Q: Which of the following is the correct limit definition of the derivative of a function $$f(x) at$$ $$x = a$$?

$$\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: Suppose $$f(x) is $$continuous at $$x = 2. $$Which of the following must be true?

$$\lim\limits_{x \to 2} f(x) = f(2)$$

Q: If $$f(x) is $$not continuous at $$x = 2$$, which of the following statements is necessarily true?

$$\lim\limits_{x \to 2} f(x) \neq f(2)$$

Question 1

Which of the following is the correct limit definition of the derivative of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 2

Given $$f(x) = x^2$ - 3x + 5$$, use the limit definition of the derivative to find $$f'(x). $$Which of the following is correct?

Accepted Answer

$$2x - 3$$

Question 3

A company’s cost to produce $$x$$ thousand calculators is $$C(x) = x^3$ - 3x^2$ + 28 (in $$thousands of dollars). What is the marginal cost when $$x = 4$$?

Accepted Answer

$$24$$ thousand dollars per thousand calculators

Question 4

Find the average rate of change of $$f(x) = x^2$ - 3x + 5$$ from $$x = 1 to$$ $$x = 2.$$

Accepted Answer

$$-1$$

Question 5

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

Accepted Answer

$$3$$

Question 6

Given the function $$f(x) = \frac{2x+3}{x^2$}$$, for which value of $$x is$$ $$f(x)$$ not defined?

Accepted Answer

$$x = 0$$

Question 7

Suppose $$f(x) is $$continuous at $$x = 2. $$Which of the following must be true?

Accepted Answer

$$\lim\limits_{x 	o 2} f(x) = f(2)$$

Question 8

If $$f(x) = 3x^2$ - 12x + 8$$, at which $$x-$$value(s) is the tangent line to the curve horizontal?

Accepted Answer

$$x = 2$$

Question 9

Given $$f(x) = x^2$ - 3x + 5$$, what is the equation of the tangent line at $$x = 1$$?

Accepted Answer

$$y = -1x + 3$$

Question 10

A company’s total cost to produce $$x$$ thousand calculators is $$C(x) = x^3$ - 3x^2$ + 28. $$What is the average cost per thousand calculators when $$x = 4$$?

Accepted Answer

$$3.25$$ thousand dollars

Question 11

Given $$f(x) = x^2$ - 3x + 5$$, what is $$f''(x)$$, the second derivative?

Accepted Answer

$$2$$

Question 12

If $$f(x) is $$not continuous at $$x = 2$$, which of the following statements is necessarily true?

Accepted Answer

$$\lim\limits_{x 	o 2} f(x) 
eq f(2)$$

Question 13

Given $$f(x) = x^2$ - 3x + 5$$, what is the instantaneous rate of change at $$x = 3$$?

Accepted Answer

$$3$$

Question 14

For the function $$f(x) = \sqrt{x-4}$$, what is the domain?

Accepted Answer

$$x \geq 4$$

Question 15

If $$f(x) = x^2$ - 3x + 5$$, what is $$f(0)$$?

Accepted Answer

$$5$$

Question 16

Given $$f(x) = x^2$ - 3x + 5$$, what is the $$x-$$intercept of the function?

Accepted Answer

$$\frac{3 \pm \sqrt{9 - 20}}{2}$$

Question 17

If $$f(x) = x^2$ - 3x + 5$$, what is the minimum value of $$f(x)$$?

Accepted Answer

$$\frac{7}{4}$$

Question 18

Given $$f(x) = x^2$ - 3x + 5$$, what is the vertex of the parabola?

Accepted Answer

$$(\frac{3}{2}, \frac{7}{4})$$

Question 19

If $$f(x) = x^2$ - 3x + 5$$, what is the slope of the tangent line at $$x = 0$$?

Accepted Answer

$$-3$$

Question 20

A company’s cost to produce $$x$$ thousand calculators is $$C(x) = x^3$ - 3x^2$ + 28. At $$what production level is the total cost increasing at a rate of $$9$$ thousand dollars per one thousand calculators?

Accepted Answer

$$x = 3$$

Business Calculus: Limits, Derivatives, and Applications Study Guide

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