Question 1

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

Accepted Answer

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

Question 2

Given the definition $$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$, what does the existence of this limit guarantee about $$f at$$ $$x = a$$?

Accepted Answer

That $$f is $$differentiable at $$x = a$$ and therefore also continuous at $$x = a.$$

Question 3

Suppose the position of an object is given by $$s(t) = -4.9t^2$ + 19.6t + 24.5. $$What is the velocity of the object at time $$t$$?

Accepted Answer

$$v(t) = -9.8t + 19.6$$

Question 4

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

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

Question 5

A cost function is given by $$C(x). $$What does the marginal cost $$C'(x)$$ represent?

Accepted Answer

The approximate cost to produce one additional item after $$x$$ items.

Question 6

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 7

Given $$f(x) = e$$^{2x}$$, what is f$$''(x)$$?

Accepted Answer

$$4e^{2x}$$

Question 8

Suppose $$y = x \cos(x^2$). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\cos(x^2$) - 2x^2$ \sin(x^2$)$$

Question 9

If $$f(x) = \sin(x)$$, what is the fourth derivative $$f^{(4)}(x$$)$$?

Accepted Answer

$$\sin(x)$$

Question 10

A store manager estimates that the demand for an energy drink is $$d(p) = \frac{100}{p^2$ + 1}. $$What is the revenue function $$R(p)$$?

Accepted Answer

$$R(p) = \frac{100p}{p^2$ + 1}$$

Question 11

Which of the following is NOT a reason why a function might fail to be differentiable at a point $$a$$?

Accepted Answer

The function is increasing at $$a.$$

Question 12

If $$y = \arctan(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 13

Suppose $$y = x^2$ y^2$ - 2x = 3y - 4. $$Using implicit differentiation, which of the following best describes the process to find $$\frac{dy}{dx}$$?

Accepted Answer

Differentiate both sides with respect to $$x$$, applying the product and chain rules as needed.

Question 14

If $$f(x) = \ln|x|$$, what is $$f'(x)$$ for $$x 
eq 0$$?

Accepted Answer

$$\frac{1}{x}$$

Question 15

A ball is dropped from rest and its position is $$d(t) = 16t^2$. What is the velocity at t = 4$ seconds?

Accepted Answer

$$v(4) = 128$$

Question 16

Given $$f(x) = x^3$ - 2x^2$ + 5x - 7$$, what is $$f''(x)$$?

Accepted Answer

$$6x - 4$$

Question 17

If $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x \cos(x^2$)$$

Question 18

Suppose $$f(x) = x^2$ and g(x$$) = \sin(x)$$. What is $$\frac{d}{dx}[f(g(x))]$$?

Accepted Answer

$$2 \sin(x) \cos(x)$$

Question 19

If $$y = \arcsin(x)$$, what is $$\frac{dy}{dx}$$ for $$-1 \u003c x \u003c 1$$?

Accepted Answer

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

Comprehensive Study Notes: The Derivative and Its Applications

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