Q: A population of bacteria grows according to $P(t) = 100e^{0.05t}$. What is the instantaneous rate of change of the population at $t = 10$?

$5 \times 100e^{0.05 \times 10}$

Question 1

Given the quadratic function $f(x) = 2x^2 - 3x + 1$, what is the value of the difference quotient $\frac{f(x+h) - f(x)}{h}$?

Accepted Answer

$4x + 2h - 3$

Question 2

Use the limit definition of the derivative to find $f'(x)$ for $f(x) = x^2$. Which of the following is correct?

Accepted Answer

$2x$

Question 3

The position $s(t)$ of a car in meters is given by $s(t) = 5t^2 + 2t$. What does the derivative $s'(t)$ represent in this context?

Accepted Answer

The instantaneous velocity of the car at time $t$

Question 4

A population of bacteria grows according to $P(t) = 100e^{0.05t}$. What is the instantaneous rate of change of the population at $t = 10$?

Accepted Answer

$5 	imes 100e^{0.05 	imes 10}$

Question 5

Given $f(x) = 3x^4 - 2x^2 + 7$, what is $f''(x)$, the second derivative?

Accepted Answer

$12x^2 - 4$

Question 6

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

Accepted Answer

$\frac{6x}{3x^2 + 1}$

Question 7

A medication's concentration in the bloodstream is modeled by $C(t) = 8e^{-0.2t}$. What is the rate at which the concentration is changing at $t = 5$ hours?

Accepted Answer

$-1.6e^{-1}$

Question 8

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

Question 9

Let $f(x) = x^2$ and $g(x) = e^x$. Using the product rule, what is $\frac{d}{dx}[f(x)g(x)]$?

Accepted Answer

$x^2e^x + 2xe^x$

Question 10

Given $h(x) = \frac{2x^3 + 1}{x^2}$, what is $h'(x)$ using the quotient rule?

Accepted Answer

$\frac{6x^2x^2 - (2x^3 + 1)2x}{x^4}$

Question 11

If $f(x) = 5x^2 + 3x$ and $g(x) = 2x - 1$, what is $\frac{d}{dx}[f(x) + g(x)]$?

Accepted Answer

$10x + 3 + 2$

Question 12

Let $f(x) = e^{2x}$. What is $f'(x)$?

Accepted Answer

$2e^{2x}$

Question 13

Given $F(x) = 7$, what is $F'(x)$?

Accepted Answer

$0$

Question 14

Suppose $f(x) = (3x^2 + 2x)^5$. Using the general power rule, what is $f'(x)$?

Accepted Answer

$5(3x^2 + 2x)^4 \cdot (6x + 2)$

Question 15

A function $f(x, y) = x^2y + 3y^2$. What is the partial derivative $\frac{\partial f}{\partial x}$?

Accepted Answer

$2xy$

Question 16

Given $f(x, y) = x^2y + 3y^2$, what is $\frac{\partial f}{\partial y}$ evaluated at $(1, 2)$?

Accepted Answer

$13$

Question 17

A chemical reaction's rate is modeled by $R(t) = 4t^3 - 2t + 5$. What is the instantaneous rate of change of the reaction at $t = 2$?

Accepted Answer

$92$

Question 18

If $f(x) = 2x^3 - 5x^2 + 4x - 7$, what is the slope of the tangent line at $x = 1$?

Accepted Answer

$-3$

Question 19

Let $f(x) = \ln(x^2 + 1)$. What is $f''(x)$, the second derivative?

Accepted Answer

$\frac{2 - 2x^2}{(x^2 + 1)^2}$

Question 20

Suppose the temperature $T$ at a point $(x, y)$ on a metal plate is given by $T(x, y) = 2x^2y + 3y^3$. What is the rate of change of temperature with respect to $y$ at the point $(1, 1)$?

Accepted Answer

$8$

Calculus I: Derivatives and Their Applications

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