Question 1

Given the graph below, at which labeled points $L_1$, $L_2$, $L_3$, or $L_4$ is the function NOT differentiable?

Accepted Answer

$L_1$ and $L_3$

Question 2

Consider the function $G(x) = \begin{cases} -x+5 & x \leq 2 \ x+2 & x > 2 \end{cases}$. Is $G(x)$ continuous at $x=2$?

Accepted Answer

No, because the left and right limits at $x=2$ are not equal.

Question 3

If the cost function is $C(x) = 13x + 900$, where $x$ is the number of hot dogs sold, what is the average rate of change from $x=2$ to $x=13$?

Accepted Answer

$13$

Question 4

For the function $f(x) = \dfrac{x^2 - 36}{x - 6}$, for what value of $x$ is the function not differentiable?

Accepted Answer

$x = 6$

Question 5

Differentiate $y = \sqrt[4]{x^5 + 3x^2 - 2}$ with respect to $x$.

Accepted Answer

$y' = \dfrac{1}{4}(x^5 + 3x^2 - 2)^{-3/4}(5x^4 + 6x)$

Question 6

Given $f(x) = x + \dfrac{5}{x^6}$, find $f'(2)$.

Accepted Answer

$1 - \dfrac{30}{64}$

Question 7

Find the derivative of $F(x) = \left( x^2 - 2x \right)^3 \div 36$.

Accepted Answer

$F'(x) = \dfrac{3(x^2 - 2x)^2 (2x - 2)}{36}$

Question 8

Find all points on the graph $f(x) = x^3 - 3x^2$ where the slope of the tangent line is horizontal.

Accepted Answer

$x = 0$ 	ext{ and } x = 2$

Question 9

Find the equation of the tangent line to the curve $f(x) = 2x^3 - 8x$ at the point $(-2, 0)$.

Accepted Answer

$y = -16x - 32$

Question 10

Find the equation of the tangent line to the graph of $y = \dfrac{8x}{x^2 + 1}$ at the point $(1, 4)$.

Accepted Answer

$y = -2x + 6$

Question 11

Using the formal definition of the derivative, $f'(x) = \lim_{h 	o 0} \dfrac{f(x+h) - f(x)}{h}$, find $f'(2)$ for $f(x) = 3x^2 - 4x + 1$.

Accepted Answer

$f'(2) = 8$

Question 12

Differentiate $F(x) = (x^2 + 3)(x^3 - 4)$.

Accepted Answer

$F'(x) = 2x(x^3 - 4) + 3x^2(x^2 + 3)$

Question 13

If $y = \sqrt{u}$ and $u = x^2 + 4$, find $\dfrac{dy}{dx}$.

Accepted Answer

$\dfrac{dy}{dx} = \dfrac{x}{\sqrt{x^2 + 4}}$

Question 14

Suppose the profit function for a business is $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of units sold. At what value of $x$ is the profit maximized?

Accepted Answer

$x = 10$

Question 15

A company’s revenue function is $R(x) = 50x - x^2$, where $x$ is the number of items sold. What is the marginal revenue when $x = 10$?

Accepted Answer

$R'(10) = 30$

Question 16

If the demand function for a product is $p(x) = 100 - 2x$, where $p$ is the price and $x$ is the quantity, what is the elasticity of demand when $x = 20$?

Accepted Answer

$E = \dfrac{-2 \cdot 20}{100 - 2 \cdot 20} = -0.5$

Business Calculus: Differentiation, Continuity, and Tangent Lines Study Guide

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