$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Question 1

Given the function $f(x) = x^2$, what is the slope of the tangent line to the curve at $x = 2$?

Accepted Answer

$4$

Question 2

Which of the following is the correct definition of the derivative of a function $f(x)$ at a point $x=a$?

Accepted Answer

$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 3

For the function $f(x) = |x|$, which statement is true about its derivative at $x=0$?

Accepted Answer

The derivative does not exist at $x=0$ because the left and right derivatives are not equal.

Question 4

If $f(x) = 3x^2 + 2x + 1$, what is $f'(x)$?

Accepted Answer

$6x + 2$

Question 5

Find the equation of the tangent line to $f(x) = 3x^2 + 2x + 1$ at $x = -2$.

Accepted Answer

$y = -10x + 15$

Question 6

Which notation does NOT represent the derivative of $f(x)$?

Accepted Answer

$f(x)'$

Question 7

Suppose $f(x)$ is continuous at $x=a$ and $\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$ exists. What does this imply?

Accepted Answer

The function $f(x)$ is differentiable at $x=a$.

Question 8

A car's position at time $t$ is given by $s(t) = t^2 + 2t$. What is the instantaneous velocity at $t = 3$ seconds?

Accepted Answer

$10$ m/s

Question 9

Given $f(x) = 2x - 1$, use the definition of the derivative to find $f'(x)$.

Accepted Answer

$2$

Question 10

If $\lim_{x 	o a^-} f(x) = L_1$ and $\lim_{x 	o a^+} f(x) = L_2$ with $L_1 
eq L_2$, what can be said about $\lim_{x 	o a} f(x)$?

Accepted Answer

The limit does not exist at $x=a$.

Question 11

Which of the following is true about the left and right derivatives at a point $x=a$?

Accepted Answer

The derivative exists at $x=a$ only if both left and right derivatives exist and are equal.

Question 12

For $f(x) = |x|$, what is the value of the left-hand derivative at $x=0$?

Accepted Answer

$-1$

Question 13

If $f(x) = x^3$, what is the slope of the tangent line at $x = 1$?

Accepted Answer

$3$

Question 14

A function $f(x)$ is said to be differentiable at $x=a$ if:

Accepted Answer

Both the left and right derivatives exist and are equal at $x=a$.

Question 15

Given $f(x) = 3x^2 + 2x + 1$, what is $f'(-2)$?

Accepted Answer

$-10$

Question 16

Which of the following best describes the geometric meaning of the derivative at a point?

Accepted Answer

It is the slope of the tangent line to the curve at that point.

Question 17

If $f(x)$ is not continuous at $x=a$, which statement is true about $f'(a)$?

Accepted Answer

$f'(a)$ does not exist.

Question 18

Suppose $f(x)$ models the height of a plant in centimeters over time $x$ in days, and $f'(10) = 2$. What does this mean in practical terms?

Accepted Answer

On day 10, the plant is growing at a rate of $2$ cm per day.

Question 19

If $f(x) = x^2$, use the definition of the derivative to compute $f'(x)$.

Accepted Answer

$2x$

Introduction to Derivatives and Limits in Calculus

Study Guide - Practice Questions