Q: A function $$f(x) is $$continuous at $$x = a if$$:

$$\lim\limits_{x \to a} f(x)$$ exists and equals $$f(a)$$

Question 1

A company’s cost to produce x units is given by $$C(x) = 2x^2$ + 5x + 100. $$What is the marginal cost when $$x = 10$$?

Accepted Answer

$$25$$

Question 2

Which of the following best describes the domain of the function $$f(x) = \dfrac{1}{x-3}$$?

Accepted Answer

All real numbers except $$x = 3$$

Question 3

Given the function $$f(x) = x^2$ - 4x + 3$$, what is the $$y-$$intercept of the graph?

Accepted Answer

$$(0, 3)$$

Question 4

A business park charges $$10$$ per day for the location of a day to rent a site, plus a fixed $$25$$ maintenance fee. Which function models the total cost $$C$$ for $$d$$ days?

Accepted Answer

$$C(d) = 10d + 25$$

Question 5

Evaluate the limit: $$\lim\limits_{x 	o 2} \dfrac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 6

Which of the following functions is even?

Accepted Answer

$$f(x) = x^2$

Question 7

Find the slope of the tangent line to the graph of $$f(x) = 3x^2$ - 2x + 1 at$$ $$x = 2.$$

Accepted Answer

$$8$$

Question 8

Which of the following is the equation of the line perpendicular to $$y = 2x + 5$$ and passing through $$(0, 3)$$?

Accepted Answer

$$y = -\dfrac{1}{2}x + 3$$

Question 9

If $$f(x) = \sqrt{x-1}$$, what is the domain of $$f$$?

Accepted Answer

$$x \geq 1$$

Question 10

Given $$f(x) = \dfrac{2x}{x^2$ - 1}$$, which of the following is a vertical asymptote?

Accepted Answer

$$x = 1$$

Question 11

Find the derivative of $$f(x) = 5x^3$ - 4x^2$ + 2x - 7.$$

Accepted Answer

$$f'(x) = 15x^2$ - 8x + 2$$

Question 12

A company’s revenue is given by $$R(x) = 100x - x^2$. What is the maximum revenue, and for what value of x$ does it occur?

Accepted Answer

Maximum revenue is $$2500 at$$ $$x = 50$$

Question 13

Evaluate $$\lim\limits_{x 	o 0} \dfrac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 14

Which of the following is the correct equation for the tangent line to $$f(x) = x^2$ at x = 1$?

Accepted Answer

$$y = 2x - 1$$

Question 15

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

Accepted Answer

$$\dfrac{2}{x}$$

Question 16

Which of the following best describes the end behavior of $$f(x) = -2x^3$ + 5x as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o -\infty$$

Question 17

Find the instantaneous rate of change of $$f(x) = x^2$ + 3x at$$ $$x = 2.$$

Accepted Answer

$$7$$

Question 18

If $$f(x) = \dfrac{1}{x}$$, which of the following statements is true?

Accepted Answer

$$f(x) is $$odd

Question 19

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 20

Given $$f(x) = x^2$ - 4$$, which of the following is a zero of the function?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Business Calculus Midterm Study Guide: Functions, Limits, Derivatives, and Applications

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