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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