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 $$G(x) = \begin{cases} -x+5 & x \leq 2 \ x+2 & x \u003e 2 \end{cases}. Is$$ $$G(x)$$ 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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