Question 1

Solve for $$x in $$the equation $$2x + 3 = 16. $$What is the exact value of $$x$$?

Accepted Answer

Question 2

Solve for $$x in $$the equation $$\ln(x) - \ln(x-1) = 2. $$What is the exact value of $$x$$?

Accepted Answer

$$x = e^2$ + 1$$

Question 3

Evaluate the following limit: $$\displaystyle \lim_{x 	o -2} \frac{x+3}{x^2$ + 3x + 2}$$

Accepted Answer

The limit does not exist.

Question 4

Evaluate the following limit: $$\displaystyle \lim_{x 	o \infty} \frac{3x^2$ + 3x + 1}{5x^2$ - 10x + 8}$$

Accepted Answer

$$\frac{3}{5}$$

Question 5

Suppose you invest $$5,000 in an account with an annual interest rate of 7% compounded continuously. How many years will it take for the account to reach 15$$,000? (Round your answer to two decimal places.)

Accepted Answer

$$t = \frac{\ln(3)}{0.07}$$

Question 6

Given $$P(t) = \frac{2t}{t-3}$$ describes the population (in hundreds) of zombies after $$t \u003e 3$$ years, what is the instantaneous rate of change of the population after 5 years?

Accepted Answer

$$P'(5) = 1.25$$

Question 7

Use the limit definition of the derivative to find $$f'(x) if$$ $$f(x) = x^2$ + 3x + 2. $$What is the equation of the tangent line to the graph of $$f at$$ $$x = 3$$?

Accepted Answer

Tangent line: $$y = 9x - 4$$

Question 8

Compute the derivative of $$y = 3x - 4x^3$.

Accepted Answer

$$y' = 3 - 12x^2$

Question 9

Compute the derivative of $$f(x) = \sqrt{x} + \frac{16x}{8x^2$ + 4}.$$

Accepted Answer

Question 10

Let $$f(x) = \frac{2x}{x^2$ + 9}. $$Compute $$f'(x). $$For what value(s) of $$x$$ does $$f'(x) = 0$$?

Accepted Answer

$$x = 0$$

Question 11

Given $$g(x) = \frac{(2-3x)(x^2$+5)}{x^3$+2}$$, which rule(s) must be used to compute $$g'(x)$$?

Accepted Answer

Product and quotient rules

Question 12

Suppose a company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 3x - 4x^3$, where x$ is the number of hundreds of units sold. What is the marginal profit when x = 2$?

Accepted Answer

$$P'(2) = -45$$

Question 13

A medication’s concentration in the bloodstream is modeled by $$C(t) = \frac{2t}{t-3}$$ for $$t \u003e 3$$ hours. What is the rate of change of concentration at $$t = 4$$ hours?

Accepted Answer

Question 14

If $$f(x) = 4^x$$, use the limit definition of the derivative to find $$f'(x). $$Which of the following is correct?

Accepted Answer

$$f'(x) = 4^x \ln(4)$$

Question 15

Evaluate $$\displaystyle \lim_{x 	o 2} \frac{3}{x-2}.$$

Accepted Answer

The limit does not exist (infinite discontinuity).

Question 16

A chemical reaction’s rate is modeled by $$R(t) = \frac{2t}{t-3}$$ for $$t \u003e 3$$ minutes. What is the instantaneous rate of change of the reaction at $$t = 6$$ minutes?

Accepted Answer

$$R'(6) = 0.67$$

Question 17

Suppose $$f(x) = x^2$ + 3x + 2. $$What is the slope of the tangent line to $$f at$$ $$x = 3$$?

Accepted Answer

$$9$$

Question 18

If $$f(x) = \frac{2x}{x^2$ + 9}$$, for which value(s) of $$x$$ does $$f'(x) = 0$$?

Accepted Answer

$$x = 0$$

Business Calculus Exam 1 Practice Guidance

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