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

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

Question 1

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

Accepted Answer

It is the slope of the tangent line to the graph of $$f(x) at$$ $$x = a.$$

Question 2

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

Accepted Answer

$$y = -2x + 3$$

Question 3

For the function $$f(x) = 4/x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -4/x^2$

Question 4

Suppose the cost to manufacture $$x$$ calculators is $$C(x) = -0.005x^2$ + 20x + 150. $$What is the rate of change of cost when 100 calculators are made?

Accepted Answer

$$C'(100) = 19$$

Question 5

Which of the following is NOT a condition for the existence of the derivative at a point?

Accepted Answer

The function has a vertical tangent at the point.

Question 6

Given $$f(x) = 2x^2$ + 4x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 4x + 4$$

Question 7

Refer to the first image (chromosomal abnormalities vs. age). What does the slope of the tangent line at age 40 represent?

Accepted Answer

The rate at which the risk of chromosomal abnormalities is increasing at age 40.

Question 8

If $$f(x) is $$not differentiable at $$x = a$$, which of the following could be true?

Accepted Answer

The function has a sharp corner at $$x = a.$$

Question 9

Given $$f(x) = x^2$, what is f$$'(3)$$ and what does it represent?

Accepted Answer

$$f'(3) = 6$$, which is the slope of the tangent line at $$x = 3.$$

Question 10

Using the table in the second image, at which altitude $$h is $$the temperature $$T(h)$$ increasing most rapidly?

Accepted Answer

$$h = 2000$$

Question 11

Given $$f(x) = 4/x$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = -x + 4$$

Question 12

Which of the following statements about the graph of a derivative is TRUE?

Accepted Answer

The graph of the derivative shows the slope of the original function at each point.

Question 13

In Figure 54 (third image), which curve is most likely the derivative of the other?

Accepted Answer

$$Y_2 is $$the derivative of $$Y_1.$$

Question 14

If a function $$f(x)$$ has a vertical tangent at $$x = a$$, what can you conclude about $$f'(a)$$?

Accepted Answer

$$f'(a)$$ does not exist.

Question 15

A business's total production $$TP as a $$function of labor $$L is $$shown in a graph. The marginal production $$MP is $$the derivative of $$TP. $$What does a negative value of $$MP$$ indicate?

Accepted Answer

Total production is decreasing.

Question 16

If $$f(x) is $$continuous but not differentiable at $$x = a$$, which of the following is a possible reason?

Accepted Answer

There is a sharp corner at $$x = a.$$

Question 17

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

Accepted Answer

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

Question 18

If the derivative of a function is positive over an interval, what can you conclude about the function on that interval?

Accepted Answer

The function is increasing.

Question 19

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

Accepted Answer

$$f(x) is $$continuous at $$x = 2.$$

Definition and Graphical Interpretation of the Derivative in Business Calculus

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