Deck 8: Appendix: Algebra Review

Question 1

To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time required for the rat to traverse the maze on the nth trial was approximately $$T ( n ) = 5 + \frac { 2 } { n } - \frac { 4 } { n ^ { 2 } }$$ minutes. How many minutes does it take the rat to traverse the maze on the 2nd trial?

Answer 1

T(2) = 5 min

Question 2

Find the points of intersection (if any) of the given pair of curves.y = x + 7 and y = 2x + 4

Answer 2

(3, 10)

Question 3

Find the slope (if possible) of the line that passes through the given pair of points. (1, 0) and (18, 12)

Answer 3

$\frac { 12 } { 17 }$

Question 4

Write an equation for the line with the given properties.Through (3, -1) with slope 2

Question 5

An appliance manufacturer can sell refrigerators for $600 apiece. The manufacturer's total cost consists of a fixed overhead of $12,000 plus production cost of $400 per refrigerator. How many refrigerators must be sold for the manufacturer to break even?

Question 6

Find the indicated limit if it exists. $$\lim _ { x \rightarrow 6 } \frac { x - 6 } { x ^ { 2 } - 36 }$$

Question 7

Decide if the given function is continuous at the specified value of x. $$f ( x ) = \left\{ \begin{array} { r l } x + 5 & \text { if } x < 2 \\3 x + 1 & \text { if } x \geq 2\end{array} ; \quad x = 2 \right.$$

Question 8

Differentiate: $f ( x ) = x ^ { 6 } + 5$

Question 9

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 4$ that passes through (1, 5) is y = 2x + 4.

Question 10

What is the rate of change of $f ( t ) = \frac { 8 t - 7 } { t + 5 }$ with respect to t when t = 42?

Question 11

An equation for the tangent line to the curve $y = \left( x ^ { 7 } + x - 1 \right) ^ { 8 }$ at the point where x = 1 is

Question 12

If $x ^ { 2 } y + x y ^ { 2 } = 7$ , then $\frac { d y } { d x } = 2 x y + y ^ { 2 }$

Question 13

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

Question 14

Find all intervals where the derivative of the function shown below is negative.

Question 15

Find the intervals of increase and decrease for $f ( x ) = \frac { 6 x - 1 } { - 2 x + 10 }$ Round numbers to two decimal places, if necessary.

Question 16

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Question 17

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Question 18

Find the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 12$ on the interval -1 $\le$ x $\le$ 2.

Question 19

Find two non-negative numbers whose sum is 12 for which the product of their squares is as large as possible.

Question 20

The owner of an appliance store expects to sell 600 toasters this year. Each toaster costs her $7 dollars to purchase, and each time she orders a shipment of toasters, it costs $24. In addition, it costs $4 a year to store each toaster. Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters, how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.)

Question 21

Find the indicated composite function.f (5x - 4) where $f ( x ) = \frac { 1 } { x } - x$

Question 22

Find the points of intersection (if any) of the given pair of curves.y = 7x - 8 and y = 2x - 6

Question 23

Find the slope and y-intercept of the line whose equation is given. $$\frac { x } { 3 } + \frac { y } { 5 } = 1$$

Question 24

Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is + $\infty$ or - $\infty$ . $$\lim _ { x \rightarrow 5^- } f ( x )$$ where $$f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & \text { if } x \leq 5 \\x + 8 & \text { if } x > 5\end{array} \right.$$

Question 25

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1.f (x) =3x³ + 2x² - 8

Question 26

The equation of the tangent line to the curve $f ( x ) = \left( 7 x ^ { 5 } - 5 x ^ { 2 } + 2 \right) \left( x ^ { 3 } + x - 1 \right)$ at the point (0, -2) is y = 2x - 2.

Question 27

Find an equation for the tangent line to the curve $y = \sqrt { 3 + \frac { x } { 4 } }$ at the point where x = -1. Round numbers to two decimal places.

Question 28

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

Question 29

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 9$ .

Question 30

Let $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13$ . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Question 31

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?

Question 32

Simplify: $\frac { 2 ^ { 3 } } { 2 ^ { 2 } }$

Question 33

The fraction of television sets manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.2 t }$ . What fraction can be expected to fail before 4 years of use? Round your answer to two decimal places, if necessary.

Question 34

Solve the given equation for x. $4 = 3 + 2 e ^ { - 7 x }$

Question 35

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 7 }$ 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.

Question 36

If $f ( x ) = \left( x + 2 e ^ { - x } \right) ^ { 3 }$ , then $f ^ { \prime } ( x ) = x + 2 e ^ { - x }$

Question 37

The equation of the tangent line to f (x) = ln x + 5 at x = 1 is

Question 38

The equation of the tangent line to $f ( x ) = 8 \ln x ^ { 3 }$ at x = e is

Question 39

The function y = ln 2x is concave downward everywhere.

Question 40

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 8 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 8 \ln | x | + C$

Question 41

Evaluate the following integral: $\int 7 e ^ { 4 x } d x$

Question 42

Evaluate $\int e ^ { 4 x - 6 } d x$ .

Question 43

Evaluate $\int x \sqrt { x ^ { 2 } + 6 } d x$ .

Question 44

Sketch the region R and then use calculus to find the area of R. R is the region between the curve $y = x ^ { 3 }$ and the line y = 20x for x $\ge$ 0.

Question 45

Records indicate that t hours past midnight, the temperature at the local airport was $f ( t ) = - 0.3 t ^ { 2 } + 8 t + 10$ degrees Fahrenheit. What was the average temperature at the airport between 3:00 A.M. and noon? Round your answer to one decimal place, if necessary.

Question 46

An investment will generate income continuously at the constant rate of $2,300 per year for 5 years. If the prevailing annual interest rate remains fixed at 11% compounded continuously, what is the present value of the investment?

Question 47

Money is transferred continuously into an account at the constant rate of $1,600 per year. Assume the account earns interest at the annual rate of 5% compounded continuously. Compute the future value of the income stream over a 16 year period.

Question 48

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x$ .

Question 49

Evaluate $$\int \frac { \ln 3 x } { x ^ { 2 } } d x$$ .

Question 50

After t weeks, a charity is raising money at the rate of 5,000 t ln(t + 1) dollars per week. How much money is raised during the first 10 weeks? Round to the nearest ten dollars.

Question 51

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

Question 52

Evaluate the improper integral: $$\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 6 } / 4 } d x$$ Round to two decimal places, if necessary.

Question 53

A certain nuclear power plant produces radioactive waste at the rate of 500 pounds per year. The waste decays exponentially at the rate of 1.5% per year. How many pounds of radioactive waste from the plant will be present in the long run? Round to two decimal places, if necessary.

Question 54

$$f ( x ) = \left\{ \begin{array} { l l } \frac { 7 } { x ^ { 8 } } & \text { if } x \geq 1 \\0 & \text { if } x < 1\end{array} \right.$$ is a probability density function for a particular random variable X Use integration to find $P ( X \geq 3 )$

Question 55

The clothes dryers at a laundromat run for 45 minutes. You arrive at the laundromat and find that all of the dryers are being used. Use an appropriate uniform density function to find the probability that a dryer chosen at random will finish its cycle within 5 minutes.

Question 56

Compute f (1, 5) if $f ( x , y ) = 7 x y ^ { 3 }$ .

Question 57

Compute f_x for f (x, y) = 3x⁵y - 8x +e^xy.

Question 58

Compute $f _ { x }$ for $f ( x , y ) = 9 x y ^ { 3 }$ .

Question 59

Let f (x, y) = x ln(1 + 2x - 5y). Find $f _ { x x } ( x , y )$ .

Question 60

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 $45,000 apiece, how many should be supplied to each market to generate the largest possible profit?

Question 61

The accompanying table lists the high-school GPA and college GPA for a number of students: $$\begin{array} { l l l l l l l } \text { High school GPA } & 2.0 & 2.5 & 3.1 & 3.7 & 3.7 & 4.0 \\\text { College GPA } & 3.2 & 3.1 & 3.0 & 3.6 & 3.8 & 3.2\end{array}$$
Using the best fit straight line, predict the college GPA (to one decimal place) for a student whose high school GPA was 3.5.

Question 62

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

Question 63

Use Lagrange multipliers to find the maximum value of f (x, y, z) = 3xyz subject to 5x + 5y + 2z = 150.

Question 64

Use a double integral to find the area of R.R is the region bounded by y = 5x, y = ln x, y = 0, and y = 1.

Question 65

Find the points of intersection (if any) of the given pair of curves.y = x + 9 and y = 2x + 4

A) (1, -4)
B) (0, 6)
C) (14, 23)
D) (5, 14)

Question 66

Find the slope (if possible) of the line that passes through the given pair of points. (17, 0) and (20, 5)

A) $- \frac { 5 } { 3 }$
B) $- \frac { 3 } { 5 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 3 } { 5 }$

Question 67

Write an equation for the line through (3, 0) with slope 2.

A) y = 2x + 6
B) y = 2x - 6
C) y = 2x + 3
D) y = 2x - 3

Question 68

Find the indicated limit if it exists. $\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

A) $\frac { 1 } { 4 }$
B) Does not exist
C) 4
D) $- \frac { 1 } { 4 }$

Question 69

Decide if the given function is continuous at the specified value of x. $$f ( x ) = \left\{ \begin{array} { r l } x + 5 & \text { if } x < 1 \\3 x + 3 & \text { if } x \geq 1\end{array} ; \quad x = 1 \right.$$

A) Yes, the function is continuous at x = 1.
B) No, the function is not continuous at x = 1.

Question 70

Differentiate: $f ( x ) = x ^ { 6 } + 7$

A) $6 x ^ { 7 } + 7 x$
B) $6 x ^ { 5 }$
C) $5 x ^ { 5 }$
D) $6 x ^ { 5 } + 7$

Question 71

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 1$ that passes through (9, 4) is y = 2x + 1.

Question 72

What is the rate of change of $f ( t ) = \frac { 4 t - 5 } { t + 4 }$ with respect to t when t = 17?

A) 21
B) -21
C) $- \frac { 1 } { 21 }$
D) $\frac { 1 } { 21 }$

Question 73

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 6 }$ at the point where x = 1 is

A) y = 18x
B) y = 18x - 1
C) y = 18x - 17
D) y = 2x + 1

Question 74

If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$ , then $\frac { d y } { d x } = 2 x + 3 y$

Question 75

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$

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

Question 76

Find the intervals of increase and decrease for $f ( x ) = \frac { 10 x - 5 } { - 2 x + 10 }$ Round numbers to two decimal places, if necessary.

A) Increasing on x $\le$ 0.50 and on x > 5, decreasing on 0.50 < x $\le$ 5
B) Increasing on x < 5, decreasing on x > 5
C) Increasing on 0.50 < x $\le$ 5, decreasing on x $\le$ 0.50 and on x > 5
D) Increasing on x < 5 and x > 5

Question 77

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

A) x < 1
B) x < -1
C) x > 1
D) x > -1

Question 78

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

A) a relative minimum only.
B) a relative maximum and two relative minima.
C) neither a relative maximum nor a relative minimum.
D) a relative maximum only.

Question 79

Find the absolute maximum of the function $f ( x ) = x ^ { 3 }$ on the interval $- \frac { 1 } { 2 } \leq x \leq 1$

A) 1
B) -1
C) 0
D) $\frac { 1 } { 8 }$

Question 80

Find two non-negative numbers whose sum is 14 for which the product of their squares is as large as possible.

A) 0 and 14
B) 5 and 9
C) 7 and 7
D) 1 and 13