Question 1

Which of the following best describes the geometric meaning of the derivative $$f'(x_0$) at a $$point $$x_0$$ for a differentiable function $$f(x)$$?

Accepted Answer

It is the slope of the tangent line to the curve $$y = f(x) at$$ $$x = x_0$.

Question 2

Suppose the position of a car along a straight road is given by $$s(t) = 4t^2$ + 2t (in $$meters, $$t in $$seconds). What is the car's instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$v(3) = 26 m/s$$

Question 3

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 4

A manufacturer’s cost function is $$C(x) = 5x^2$ + 10x + 100. $$What is the marginal cost when producing 20 units?

Accepted Answer

$$C'(20) = 210$$

Question 5

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

Accepted Answer

$$f''(x) = 6x - 4$$

Question 6

Which of the following statements is TRUE?

Accepted Answer

If $$f is $$differentiable at $$x = c$$, then $$f is $$continuous at $$x = c.$$

Question 7

Let $$y = \sin(x) \cos(x). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 8

A ball is dropped from rest and its position is given by $$s(t) = 4.9t^2$ (in meters). What is its velocity and acceleration at t = 2$ seconds?

Accepted Answer

Velocity: $$19.6 m/s$$, Acceleration: $$9.8 m/s$^2$$

Question 9

Which of the following is the correct application of the Chain Rule for $$y = e$$^{\cos x}$$?

Accepted Answer

$$\frac{dy}{dx} = -e$$^{\cos x}$$ \sin x$$

Question 10

If $$f(x) = \sqrt{x^2$ + \pi x}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{2x + \pi}{2\sqrt{x^2$ + \pi x}}$$

Question 11

Which of the following is NOT a possible reason for a function to fail to have a derivative at a point?

Accepted Answer

The function is constant near the point.

Question 12

Let $$y = 	an(5 - \sin^2$ x). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$-2\cos x \sec^2$(5 - \sin^2$ x) \sin x$$

Question 13

A circle's area is $$A = \frac{\pi}{4} D^2$, where D$ is the diameter. What is the rate of change of area with respect to diameter when D = 8$ m?

Accepted Answer

$$2\pi m$^2/m$$

Question 14

If $$f(x) = x^2$ + 3x$$, what is the equation of the tangent line to $$f at$$ $$x = 2$$?

Accepted Answer

$$y = 7x - 4$$

Question 15

Which of the following is the correct derivative of $$f(x) = \frac{x}{x-1}$$ using the definition of the derivative?

Accepted Answer

$$f'(x) = \frac{-1}{(x-1)^2$}$$

Question 16

Let $$y = 5\cos t$$ describe the position of a weight in simple harmonic motion. What is the acceleration at time $$t$$?

Accepted Answer

$$a(t) = -5\cos t$$

Question 17

Which of the following is the correct derivative of $$f(x) = \sin(x^2$)$$?

Accepted Answer

$$f'(x) = 2x \cos(x^2$)$$

Question 18

If $$f(x) = 3x^2$ + 4x - 5$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x + 4$$

Chapter 3: Derivatives – Tangent Lines, Rates of Change, and Differentiation Rules

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