.

Accepted Answer

$\frac{6}{1 + 36x^2}$

Question 8

Given $h(x) = f(x)g(x)$, and $f(2) = 3$, $g(2) = 2$, $f'(2) = 1$, $g'(2) = 4$, what is $h'(2)$?

Accepted Answer

$f'(2)g(2) + f(2)g'(2) = 1 	imes 2 + 3 	imes 4 = 2 + 12 = 14$

Question 9

Evaluate $\lim_{x 	o 0} \frac{\sin(3x)}{x}$.

Accepted Answer

$3$

Question 10

Which of the following functions is NOT differentiable at $x = 1$?

Accepted Answer

$f(x) = |x-1|$

Question 11

Given $y = x^2 + 2x + 1$, find the rate of change of the slope at $x = 0$.

Accepted Answer

$2$

Question 12

If $f(x) = e^{x^2}$, what is $f'(x)$?

Accepted Answer

$2x e^{x^2}$

Question 13

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

Accepted Answer

$1$

Question 14

Find the derivative of $f(x) = \log_8(3x-7)$.

Accepted Answer

$\frac{3}{(3x-7) \ln 8}$

Question 15

Given $f(x) = x^2$, $g(x) = 2x$, find $\frac{d}{dx}[g(2x-x) - (2x-x)]$ at $x = 1$.

Accepted Answer

$2$

Question 16

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

Accepted Answer

$\frac{1}{\cos(f^{-1}(x))}$

Question 17

Find the $x$-coordinate(s) where the curve $y = x^4 - 12x^2 + 10$ has a horizontal tangent.

Accepted Answer

$x = -\sqrt{6}, \sqrt{6}$

Question 18

Evaluate $\lim_{x 	o \infty} \frac{6x^2 + 2x}{3x^2 - x}$.

Accepted Answer

$2$

Question 19

Which of the following is a removable discontinuity?

Accepted Answer

$f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$

Calculus I: Derivatives, Tangents, and Continuity – Unit 3 Review Study Guide

Study Guide - Practice Questions