Deck 3: Applications of the Derivative

Question 1

Name the horizontal and vertical asymptotes of the function $f(x)=\frac{3 x}{x^{2}-16}$ .

A) Horizontal: $\mathrm{y}=3$ ; vertical: $\mathrm{x}=4$ and $\mathrm{x}=-4$ .
B) Horizontal: $\mathrm{y}=0$ ; vertical: $\mathrm{x}=4$ and $\mathrm{x}=-4$ .
C) Horizontal: $\mathrm{y}=0$ ; vertical: $\mathrm{x}=4$ .
D) Horizontal: $\mathrm{y}=3$ ; vertical: $\mathrm{x}=4$ .

Answer 1

Horizontal: $\mathrm{y}=0$ ; vertical: $\mathrm{x}=4$ and $\mathrm{x}=-4$ .

Question 2

Find the relative maximum and minimum points of the function $f(x)=x^{3}-x^{2}+15 x+8$ .

A) Relative maximum at $(1,15)$ ; relative minimum at $(5,-17)$ .
B) Relative maximum at (7, 15); relative minimum at ( $-1,-17$ ).
C) Relative maximum at $(8,64)$ ; relative minimum at $(-2,-66)$ .
D) Relative maximum at $(5,15)$ ; relative minimum at $(1,-17)$ .

Answer 2

Relative maximum at $(1,15)$ ; relative minimum at $(5,-17)$ .

Question 3

Find the inflection point(s) of the function $f(x)=x^{3}-9 x^{2}+15 x+8$ .

A) $(0,8)$
B) $(-3,-145)$
C) $(3,-1)$ and $(-3,-145)$
D) $(3,-1)$

Answer 3

$(3,-1)$

Question 4

Use Newton's Method to improve the given estimated solution to at least six significant digits.

- $\mathrm{x}^{3}$ - $12=0 ; \mathrm{x}=2.4$

A) 2.28944
B) 2.28942
C) 2.34043
D) 2.28943

Question 5

Use Newton's Method to improve the given estimated solution to at least six significant digits.

- $x^{3}-3 x+4=0 ; x=-2$

A) -2.19584
B) -2.19582
C) -2.19583
D) -2.19622

Question 6

Use Newton's Method to improve the given estimated solution to at least six significant digits.

- $x+\cos x=0 ; x=-0.5$

A) -0.739142
B) -0.739141
C) -0.739084
D) -0.739085

Question 7

A rectangular, $450 \mathrm{ft}^{2}$ family room addition is to be built using the existing house for one of its walls. Find a function $\mathrm{f}(\mathrm{x})$ that would represent the total length of the other 3 walls that need to be built, where $x$ represents the length of the wall that will be parallel to the existing wall.

A) $f(x)=2 x+\frac{900}{x}$
B) $f(x)=x+\frac{450}{x}$
C) $f(x)=x+\frac{900}{x}$
D) $f(x)=3 x-\frac{450}{x}$

Question 8

A brush fire is burning a circular area whose radius is increasing at a rate of 3 feet per minute. How fast is the area of the fire increasing when its radius is 25 feet?

A) $225 \pi \mathrm{ft} 2 / \mathrm{min}$
B) $625 \pi \mathrm{ft} 2 / \mathrm{min}$
C) $75 \pi \mathrm{ft}^{2} / \mathrm{min}$
D) $150 \pi \mathrm{ft} 2 / \mathrm{min}$

Question 9

The area of a square in terms of its diagonal $z$ is given by the formula $A=\frac{1}{2} z^{2}$ . Use a differential to estimate the change in the area of a square whose diagonal has been increased from $6.0 \mathrm{~cm}$ to 6.2 $\mathrm{cm}$ .

A) $\mathrm{dA}=1.2 \mathrm{~cm}^{2}$
B) $\mathrm{dA}=1.8 \mathrm{~cm}^{2}$
C) $\mathrm{dA}=0.4 \mathrm{~cm}^{2}$
D) $\mathrm{dA}=0.2 \mathrm{~cm}^{2}$

Question 10

In the problems below, determine the intercepts, the intervals in which the curve is above or touching the $\mathrm{x}$ - axis, symmetry, and asymptotes. Graph each equation.

- $y^{2}=\frac{1}{x-3}$

Question 11

In the problems below, determine the intercepts, the intervals in which the curve is above or touching the $\mathrm{x}$ - axis, symmetry, and asymptotes. Graph each equation.

- $y=x^{3}-5 x^{2}-12 x+36$

Question 12

In the problems below, determine the intercepts, the intervals in which the curve is above or touching the $\mathrm{x}$ - axis, symmetry, and asymptotes. Graph each equation.

- $y=\frac{4}{5 x-3}$

Question 13

In the problems below, determine the intercepts, the intervals in which the curve is above or touching the $\mathrm{x}$ - axis, symmetry, and asymptotes. Graph each equation.

- $y=\frac{x}{(x+4)(x-5)}$

Question 14

In the problems below, find any relative maximum or minimum values of each function.

- $y=x^{3}+2 x^{2}-7 x+4$

Question 15

In the problems below, find any relative maximum or minimum values of each function.

- $y=x^{4}-6 x^{2}+5$

Question 16

In the problems below, find any relative maximum or minimum values of each function.
-Given $f(x)=\frac{6}{x^{2}-9}$ , find a) the intervals for which $f(x)$ is increasing and decreasing, b) relative maximums and relative minimums, $\mathrm{c}$ ) intervals for which $\mathrm{f}(\mathrm{x})$ is concave upward and concave downward, and d) points of inflection. Sketch the curve.

Question 17

Use Newton's Method to find a solution to six significant digits in the given interval.

- $x^{3}-2 x^{2}=8 x-6 ; 3 \leq x \leq 5$

Question 18

Use Newton's Method to find a solution to six significant digits in the given interval.

- $4 \sin x=\cos x ; 3 \leq x \leq 5$

Question 19

Use Newton's Method to find a solution to six significant digits in the given interval.

- $x^{4}-5 x^{2}-6=0 ; 2 \leq x \leq 4$

Question 20

A rectangular box, open at the top, with a square base is to have a volume of $13,500 \mathrm{~cm}^{3}$ . Find the dimensions if the box is to contain the least amount of material.

Question 21

The power $P$ in watts (W) in a circuit with resistance $R$ in ohms $(\Omega)$ varies according to $\mathrm{P}=\frac{49 \mathrm{R}}{(\mathrm{R}+9)^{2}}$ . Find the resistance that gives the maximum power.

Question 22

Given $y=5 x^{3}$ and $\frac{d x}{d t}=3$ at $x=2$ , find $\frac{d y}{d t}$ .

Question 23

A circular plate is being heated so that its radius is increasing at a rate of $0.09 \mathrm{in}$ ./h. How fast is its area increasing when its radius is 15 inches?

Question 24

A cylinder with an inside radius of $35 \mathrm{~mm}$ is sealed at one end and a piston is at the other end. At what rate is the piston moving if fluid is being pumped into the cylinder at the rate of $60 \mathrm{~cm}^{3} / \mathrm{s}$ ? (Hint: Pay attention to units.)

Question 25

Find dy for the expression $9 x^{2}+25 y^{2}=10$ .

Question 26

Using a differential expression, find the change in $V=\frac{4}{3} \pi r^{3}$ from $r=25.00$ in. to 25.10 in.

Question 27

Using a differential expression, find the percentage error in $V=\frac{4}{3} \pi r^{3}$ from $r=25.00$ in. to $25.10 \mathrm{in}$ .

Question 28

The town of Darlington plans to build a spherical water tower with an inner diameter of $30.00 \mathrm{~m}$ and sides of thickness $5.00 \mathrm{~cm}$ . Find the approximate volume of steel needed using differentials. If the density of steel is $7800 \mathrm{~kg} / \mathrm{m}^{3}$ , find the approximate amount of steel needed.