$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

A horizontal asymptote occurs at $$y = L if$$ $$\lim_{x \to \infty} f(x) = L.$$

Question 1

Which of the following best describes the meaning of the derivative $$f'(x) at a $$point $$x=a$$?

Accepted Answer

It is the instantaneous rate of change of $$f(x) at$$ $$x=a$$, which is the slope of the tangent line at that point.

Question 2

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

Accepted Answer

$$y = 5x - 3$$

Question 3

A company’s total cost function is $$C(x) = 25x + 1500$$, where $$x is $$the number of units produced. What is the marginal cost?

Accepted Answer

$$25$$

Question 4

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

Accepted Answer

$$2.5$$

Question 5

Which of the following statements about continuity at $$x=c is $$correct?

Accepted Answer

$$f(x) is $$continuous at $$x=c if$$ $$f(c) is $$defined, $$\lim_{x 	o c} f(x)$$ exists, and $$\lim_{x 	o c} f(x) = f(c).$$

Question 6

Find $$\lim_{x 	o 0} \frac{1}{x-1}$$ using a table of values approaching $$x=1$$ from both sides.

Accepted Answer

The limit does not exist because the function approaches infinity from both sides.

Question 7

Given $$f(x) = \frac{1 + x^2$}{x + 4}$$ for $$x  -1$$, and $$f(-1) = 2$$, what is $$\lim_{x 	o -1^-} f(x)$$?

Accepted Answer

$$\frac{2}{3}$$

Question 8

Which of the following is the correct definition of the derivative using limits?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 9

If $$f(x) = \ln(x^2$ + 5)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 5}$$

Question 10

A company’s revenue function is $$R(x) = 30x$$, and its cost function is $$C(x) = 25x + 1500. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 30x - (25x + 1500)$$

Question 11

Which of the following is true about horizontal asymptotes for rational functions?

Accepted Answer

A horizontal asymptote occurs at $$y = L if$$ $$\lim_{x 	o \infty} f(x) = L.$$

Question 12

Evaluate $$\lim_{x 	o 5} \frac{\sqrt{x^2$ - 9} - 2}{x - 5}.$$

Accepted Answer

$$\frac{1}{3}$$

Question 13

Which of the following is the correct power rule for derivatives?

Accepted Answer

$$\frac{d}{dx} x^n = n x$$^{n-1}$$

Question 14

For the function $$f(x) = \frac{3x^2$ - 1}{x}$$, what is the algebraic form before taking the derivative?

Accepted Answer

$$3x - \frac{1}{x}$$

Question 15

If $$f(x)$$ has a vertical asymptote at $$x=c$$, what does this mean for $$\lim_{x 	o c} f(x)$$?

Accepted Answer

The limit does not exist because $$f(x)$$ approaches infinity or negative infinity as $$x$$ approaches $$c.$$

Question 16

Given $$f(x) = 3x^2$ - x + 1$$, at which $$x-$$values does the function have horizontal tangent lines?

Accepted Answer

$$x = \frac{1}{6}$$ and $$x = -\frac{1}{6}$$

Question 17

Which of the following is the correct interpretation of the secant line between $$x_1$$ and $$x_2 on a $$graph of $$f(x)$$?

Accepted Answer

It represents the average rate of change of $$f(x)$$ between $$x_1$$ and $$x_2.$$

Question 18

If $$f(x) = 3x^2$ - x + 1$$, what is $$f'(x)$$?

Accepted Answer

$$6x - 1$$

Question 19

Suppose the cost per unit to produce $$x$$ items is $$C(x) = 34x$$^{-1}$$. What is the marginal cost at x=2$?

Accepted Answer

$$-17$$

Limits, Continuity, and Derivatives: Study Notes for Applied Calculus (MAC 2233)

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