Deck 11: Polynomial and Rational Functions

Question 1

The following table represents the amount of monthly data handled by an Internet server, sampled at six month intervals for the years 1993-1995.
Let d be the number of data packets handled monthly (in billions), and t be time in years since December, 1992. The best exponential model for this data is of the form , where a = _____ and b = _____. Give both answers to 6 decimal places.

Answer 1

a)1.205638
b)2.081993

Question 2

The following table represents the amount of monthly data handled by MichNet, the University of Michigan's gateway to the Internet, sampled at six month intervals for the years 1993-1995. (The data is from the Michigan Daily, March 30, 1996, p.4.)

Let d be the number of data packets handled monthly (in billions), and t be time in years since December, 1992. Find equations for the best fit exponential function and the best fit power function that model these data (you do not have to enter your equations). By comparing their correlation coefficients, choose which model gives a better fit.

A) The exponential model
B) The power function model

Answer 2

A

Question 3

The following table represents the amount of monthly data handled by MichNet, the University of Michigan's gateway to the Internet, sampled at six month intervals for the years 1993-1995. (The data is from the Michigan Daily, March 30, 1996, p.4.) The article also stated that the data handling capacity of the MichNet system was about 40 billion data packets per month.

Let d be the number of data packets handled monthly (in billions), and t be time in years since December, 1992. Find an equation for the best fit exponential function that models these data (you do not have to enter your equation). Use this equation to determine in what year the network became overloaded.

Answer 3

1997

Question 4

The following table contains the names of the nine known planets that orbit about the sun, along with their distance d from the sun in millions of miles and the period P of the revolution about the sun in standard earth years, that is, the time it takes the planet to go around the sun. (For this problem ,we will assume that Pluto is a planet).

Let
Suppose a tenth planet is discovered at a distance of 5.4 billion (i.e. 5400 million) miles from the sun. Use the best-fit power model for the data to estimate the number of years in the period of the new planet to the nearest year.

Question 5

George measures the force exerted by a certain spring when it is stretched various distances beyond its natural length. The data he gathers is compiled in the following table.

-He wishes to describe the force as a function of the stretched distance, and decides to use a power function model. After fitting a power function to the data, he uses it to estimate the force when the spring is stretched 22 inches. What is his approximation (to three decimal places) of this force?

Question 6

George measures the force exerted by a certain spring when it is stretched various distances beyond its natural length. The data he gathers is compiled in the following table.

- He wishes to describe the force as a function of the stretched distance, and decides to use a cubic function model. After fitting a cubic function to the data, he uses it to estimate the force when the spring is stretched 22 inches. What is his approximation (to three decimal places) of this force?

Question 7

George measures the force exerted by a certain spring when it is stretched various distances beyond its natural length. The data he gathers is compiled in the following table.

- He wishes to describe the force as a function of the stretched distance, and decides to use a quadratic function model. After fitting a quadratic function to the data, he uses it to estimate the force when the spring is stretched 19 inches. What is his approximation (to three decimal places) of this force?

Question 8

George measures the force exerted by a certain spring when it is stretched various distances beyond its natural length. The data he gathers is compiled in the following table.

- He wishes to describe the force as a function of the stretched distance, and decides to use an exponential function model. After fitting an exponential function to the data, he uses it to estimate the force when the spring is stretched 21 inches. What is his approximation (to three decimal places) of this force?

Question 9

The number of yearly childbirths for a growing town are recorded in the table below. Suppose the number of childbirths each year is a function of the number of years since 1996. Fit a power function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2012.

Question 10

The number of yearly childbirths for a growing town are recorded in the table below. Suppose the number of childbirths each year is a function of the number of years since 1996.

- Fit a cubic function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2011.

Question 11

The number of yearly childbirths for a growing town are recorded in the table below. Suppose the number of childbirths each year is a function of the number of years since 1996.

-Fit a quadratic function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2017.

Question 12

The number of yearly childbirths for a growing town are recorded in the table below. Suppose the number of childbirths each year is a function of the number of years since 1996.

- Fit an exponential function to the data and use it to estimate the number of childbirths (rounded to the nearest whole number) in 2017.

Question 13

The squirrel population in a certain area is estimated each year and recorded in the table below. Suppose the squirrel population in a given year is a function of the number of years since 1998.

-Fit a power function to the data and use it to estimate the squirrel population (rounded to the nearest whole number) in 2018.

Question 14

The squirrel population in a certain area is estimated each year and recorded in the table below. Suppose the squirrel population in a given year is a function of the number of years since 1998.

-Fit a cubic function to the data and use it to estimate the squirrel population (rounded to the nearest whole number) in 2015.

Question 15

The squirrel population in a certain area is estimated each year and recorded in the table below. Suppose the squirrel population in a given year is a function of the number of years since 1998.

-Fit a quadratic function to the data and use it to estimate the squirrel population (rounded to the nearest whole number) in 2012.

Question 16

The squirrel population in a certain area is estimated each year and recorded in the table below. Suppose the squirrel population in a given year is a function of the number of years since 1998. Fit an exponential function to the data and use it to estimate the squirrel population (rounded to the nearest whole number) in 2013.

Question 17

The following table represents the amount of monthly data handled by an Internet server, sampled at six month intervals for the years 1993-1995.
Let d be the number of data packets handled monthly (in billions), and t be time in years since December, 1992. The best power model for this data is of the form where a = _____ and p= _____. Give both answers to 6 decimal places.

Question 18

A)
B)

Question 19

Which of the following functions dominates as ?

A)
B)

Question 20

Can be written in the form of an exponential function, a power function, or neither?

A) A power function
B) An exponential function
C) Neither

Question 21

Can be written in the form of an exponential function, a power function, or neither?

A) Neither
B) A power function
C) An exponential function

Question 22

Can be written in the form of an exponential function, a power function, or neither?

A) Neither
B) An exponential function
C) A power function

Question 23

The formula for the power function shown in the following table is , where k = _____ and p = _____. Round both answers to 1 decimal place.

Question 24

The formula for the exponential function shown in the following table is , where a = _____ and b = _____. Round both answers to 2 decimal places.

Question 25

The formula for the polynomial function of degree 2 shown in the following table is , where a = _____, b = _____, and c = _____.

Question 26

Does have an inverse that is a function?

Question 27

Question 28

Question 29

Question 30

Question 31

Question 32

Suppose that a, b, and c are integers with a negative and odd, b positive and even, c positive and odd, and d positive and odd. Which of the following functions corresponds to the graph below?

A)
B)
C)
D)
E)
F)

Question 33

One of the following tables of data comes from a linear function, one from an exponential function, and one from a power function. Which one is the power function?

A) h
B) g
C) f

Question 34

One of the following tables of data comes from a linear function, one from an exponential function, and one from a power function. The formula for the power function is with a = _____ and b = _____.

Question 35

Given the following graph, what would the equation of the line be if the horizontal axis is labeled ln x and the vertical axis is labeled ln y?

A)
B)
C)
D)

Question 36

What is the long-run behavior of the function

Question 37

What is the long-run behavior of the function

Question 38

What is the long-run behavior of the function

Question 39

Which of the following statements are true for ?
Mark all that apply.

A) The x- intercepts are 0 and -3.
B) The x- intercepts are -5 and 5.
C) There are no x-intercepts.
D) The y- intercepts are 0 and -3.
E) The y- intercepts are -5 and 5.
F) There are no y-intercepts.

Question 40

Which of the following statements are true for ? Mark all that apply.

A) There is a horizontal asymptote at y = 1.
B) There are horizontal asymptotes at y = -2 and 2.
C) There are no horizontal asymptotes.
D) There is a vertical asymptote at x = 1.
E) There are vertical asymptotes at x = 0 and -3.
F) There are no vertical asymptotes.

Question 41

Which of the following statements are true for ? Mark all that apply.

A) There is a horizontal asymptote at y = 0.
B) There are horizontal asymptotes at y = 5 and 6.
C) There are no horizontal asymptotes.
D) There is a vertical asymptote at x = 0.
E) There are vertical asymptotes at x = 5 and 6.
F) There are no vertical asymptotes.

Question 42

Which of the following statements are true for ? Mark all that apply.

A) The x- intercepts are -6 and 1.
B) The x- intercepts are -10 and 5.
C) There are no x-intercepts.
D) The y- intercept is

E) The y- intercepts are -10 and 5.
F) There are no y-intercepts.

Question 43

Which of the following statements are true for ? Mark all that apply.

A) There is a horizontal asymptote at y = -10.
B) There are horizontal asymptotes at y = -1 and 1.
C) There are no horizontal asymptotes.
D) There are vertical asymptotes at x = -1 and 1.
E) There are vertical asymptotes at x = -3 and 5.
F) There are no vertical asymptotes.

Question 44

Which of the following could be a formula for the graph shown below?

A)
B)
C)
D)
E)

Question 45

Which of the following could be a formula for the graph shown below?

A)
B)
C)
D)

Question 46

Which of the following statements describe the graph of ? Mark all that apply.

A) There is a vertical asymptote at x = -4.
B) There is a vertical asymptote at x = 1.
C) There is a horizontal asymptote at y = -8.
D) There is a horizontal asymptote at y = 1.
E) There is an x-intercept at x = -8.
F) There is an x-intercept at x = 0.
G) There is a y-intercept at y = 2.
H) There is a y-intercept at y = 0.

Question 47

Which of the following formulas has a graph with vertical asymptotes at x = 1 and x = 4, a horizontal asymptote at y = 1, and touches the x-axis at x = 2?

A)

B)

C)

D)

Question 48

Is a possible formula for the graph below?

Question 49

The following figure is a graph of , where and all of the constants appearing in the function are positive. What is the value of b (in terms of P, Q, etc.)?

Question 50

Which of the following figures shows the graph of ?

A)

B)

C)

Question 51

The following graphs are of functions of the form , where M and N are constants. Which graph has M > 0 and N < 0?

A)

B)

C)

D)

Question 52

Question 53

Question 54

The graph of is a transformation of the graph of for some . Which of the following correctly describes the transformation?

A)
B)

C)

D)

Question 55

What are the -intercepts, -intercepts, and horizontal asymptotes of thefunction

Question 56

Estimate the following limits:

A)

B)

Question 57

Write the equation for the following graph:
The graph of shifted to the left 4 units and up 1 units.

Question 58

It is possible for a rational function to have horizontal asymptotes at and

Question 59

It is possible for a rational function to have no zeros, such that for and for

Question 60

Question 61

Question 62

Does the figure below show an accurate graph of ?

Question 63

Is a rational function?

Question 64

If the function is written in the form , a ratio of polynomials, which of the following could be ?

A)
B)
C)
D)

Question 65

If the function is written in the form , a ratio of polynomials, which of the following could be ?

A)
B)
C)
D) -8

Question 66

The horizontal asymptote of is y = _____. Enter "none" if there is no horizontal asymptote.

Question 67

The following figure gives the graphs of four power functions. Which one could be the graph of ? If none of the graphs match, enter "none".

A)

B)

C)

D)

Question 68

The following figure gives the graphs of four power functions. Which one could be the graph of ? If none of the graphs match, enter "none".

A)

B)

C)

D)

E) none

Question 69

Must the sum of two functions with horizontal asymptotes also have a horizontal asymptote?

Question 70

A 16 kg sample of a certain alloy (mixture of metals) contains 4 kg of tin and 12 kg of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose x represents the quantity of tin, in kg, the chemist adds to the sample. Let represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of x represents a quantity of tin removed from the original 16 kg sample . = ______ (kg tin)/(kg mixture). Round to 2 decimal places.

Question 71

A 14 kg sample of a certain alloy (mixture of metals) contains 5 kg of tin and 9 kg of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose x represents the quantity of tin, in kg, the chemist adds to the sample. Let represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of x represents a quantity of tin removed from the original 14 kg sample.
= ______ kg. Round to 2 decimal places.

Question 72

The infant mortality, I, in a country is related to the country's GNP (gross national product), g. Some authors (Weld and Helms, 1971) have argued that the relationship is of the form , where , k, and a are positive constants and . For , k = 4, and a = 2, the vertical intercept of the graph of I against g is at I = _____. If there is no vertical intercept, enter "none".

Question 73

Suppose that is a power function and tha t . Must
?

Question 74

Which of the following are rational functions:

A)
B)
C)
D)

Question 75

Let . Find .

Question 76

Find the long-run behavior of the function .

Question 77

Let be a polynomial of degree 5. Let be a polynomial of degree 4. Which of the following are true about ?

A)
B)
C) The graph has no horizontal asymptote.
D) The graph has a horizontal asymptote.

Question 78

The profit earned by a producer to manufacture and sell units of a good is given by . The average profit for units is given by

A) Compute
B) Interpret the economic significance of the values in part (A).
C) What trend do you notice in the values of as gets large? Explain this trend in economic terms.

Question 79

Determine the vertical and horizontal asymptotes, if they exist, of the function

Question 80

Simplify , if possible.