Q: Which of the following is an alternative definition for the derivative of $f(x)$ at $x = a$?

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

Q: Which of the following expressions represents the derivative of $f(x)$ at $x = a$ using the difference quotient?

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

Q: Which of the following is the correct limit definition for the slope of the tangent line at $x = a$?

$m_{tan} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Question 1

Which of the following best describes the difference between the slope of a secant line and the slope of a tangent line to the curve $y = f(x)$ at $x = a$?

Accepted Answer

The secant line slope is given by $m_{sec} = \frac{f(x) - f(a)}{x - a}$, while the tangent line slope is $m_{tan} = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$.

Question 2

Given $f(x) = 3x + 4$, what is the slope of the tangent line to the graph at the point $P(2, 10)$?

Accepted Answer

$3$

Question 3

For $f(x) = x^2 + 6x$, find the equation of the tangent line at the point $P(1, 7)$.

Accepted Answer

$y = 8x - 1$

Question 4

Which of the following is an alternative definition for the derivative of $f(x)$ at $x = a$?

Accepted Answer

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

Question 5

Find the slope of the tangent line to the graph of $f(x) = -2x + 7$ at the point $(2, 3)$ using the limit definition.

Accepted Answer

$-2$

Question 6

A chemical reaction's rate at time $t$ is modeled by $f(t) = t^2 + 4t$. What is the equation of the tangent line to the rate curve at $t = 1$?

Accepted Answer

$y = 6t - 1$

Question 7

Given $f(x) = 2x^2 - 3x$, compute $f'(3)$ using the limit definition of the derivative.

Accepted Answer

$9$

Question 8

Which of the following expressions represents the derivative of $f(x)$ at $x = a$ using the difference quotient?

Accepted Answer

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

Question 9

A car's position is given by $s(t) = t^2 + 6t$. What is the instantaneous velocity at $t = 1$?

Accepted Answer

$8$

Question 10

Suppose $f(x) = x^2 + 4x$. Using the limit definition, what is the slope of the tangent line at $x = 1$?

Accepted Answer

$6$

Question 11

Which of the following is the correct equation for the tangent line to $f(x) = -2x + 7$ at $(2, 3)$?

Accepted Answer

$y = -2x + 7$

Question 12

A biologist models the growth of bacteria with $f(t) = 3t + 4$. What is the instantaneous rate of growth at $t = 2$?

Accepted Answer

$3$

Question 13

Which of the following is NOT a correct way to write the derivative of $f(x)$ at $x = a$?

Accepted Answer

$f'(a) = \frac{f(a + h) - f(a)}{h}$$

Question 14

If $f(x) = x^2 + bx$ and $f(1) = 7$, what is the value of $b$?

Accepted Answer

$6$

Question 15

A medical researcher models the concentration of a drug in the bloodstream as $C(t) = 2t^2 - 3t$. What is the rate of change of concentration at $t = 3$?

Accepted Answer

$9$

Question 16

Which of the following best explains why the tangent line at a point on a curve represents the instantaneous rate of change?

Accepted Answer

Because the tangent line's slope is the limit of the secant line's slope as the two points become infinitely close.

Question 17

Given $f(x) = x^2 + 6x$, what is the value of $f'(1)$ using the limit definition?

Accepted Answer

$8$

Question 18

A physicist measures the position of a particle as $s(t) = 2t^2 - 3t$. What is the instantaneous velocity at $t = 3$?

Accepted Answer

$9$

Question 19

Which of the following is the correct limit definition for the slope of the tangent line at $x = a$?

Accepted Answer

$m_{tan} = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$

Introducing the Derivative 3.1

Study Guide - Practice Questions

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