$f'(x_0) = \lim_{h \to 0} \dfrac{f(x_0 + h) - f(x_0)}{h}$

$\lim_{h \to 0^+} \dfrac{f(a + h) - f(a)}{h}$ and $\lim_{h \to 0^-} \dfrac{f(b + h) - f(b)}{h}$

Question 1

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

Accepted Answer

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

Question 2

Given the definition of the derivative at a point, which of the following is the correct formula for $f'(x_0)$?

Accepted Answer

$f'(x_0) = \lim_{h 	o 0} \dfrac{f(x_0 + h) - f(x_0)}{h}$

Question 3

Suppose $y = \dfrac{1}{x}$ for $x > 0$. What is the slope of the tangent line at $x = 2$?

Accepted Answer

$-\dfrac{1}{4}$

Question 4

For the function $f(x) = \sqrt{x}$, which of the following is the correct expression for its derivative for $x > 0$?

Accepted Answer

$f'(x) = \dfrac{1}{2\sqrt{x}}$

Question 5

Which of the following statements is TRUE about differentiability and continuity at a point $x = c$?

Accepted Answer

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

Question 6

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

Accepted Answer

The function is linear at the point.

Question 7

Consider the function $f(x) = x^n$ where $n$ is a positive integer. What is the derivative $f'(x)$?

Accepted Answer

$f'(x) = n x^{n-1}$

Question 8

If $f(x)$ is a constant function, what is its derivative with respect to $x$?

Accepted Answer

$0$

Question 9

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

Accepted Answer

$3x^2 - 4x + 4$

Question 10

Which of the following is the correct application of the sum rule for derivatives?

Accepted Answer

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

Question 11

A function $f(x)$ is differentiable on the open interval $(a, b)$ and at the endpoints $a$ and $b$ if which of the following limits exist?

Accepted Answer

$\lim_{h 	o 0^+} \dfrac{f(a + h) - f(a)}{h}$ and $\lim_{h 	o 0^-} \dfrac{f(b + h) - f(b)}{h}$

Question 12

For the function $f(x) = \sqrt{x}$, why does the derivative not exist at $x = 0$?

Accepted Answer

Because the right-hand limit of $\dfrac{1}{2\sqrt{x}}$ as $x 	o 0^+$ is infinite.

Question 13

Which of the following functions has a point where the derivative does NOT exist?

Accepted Answer

$f(x) = |x|$

Question 14

A function $y = f(x)$ has a cusp at $x = c$. What does this mean for the derivative at $x = c$?

Accepted Answer

The derivative does not exist at $x = c$ because the slope approaches $+\infty$ from one side and $-\infty$ from the other.

Question 15

Which of the following is an example of a function whose derivative fails to exist due to rapid oscillation near a point?

Accepted Answer

$f(x) = \sin\left(\dfrac{1}{x}\right)$ at $x = 0$

Question 16

If $f(x) = 3x^2$, what is the effect on the derivative if we multiply the function by a constant $c$?

Accepted Answer

The derivative is multiplied by $c$; $\dfrac{d}{dx}[c f(x)] = c f'(x)$.

Question 17

Which of the following is the correct difference quotient for the secant slope between points $x$ and $z$ on the graph of $y = f(x)$?

Accepted Answer

$\dfrac{f(z) - f(x)}{z - x}$

Question 18

Which of the following is NOT an interpretation of the limit $\lim_{h 	o 0} \dfrac{f(x_0 + h) - f(x_0)}{h}$?

Accepted Answer

The area under the curve $y = f(x)$ from $x_0$ to $x_0 + h$.

Derivatives: Tangents, Rates of Change, and Basic Rules

Study Guide - Practice Questions