Deck 13: Mathematical Problem Set: Calculus, Algebra, and Optimization

Question 1

Simplify: $\frac { 3 ^ { 8 } } { 3 ^ { 7 } }$

A) $\frac { 1 } { 9 }$
B)3
C) $\frac { 1 } { 3 }$
D) $3 ^ { 15 }$

Answer 1

3

Question 2

Find the slope and y-intercept of the line whose equation is given. $\frac { x } { 2 } + \frac { y } { 3 } = 1$

A)Slope: $\frac { 1 } { 2 }$ ; y-intercept: 1
B)Slope: $\frac { 3 } { 2 }$ ; y-intercept: $\frac { 1 } { 2 }$
C)Slope: $\frac { 3 } { 2 }$ ; y-intercept: 1
D)Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3

Answer 2

Slope: $- \frac { 3 } { 2 }$ ; y-intercept: 3

Question 3

Find the points of intersection (if any)of the given pair of curves. y = x + 3 and y = 2x + 4

A)(-1,2)
B)(0,3)and (1,4)
C)(1,6)
D)(1,-4)

Answer 3

(-1,2)

Question 4

Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$ .

A)1
B) $\frac { 1 } { 3 }$
C)Diverges
D)0

Question 5

An object moving in a straight line has velocity $v ( t ) = t e ^ { \sqrt { t } }$ meters per second.Is it true that in the first 4 seconds the object will have travelled $4 e ^ { 2 } + 12$ meters?

Question 6

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 6 }$ the ratio in the atmosphere.Approximately how old is the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years.Round to the nearest whole year.

A)17,190 years
B)14,812 years
C)34,380 years
D)14,842 years

Question 7

An investment will generate income continuously at the constant rate of $1,100 per year for 5 years.If the prevailing annual interest rate remains fixed at 6% compounded continuously,what is the present value of the investment?

A)$6,044.13
B)$47.52
C)$4,751.67
D)$475.17

Question 8

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

A)y = 2x + 1
B)y = 9x - 1
C)y = 9x
D)y = 9x - 8

Question 9

To raise money,a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling.Since the project began 90 days ago,the club has collected 45,000 pounds of glass for which the glass company currently offers 1 cent per pound.However,because bottles are accumulating faster than they can be recycled,the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass.Assume that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible.What is the most advantageous time for the club to conclude its project and deliver the bottles?

A)15 days from now
B)Today
C)5 days from now
D)55 days from now

Question 10

Find the relative rate of change of f (x)with respect to x for the prescribed value x = 1. f (x)= 3x³ + x² - 8

A) $\frac { 1 } { 11 }$
B) $- \frac { 11 } { 4 }$
C)11
D) $- \frac { 4 } { 11 }$

Question 11

The second derivative test reveals that f (x)= x² - 4 has

A)nothing significant at x = 4.
B)a relative maximum at $x = \sqrt { 4 }$ .
C)a point of inflection at $x = \sqrt { 4 }$ .
D)a relative minimum at x = 0.

Question 12

Evaluate $\int e ^ { 6 x - 8 } d x$ .

A) $( 6 x - 8 ) e ^ { 6 x - 8 } + C$
B) $( 6 x - 8 ) C e ^ { 6 x - 8 }$
C) $e ^ { 6 x - 8 } + C$
D) $\frac { e ^ { 6 x - 8 } } { 6 } + C$

Question 13

Find the equation of the tangent line to f (x)= ln x + 2 at x = 1.

A)y = 1
B)x = 1
C)y = x + 1
D)y = x

Question 14

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x$ .

A) $\frac { 1 } { 2 \ln 2 }$
B) $- \frac { 1 } { 2 \ln 2 }$
C)Diverges
D)0

Question 15

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$ .

A) $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$
B) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$
C) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$
D) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$

Question 16

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 4$ .

A)Decreasing for $x < - \frac { 5 } { 2 }$ and increasing for $x > - \frac { 5 } { 2 }$
B)Decreasing for all x
C)Increasing for all x
D)Decreasing for $x > - \frac { 5 } { 2 }$ and increasing for $x < - \frac { 5 } { 2 }$

Question 17

Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$ .

A) $- 3 e ^ { - 2 x } + C$
B) $\frac { 3 } { 2 } x e ^ { - 2 x } + C$
C) $e ^ { - 2 x } \left( \frac { 3 } { 2 } x - \frac { 1 } { 4 } \right) + C$
D) $x e ^ { - 2 x } + C$

Question 18

Find the slope of the tangent line to the curve $x ^ { 2 } + 3 x y - y ^ { 2 } = 3$ at the point (1,1).

A)-5
B)5
C)3
D)1

Question 19

Find the composite function $f ( g ( x ) )$ . $f ( u ) = \frac { 1 } { u }$ , $g ( x ) = x + 3$

A)f (g(x))= $1 + \frac { 3 } { x }$
B)f (g(x))= $\frac { 1 } { x + 3 }$
C)f (g(x))= $x + 3 + \frac { 1 } { x }$
D)f (g(x))= $\frac { 1 } { x } + 3$

Question 20

Records indicate that t hours past midnight,the temperature at the local airport was $f ( t ) = - 0.3 t ^ { 2 } + 4 t + 10$ degrees Fahrenheit.What was the average temperature at the airport between 8:00 A.M.and noon? Round your answer to one decimal place,if necessary.

A)3.9 degrees F
B)9.6 degrees F
C)98.0 degrees F
D)19.6 degrees F

Question 21

Compute $f _ { x }$ for $f ( x , y ) = 7 x y ^ { 2 }$ .

A) $14 x y ^ { 1 }$
B)7x
C) $14 x y ^ { 1 } + 7 y ^ { 2 }$
D) $7 y ^ { 2 }$

Question 22

A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms.The price the manufacturer can expect to receive for the machines will depend on the number of machines made available.(For example,if only a few of the machines are placed on the market,competitive bidding among prospective purchasers will tend to drive the price up.)It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market,the machines will sell for $60 - \frac { x } { 5 } + \frac { y } { 20 }$ thousand dollars apiece at home and for $50 - \frac { y } { 10 } + \frac { x } { 20 }$ thousand dollars apiece abroad.If the manufacturer can produce the machines at a cost of $38,000 apiece,how many should be supplied to each market to generate the largest possible profit?

A)x = 41,y = 643
B)x = 40,y = 500
C)x = 100,y = 8
D)x = 80,y = 100

Question 23

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 2 )$ of the given initial value problem.Round to one decimal place. $y ^ { t } = \frac { 2 x } { y } ; y ( 1 ) = 2$

A)2.6
B)3.1
C)2.1
D)3.4

Question 24

Use Lagrange multipliers to find the maximum value of f (x,y,z)= 6xyz subject to 2x + 5y + 2z = 60.

A)f (10,4,10)= 2,400
B)f (0,0,0)= 0
C)f (11,2,10)= 1,320
D)f (13,5,8)= 3,120

Question 25

Use a double integral to find the area of R. R is the region bounded by y = 3x,y = ln x,y = 0,and y = 1.

A) $\int _ { y - 0 } ^ { y - 1 } \int _ { x - \frac { y } { 3 } } ^ { x - e ^ { y } } 1 d x d y = e - \frac { 7 } { 6 }$
B) $\int _ { y - 0 } ^ { y - 1 } \int _ { x - 0 } ^ { x - e ^ { y } } 1 d x d y = e - 1$
C) $\int _ { x = 0 } ^ { x - e } \int _ { y - \ln x } ^ { y - 1 } 1 d y d x = e$
D) $\int _ { x = 0 } ^ { x - 2 } \int _ { y - \operatorname { m } x } ^ { y - 3 x } 1 d y d x = \frac { 3 e ^ { 2 } } { 2 }$

Question 26

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ is a probability density function for a particular random variable X.Use integration to find $P ( X \geq 4 )$

A) $e ^ { - 2 } - 1$
B) $e ^ { 2 }$
C) $e ^ { - 2 }$
D) $1 + e ^ { 2 }$

Question 27

Find the general solution of $\frac { d y } { d x } = e ^ { x }$ .

A) $y = \frac { x ^ { e } } { C }$
B) $y = x e ^ { x } + C$
C) $y = C e ^ { x }$
D) $y = e ^ { x } + C$

Question 28

The given series converges. $\sum _ { k = 1 } ^ { \infty } k ^ { - 2.5 }$

Question 29

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { 7 ^ { k - 1 } x ^ { k } } { k }$

A) $R =$ 7
B) $R =$ $\infty$
C) $R =$ $\frac { 1 } { 7 }$
D) $R =$ 0

Question 30

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = 28 x ^ { 3 } y ^ { 2 }$ ; y = 6 when x = 1.

A) $y = \frac { 6 } { 43 - 42 x ^ { 4 } }$
B) $y = \frac { 6 } { 43 + 42 x ^ { 4 } }$
C) $y = - \frac { 6 } { 43 + 42 x ^ { 4 } }$
D) $y = \frac { 6 } { 43 - 42 x ^ { 3 } }$

Question 31

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 3 ^ { k } + 5 ^ { k } } { 5 ^ { k } + 7 ^ { k } }$

Question 32

Consider the random experiment of rolling a fair die two times,and let X denote the random variable that counts the number of times 6 appears.Find $P ( X = 2 )$

A) $\frac { 25 } { 36 }$
B) $\frac { 1 } { 36 }$
C) $\frac { 1 } { 12 }$
D) $\frac { 11 } { 36 }$

Question 33

Write a differential equation describing the given situation: Powdered lemonade dissolves in a pitcher at a rate that is proportional to the amount of undissolved powder remaining.Let P = the total amount of powder added,D = the amount dissolved,and t = time.

A) $\frac { d D } { d t } = k + P - D$
B) $\frac { d D } { d t } = k ( P - D )$
C) $\frac { d D } { d t } = k P ( P - D )$
D) $\frac { d D } { d t } = k P D$

Question 34

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = \ln ( 5 x ) ; a = \frac { 1 } { 5 }$

A) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 5 ^ { n } } { n } \left( x - \frac { 1 } { 5 } \right) ^ { n }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { 5 ^ { n } } { n } \left( x - \frac { 1 } { 5 } \right) ^ { n }$
C) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n 5 ^ { n } } \left( x - \frac { 1 } { 5 } \right) ^ { n }$
D) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { n 5 ^ { n } } \left( x - \frac { 1 } { 5 } \right) ^ { n }$

Question 35

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }$

A)Diverges.
B)Converges to $\frac { 5 } { 2 }$
C)Converges to $\frac { 5 } { 12 }$
D)Converges to $\frac { 7 } { 2 }$

Question 36

A wooded nature preserve supports populations of silver foxes and snowshoe hares.The growth rate for each population can be modeled by this pair of differential equations: $\begin{array} { l } \frac { d F } { d t } = - 0.24 F + 0.0012 F H \\\frac { d H } { d t } = 0.08 R - 0.005 F H\end{array}$ where H is the number of hares and F is the number of foxes.Find equilibrium populations for this model.(Hint: These are the populations for which $\frac { d F } { d t } = \frac { d H } { d t } = 0$ .)

A)12 foxes,170 hares
B)12 foxes,100 hares
C)30 foxes,220 hares
D)16 foxes,200 hares

Question 37

A factory produces MP3 players.During regular inspections,2% of the MP3 players are found to be defective.On any particular day,what is the probability that the twentieth MP3 player inspected is the first defective player found that day? Round your answer to four decimal places.

A)0.6812
B)0.0134
C)0.0136
D)0.9864

Question 38

A ball is dropped from a height of H feet and bounces indefinitely,repeatedly rebounding to 76% of its previous height.If it travels a total distance of 80 feet,what is H? Round your answer to one decimal place.

A)9.6 feet
B)30.4 feet
C)19.2 feet
D)10.9 feet

Question 39

Find the general solution of the given first-order linear differential equation. $x ^ { 3 } \frac { d y } { d x } + x ^ { 2 } y = 3$

A) $y = \frac { C } { x ^ { 2 } } - \frac { 3 } { x }$
B) $y = \frac { C } { x } - \frac { 3 } { x ^ { 2 } }$
C) $y = C - \frac { 3 } { x ^ { 2 } }$
D) $y = - \frac { C } { x ^ { 2 } }$

Question 40

Compute f (4,3)if $f ( x , y ) = 5 x y ^ { 3 }$ .

A)108
B)192
C)960
D)540

Question 41

A manufacturer will supply x units of a certain commodity to the market when the price is $p = S ( x )$ dollars per unit,where $S ( x ) = 21 + 2 x - 2.5 \sin \pi x$ . What is the average price per unit as the level of production increases from $x = 4$ to $x = 7 ?$ (Round to the nearest cent.)

A)$32.53 per unit
B)$32.00 per unit
C)$31.47 per unit
D)$42.47 per unit

Question 42

Suppose that during business hours,the time between successive wireless calls at a network switch can be represented by an exponentially distributed random variable X with expected value $E ( X ) = 0.025$ second.Using an appropriate probability density function,find the probability that the time between the arrival of successive wireless calls at the switch is between 0.03 and 0.065 second.Round to the nearest hundredth.

A)0.63
B)0.46
C)0.77
D)0.23

Question 43

Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t). $$f ( t ) = 1.5 + 5.5 \sin \left[ \frac { \pi } { 3 } ( t - 1 ) \right]$$

A) $p = 6 , b = - 5.5 , d = - 1 , a = 1.5$
B) $p = 3 , b = 1.5 , d = 1 , a = 5.5$
C) $p = 6 , b = 5.5 , d = 1 , a = 1.5$
D) $p = 6 , b = 1.5 , d = 1 , a = 5.5$

Question 44

Differentiate the given function. $y = 5 \tan ( 2 x - \sqrt { x } )$

A) $\left( 10 - \frac { 5 } { 2 \sqrt { x } } \right) \sec ^ { 2 } ( 2 x - \sqrt { x } )$
B) $5 \sec ^ { 2 } ( 2 x - \sqrt { x } )$
C) $5 \cot \left( \frac { 1 } { 2 x - \sqrt { x } } \right)$
D) $\left( 10 - \frac { 5 } { 2 \sqrt { x } } \right) \tan ( 2 x - \sqrt { x } ) \sec ( 2 x - \sqrt { x } )$

Question 45

Suppose that the number of people suffering bee stings in a rural county in May follows a Poisson distribution with a mean of 1.5 bee stings per day.Find the probability that on a randomly selected day,there will be 2 bee stings.Round to three decimal places.

A)0.151
B)0.251
C)0.201
D)0.100

Question 46

Evaluate the given expression. cos $\left( - \frac { 2 \pi } { 3 } \right)$

A) $- \frac { \sqrt { 3 } } { 2 }$
B) $- \frac { 1 } { 2 }$
C) $\frac { 1 } { 2 }$
D) $\frac { \sqrt { 3 } } { 2 }$

Question 47

A study shows that consumers in one city will buy q hundred bottles of a new perfume when the price is $p = 30 - 8 q + \sin \left( \frac { \pi q } { 3 } \right)$ dollars per bottle.Find the consumer's surplus for this product when 22,000 bottles are demanded and produced.Round to the nearest dollar.

A)$2,311
B)$1,932
C)$1,837
D)$847

Question 48

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 6 } & \text { if } 2 \leq x \leq 8 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a continuous random variable X. The expected value is 5.