Resources

Resources

Find instantaneous rates of change

One leg of a right triangle is decreasing at a rate of

5

kilometers per hour and the other leg of the triangle is increasing at a rate of

14

kilometers per hour.

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At a certain instant, the decreasing leg is

3

kilometers and the increasing leg is

9

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What is the rate of change of the area of the right triangle at that instant (in square kilometers per hour)?

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1

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1

5

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111

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-1

5

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-111

One leg of a right triangle is decreasing at a rate of

5

kilometers per hour and the other leg of the triangle is increasing at a rate of

14

kilometers per hour.

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At a certain instant, the decreasing leg is

3

kilometers and the increasing leg is

9

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What is the rate of change of the area of the right triangle at that instant (in square kilometers per hour)?

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1

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-1

5

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111

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1

5

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-111

The height of a rectangle is increasing at a rate of

11

centimeters per hour and the width of the rectangle is decreasing at a rate of

9

centimeters per hour.

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At a certain instant, the height is

3

centimeters and the width is

8

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What is the rate of change of the area of the rectangle at that instant (in square centimeters per hour)?

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1

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-61

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61

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99

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-99

f(t) = 10^t

change over the interval from

t = 9

t = 10

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f(t)

1,000\%

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f(t)

decreases by a factor of

10

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f(t)

10\%

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f(t)

t = 9

0

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y=-75 x+950

represents the line of fit for a data set. Which terms best describe the data?

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A. positive linear association

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B. negative linear association

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C. positive nonlinear association

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D. negative nonlinear association

H

is the midpoint of

I G

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J

is the midpoint of

I K

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I G=G K=10

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I J

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5

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Quantity A is greater.

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Quantity B is greater.

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The two quantities are equal.

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The relationship cannot be determined from the information given.

DeltaMath Student Application

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Content Keeper Auth

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Use technology to find points and graph the line

2 y+3 x=-2

, following the instructions below.

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done plotting points

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Plot at least two points that fit on the axes below.

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Click a point to delete it.

\pi

is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is

\pi=\frac{c}{d}

. This seems to contradict the fact that

\pi

is irrational. How will you resolve this contradiction?

N \in \mathbb{N}\setminus\{0\}

a \neq 1

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\frac{1}{1-a} = \frac{a^N}{1-a} + \sum_{n=0}^{N-1}a^n

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Use this result to find a formula for

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\sum_{n=0}^{\infty}a^n

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What assumption did you have to make to obtain a converging result?

The movement of the progress bar may be uneven because questions can be worth more or less (Including zero) depending on your answer.

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Match the line described on the left with the slope of the line on the right.

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y=0

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-3

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9 x+3 y=18

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the line through

(-5,7)

(-5,-8)

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2

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the line through

(4,-1)

(8,7)

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The base of a triangle is shrinking at a rate of

8 \mathrm{~cm} / \mathrm{s}

and the height of the triangle is increasing at a rate of

2 \mathrm{~cm} / \mathrm{s}

. Find the rate at which the area of the triangle changes when the height is

14 \mathrm{~cm}

and the base is

10 \mathrm{~cm}

An investor in Treasury securities expects inflation to be

2.0\%

1

2.6\%

2

3.75\%

each year thereafter. Assume that the real risk-free rate is

1.95\%

and that this rate will remain constant. Three-year Treasury securities yield

5.20\%

5

-year Treasury securities yield

6.00\%

. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is

\text{MRP}_5 - \text{MRP}_3

The rate of change

\frac{d P}{d t}

of the number of bacteria in a tank is modeled by the following differential equation:

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\frac{d P}{d t}=\frac{2}{9849} P(598-P)

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t=0

, the number of bacteria in the tank is

196

and is increasing at a rate of

16

bacteria per minute. At what value of

P

does the graph of

P(t)

have an inflection point?

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The rate of change

\frac{d P}{d t}

of the number of people infected by a disease is modeled by the following differential equation:

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\frac{d P}{d t}=\frac{45}{125404} P(800-P)

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t=0

, the number of people infected by the disease is

214

and is increasing at a rate of

45

people per hour. What is the limiting value for the total number of people infected by the disease as time increases?

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The rate of change

\frac{d P}{d t}

of the number of algae in a tank is modeled by the following differential equation:

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\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)

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t=0

, the number of algae in the tank is

174

and is increasing at a rate of

28

algae per minute. At what value of

P

P(t)

growing the fastest?

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