Q: Which of the following is the correct limit definition of the derivative of $f$ at $x=a$?

$f'(a) = \lim\limits_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$

Question 1

Given the function $f(x) = \sqrt{x-5}$, what is the derivative $f'(x)$?

Accepted Answer

$f'(x) = \dfrac{1}{2\sqrt{x-5}}$

Question 2

The average rate of change of $f(x)$ from $x=a$ to $x=b$ is given by which formula?

Accepted Answer

$\dfrac{f(b)-f(a)}{b-a}$

Question 3

Suppose $f(x) = 4 - x^2$. What is the slope of the tangent line to $f$ at the point $(0,4)$?

Question 4

Given $f(x) = \sqrt{5x+1}$, what is the slope of the tangent line at the point $(3,4)$?

Accepted Answer

$\dfrac{5}{8}$

Question 5

Which of the following statements is TRUE about differentiability and continuity?

Accepted Answer

If a function is differentiable at $x=a$, then it is continuous at $x=a$.

Question 6

For the function $f(x) = |x-4|$, is $f$ differentiable at $x=4$?

Accepted Answer

No, because the left and right derivatives at $x=4$ are not equal.

Question 7

Which of the following is NOT a reason for a function to be non-differentiable at a point $a$?

Accepted Answer

The function is linear at $a$.

Question 8

Given $f(x) = \dfrac{1}{3x-1}$, what is $f'(x)$?

Accepted Answer

$f'(x) = -\dfrac{3}{(3x-1)^2}$

Question 9

A function $f$ is continuous at $x=a$ but not differentiable at $x=a$. Which of the following could be true?

Accepted Answer

The graph of $f$ has a sharp corner at $x=a$.

Question 10

If $f(x)$ is differentiable at $x=a$, which of the following must be true?

Accepted Answer

$f(x)$ is continuous at $x=a$.

Question 11

Which of the following best describes the instantaneous rate of change of $f$ at $x=a$?

Accepted Answer

The slope of the tangent line to $f$ at $x=a$.

Question 12

Given $f(x) = x^2$, what is the equation of the tangent line at $x=1$?

Question 13

A function $f$ is not differentiable at $x=a$ if:

Accepted Answer

The left-hand and right-hand derivatives at $x=a$ are not equal.

Question 14

Suppose the position $s(t)$ of a car is given by $s(t) = 3t^2 + 2t$. What is the instantaneous velocity at $t=2$?

Accepted Answer

$v(2) = 14$

Question 15

If $f(x)$ has a vertical tangent at $x=a$, what can you conclude about $f'(a)$?

Accepted Answer

$f'(a)$ does not exist.

Question 16

Which of the following is the correct limit definition of the derivative of $f$ at $x=a$?

Accepted Answer

$f'(a) = \lim\limits_{h 	o 0} \dfrac{f(a+h) - f(a)}{h}$

Question 17

A medical researcher models the temperature $T$ (in $^\circ$C) of a patient as $T(t) = 37 + 0.5t - 0.02t^2$, where $t$ is hours after medication. What is the rate of change of temperature at $t=5$ hours?

Accepted Answer

$0.3$

Question 18

If $f(x)$ is not continuous at $x=a$, which of the following must be true?

Accepted Answer

$f(x)$ is not differentiable at $x=a$.

Question 19

A chemical reaction's concentration is modeled by $C(t) = 10e^{-0.1t}$. What is the instantaneous rate of change of concentration at $t=0$?

Accepted Answer

$-1$

Rates of Change, Tangent Lines, and Differentiability in Calculus

Study Guide - Practice Questions

Study Guide - Flashcards