Question 1

Which of the following best describes the meaning of the limit $$\lim\limits_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L.$$

Question 2

Given the function $$f(x) = \frac{1}{x-2}$$, what is $$\lim\limits_{x 	o 2^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 3

A car travels 350 miles on 20 gallons of gas. What is the average rate of change of miles with respect to gallons?

Accepted Answer

$$17.5$$ miles per gallon

Question 4

Which of the following is NOT a requirement for a function $$f to be $$continuous at $$x=a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x=a$$

Question 5

Let $$f(x) = x^2$. What is the difference quotient $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$2x + h$$

Question 6

If $$y = 5x^4$ - 3x^2$ + 7$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$20x^3 - 6x$

Question 7

A pebble is dropped from a hot-air balloon. Its position is $$s(t) = 16t^2$ (in feet, t$ in seconds). What is its velocity after 3.5$ seconds?

Accepted Answer

$$112 ft/sec$$

Question 8

Which of the following is the correct application of the Product Rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$$

Question 9

Let $$y = (3x^2$ + 2x)^5$. Using the Chain Rule, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$5(3x^2$$ + $$2x)^4$$ \cdot (6x + 2)$$

Question 10

For the function $$f(x) = |x|$$, at which point is $$f$$ NOT differentiable?

Accepted Answer

$$x = 0$$

Question 11

Suppose $$f(x) = x^3$ - 6x^2$. At which points does the tangent line to the graph of f$ become horizontal?

Accepted Answer

$$x = 0$$ and $$x = 4$$

Question 12

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

Accepted Answer

$$2$$

Question 13

If $$y = \frac{2x^2$ - 3x + 1}{x}$$, what is $$\frac{dy}{dx}$$ using the Quotient Rule?

Accepted Answer

$$\frac{2x^2$ - 1}{x^2$}$$

Question 14

Which of the following is the correct expression for the average rate of change of $$f(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 15

If $$f(x) = x^4$ - 2x^2$ + 3$$, what is $$f'''(x) ($$the third derivative)?

Accepted Answer

$$24x$$

Question 16

A business finds its cost function is $$C(x) = 200 + 100x^4$. What is the average cost function $$\overline{C}(x)$$?

Accepted Answer

$$\overline{C}(x) = \frac{200}{x} + 100x^3$

Question 17

Which of the following is a correct application of the Chain Rule for $$y = (5x^2$ + 3)^7$?

Accepted Answer

$$7(5x^2$$ + $$3)^6$$ \cdot 10x$$

Question 18

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

Accepted Answer

There is a corner or cusp at $$x=a.$$

Calculus and Its Applications: Chapter 1 – Differentiation (Comprehensive Study Notes)

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