Question 1

A cylindrical water tank has a height of 17 m. The radius of the tank is increasing at a constant rate of 12 cm/hr. The surface area $A$ of the cylinder is given by $A = 2\pi r^2 + 2\pi r h$. What is the rate at which the surface area is increasing when the radius is 8 cm?

Accepted Answer

$416\pi$ cm$^2$/hr

Question 2

The volume $V$ of a cone is given by $V = \frac{1}{3}\pi r^2 h$. Suppose a cone of height 4 in has a radius that is increasing at a constant rate of 3 in/hr. Find the rate at which the volume is increasing when the radius is 6 in.

Accepted Answer

$48\pi$ in$^3$/hr

Question 3

Calculate the derivative: $\frac{d}{dx}\left(\frac{5x^2 - 1}{3x^2 + 15}\right)$

Accepted Answer

$\frac{30x(3x^2 + 15) - 6x(5x^2 - 1)}{(3x^2 + 15)^2}$

Question 4

Given a particle's position $s(t) = -11t^2 + 15t - 58$, what is its velocity at $t = 9$ seconds?

Accepted Answer

$-183$ m/s

Question 5

Given $y(x) = 12x^2 e^{2x}$, find $\frac{dy}{dx}$.

Accepted Answer

$24x e^{2x} + 24x^2 e^{2x}$

Question 6

Use the limit definition of the derivative to calculate the slope of the tangent to $g(x) = \frac{10}{7}$ at $x = -1$.

Accepted Answer

$0$

Question 7

Find $\frac{d}{dx}\left(4x^2 \arctan(x)\right)$.

Accepted Answer

$8x \arctan(x) + \frac{4x^2}{1 + x^2}$

Question 8

Given $h(x) = \sqrt{x}$, use the limit definition of the derivative to find $h'(64)$.

Accepted Answer

$\frac{1}{16}$

Question 9

Find $\frac{d}{dx}\left(19x^7 \ln(x)\right)$.

Accepted Answer

$19x^7 \cdot \frac{1}{x} + 19 \ln(x) \cdot 7x^6$

Question 10

A particle moves according to $s(t) = -7 \sin(4t)$. What is its acceleration at $t = \frac{5\pi}{16}$ seconds?

Accepted Answer

$-112 \cos(4t)$

Question 11

Use implicit differentiation to find the slope of the tangent line to $-y^3 + 5xy + 10x^2 - 1625 = 0$ at the point $(5, -5)$.

Accepted Answer

$1$

Question 12

Find $\frac{d}{dx}\left(\arccos(\sqrt{3x^2})\right)$.

Accepted Answer

$-\frac{3x}{\sqrt{1 - 3x^2}}$

Question 13

Find $\frac{d}{dx}\left(\frac{x}{\sqrt{x^7}}\right)$.

Accepted Answer

$\frac{1 - 3.5}{x^{3.5}}$

Question 14

Find $\frac{d}{dx}\left(	an^{-1}(\ln(x))ight)$.

Accepted Answer

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

Question 15

Find $\frac{d}{dx}\left(\sin(x^2 + 6)\right)$.

Accepted Answer

$2x \cos(x^2 + 6)$

Question 16

Find $\frac{d}{dx}\left(-5 \cos x + 15 	an xight)$.

Accepted Answer

$5 \sin x + 15 \sec^2 x$

Question 17

Find $\frac{d}{dx}\left(11 \sqrt{x^2} - 18 \arctan x\right)$.

Accepted Answer

$\frac{11x}{|x|} - \frac{18}{1 + x^2}$

Question 18

A medication is administered to a patient and the concentration in the bloodstream is modeled by $C(t) = 15e^{-0.2t}$ mg/L, where $t$ is in hours. What is the rate of change of the concentration at $t = 3$ hours?

Accepted Answer

$-3e^{-0.6}$ mg/L/hr

Question 19

A bacteria population grows according to $P(t) = 1000e^{0.05t}$. What is the instantaneous rate of growth at $t = 10$ hours?

Accepted Answer

$50e^{0.5}$

Calculus I: Derivatives, Related Rates, and Applications Study Guide

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