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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