Question 1

Suppose $$f(x) is $$defined as $$f(x) = \frac{2x^2$ - 3x + 4}{x - 1}. $$What is $$\lim\limits_{x 	o 1} f(x)$$?

Accepted Answer

$$3$$

Question 2

Given $$g(x) = \sqrt{x^2$ + 7}$$, what is $$g'(x)$$?

Accepted Answer

$$\frac{x}{\sqrt{x^2$ + 7}}$$

Question 3

Find the equation of the tangent line to $$f(x) = 3x^2$ - 5x + 8 at$$ $$x = 2. $$Express your answer in slope-intercept form.

Accepted Answer

$$y = 7x - 4$$

Question 4

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4x + 3}{x - 2}.$$

Accepted Answer

$$-1$$

Question 5

A ball is thrown from the roof of a building. Its height after $$t$$ seconds is $$h(t) = -16t^2$ + 40t + 100. $$What is the velocity of the ball after 3 seconds?

Accepted Answer

$$-56 ft/s$$

Question 6

Find $$\lim\limits_{x 	o 0} \frac{1 - \cos x}{x^2$}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 7

Given $$f(x) = \frac{2x - 3}{4x + 5}$$, use the quotient rule to find $$f'(x).$$

Accepted Answer

$$\frac{8x + 22}{(4x + 5)^2$}$$

Question 8

The number of people playing a new game $$t$$ weeks after it is introduced is given by $$g(t) = \frac{8000t}{t^2$ + 8}. $$What is the instantaneous rate of change in the number of people playing the game exactly 4 weeks after introduction?

Accepted Answer

$$120$$

Question 9

Find the derivative of $$f(x) = \cot x.$$

Accepted Answer

$$-\csc^2$ x$$

Question 10

Find $$\lim\limits_{x 	o 3} \frac{3x^2$ + 5x - 2}{x^2$ - 3x - 10}.$$

Accepted Answer

$$\frac{22}{-1}$$

Question 11

If $$f(x) = 2x^2$ - 3x + 4$$, find and simplify the difference quotient $$\frac{f(x + h) - f(x)}{h}.$$

Accepted Answer

$$2h + 2x - 3$$

Question 12

Find the average rate of change of $$F(u)$$ from $$u = 1 to$$ $$u = 3 if$$ $$F(u) is $$given by the table: $$F(1) = 12$$, $$F(3) = 16.$$

Accepted Answer

$$2$$

Question 13

Find the derivative of $$f(x) = (5x - 8$$)^{10}$$.

Accepted Answer

$$50(5x - 8)^9$

Question 14

Find $$\lim\limits_{x 	o 2} \frac{2x^2$ + x - 10}{x - 2}.$$

Accepted Answer

$$5$$

Question 15

A car's position $$s(t) in $$feet after $$t$$ seconds is given by $$s(t) = 6t^2$. What is the average velocity from t = 0$ to t = 4$?

Accepted Answer

$$24 ft/s$$

Question 16

Find the derivative of $$f(x) = 	an(2x).$$

Accepted Answer

$$2\sec^2$(2x)$$

Question 17

Find $$\lim\limits_{x 	o 0} \frac{\sin(2x)}{x}.$$

Accepted Answer

$$2$$

Question 18

Find the derivative of $$h(x) = \sqrt{10 - x^3$}.$$

Accepted Answer

$$-\frac{3x^2$}{2\sqrt{10 - x^3$}}$$

Question 19

Find the derivative of $$f(x) = \sec^2$ x.$$

Accepted Answer

$$2\sec^2$ x 	an x$$

Business Calculus: Limits, Derivatives, and Applications Study Guide

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