1.4.1 For
the following lines, find the slopes between the two given points by
finding the change in output divided by the change in input. What is the
ratio of the output to the input at each of the points? Which are
proportional relations? Which are increasing and which are decreasing?
Sketch a graph. y = 2x + 3, using points with x = 1 and x = 3.
Get solution
1.4.2 For
the following lines, find the slopes between the two given points by
finding the change in output divided by the change in input. What is the
ratio of the output to the input at each of the points? Which are
proportional relations? Which are increasing and which are decreasing?
Sketch a graph. z=−5w, using points with w = 1 and w = 3.
Get solution
1.4.3 For
the following lines, find the slopes between the two given points by
finding the change in output divided by the change in input. What is the
ratio of the output to the input at each of the points? Which are
proportional relations? Which are increasing and which are decreasing?
Sketch a graph. z = 5(w − 2) + 8, using points with w = 1 and w = 3.
Get solution
1.4.4 For
the following lines, find the slopes between the two given points by
finding the change in output divided by the change in input. What is the
ratio of the output to the input at each of the points? Which are
proportional relations? Which are increasing and which are decreasing?
Sketch a graph. y − 5 = −3(x + 2) − 6, using points with x = 1 and x = 3.
Get solution
1.4.5 Check
that the point indicated lies on the line and find the equation of the
line in point-slope form using the given point. Multiply out to check
that the point-slope form matches the original equation. The line f (x) = 2x + 3 and the point (2, 7).
Get solution
1.4.6 Check
that the point indicated lies on the line and find the equation of the
line in point-slope form using the given point. Multiply out to check
that the point-slope form matches the original equation. The line g(y)=−2y + 7 and the point (3, 1).
Get solution
1.4.7 Find
equations in slope-intercept form for the following lines. Sketch a
graph indicating the original point from point-slope form. The line f (x)=2(x − 1) + 3.
Get solution
1.4.8 Find
equations in slope-intercept form for the following lines. Sketch a
graph indicating the original point from point-slope form. The line g(z) = −3(z + 1) − 3.
Get solution
1.4.9 Find
equations in slope-intercept form for the following lines. Sketch a
graph indicating the original point from point-slope form. A line passing through the point (1, 6) with slope −2.
Get solution
1.4.10 Find
equations in slope-intercept form for the following lines. Sketch a
graph indicating the original point from point-slope form. A line passing through the point (−1, 6) with slope 4.
Get solution
1.4.11 Find
equations in slope-intercept form for the following lines. Sketch a
graph indicating the original point from point-slope form. A line passing through the points (1, 6) and (4, 3).
Get solution
1.4.12 Find
equations in slope-intercept form for the following lines. Sketch a
graph indicating the original point from point-slope form. A line passing through the points (6, 1) and (3, 4).
Get solution
1.4.13 Check whether the following are linear functions. ...
Get solution
1.4.14 Check whether the following are linear functions. ...
Get solution
1.4.15 Check whether the following are linear functions. P ( q )=8(3q + 2) − 6.
Get solution
1.4.16 Check whether the following are linear functions. Q ( w ) = 8(3q + 2) − 6(q + 4).
Get solution
1.4.17 Check
that the following curves do not have constant slope by computing the
slopes between the points indicated. Compare with the graphs in
Exercises 5 and 6 in Section 1.2. ... at z = 1, z = 2, and z = 4, as in Exercise 5. Find the slope between z = 1 and z = 2, and the slope between z = 2 and z =4. Exercise 5 Compute the values of the following functions at the points indicated and sketch a graph of the function. h ( z )= ... at z = 1, z = 2, and z = 4.
Get solution
1.4.18 Check
that the following curves do not have constant slope by computing the
slopes between the points indicated. Compare with the graphs in
Exercises 5 and 6 in Section 1.2. F ( r )= ...+ 5 at r =0, r =1, and r =4, as in Exercise 6. Find the slope between r = 0 and r = 1, and the slope between r =1 and r =4.
Exercises 6 Compute the values of the following functions at the
points indicated and sketch a graph of the function. F ( r )= ...+ 5 at r =0, r =1, and r =4.
Get solution
1.4.19 Solve the following equations. Check your answer by plugging in the value you found. 2x + 3=7.
Get solution
1.4.20 Solve the following equations. Check your answer by plugging in the value you found. ...
Get solution
1.4.21 Solve the following equations. Check your answer by plugging in the value you found. 2x + 3=3x + 7.
Get solution
1.4.22 Solve the following equations. Check your answer by plugging in the value you found. −3y + 5=8 + 2y.
Get solution
1.4.23 Solve the following equations. Check your answer by plugging in the value you found. 2(5(x − 1) + 3)=5(2(x − 2) + 5).
Get solution
1.4.24 Solve the following equations. Check your answer by plugging in the value you found. 2(4(x − 1) + 3) = 5(2(x − 2) + 5).
Get solution
1.4.25 Solve the following equations for the given variable, treating the other letters as constant parameters. Solve 2x + b =7 for x.
Get solution
1.4.26 Solve the following equations for the given variable, treating the other letters as constant parameters. Solve mx + 3=7 for x.
Get solution
1.4.27 Solve the following equations for the given variable, treating the other letters as constant parameters. Solve 2x + b = mx + 7 for x. Are there any values of b or m for which this has no solution?
Get solution
1.4.28 Solve the following equations for the given variable, treating the other letters as constant parameters. Solve mx + b = 3x + 7 for x. Are there any values of b or m for which this has no solution?
Get solution
1.4.29 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units. Graph
the length of a fish in inches on the horizontal axis and centimeters
on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point
corresponding to a length of 1 in.
Get solution
1.4.30 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units. Graph
the length of a fish in centimeters on the horizontal axis and inches
on the vertical axis. Use the fact that 1 in. = 2.54 cm. Mark the point
corresponding to a length of 1 in.
Get solution
1.4.31 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units. Graph
the mass of a fish in grams on the horizontal axis and its weight in
pounds on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the point corresponding to a weight of 1 lb.
Get solution
1.4.32 Most unit conversions are proportional relations. Find the slope and graph the relations between the following units. Graph
the weight of a fish in pounds on the horizontal axis and its mass in
grams on the vertical axis. Use the identity 1 lb = 453.6 g. Mark the
point corresponding to a weight of 1 lb.
Get solution
1.4.33 Not
very many functions commute with each other. The following problems ask
you to find all linear functions that commute with the given linear
function. Find all functions of the form g(x)=mx + b that commute with the function f (x) = x + 1. Can you explain your answer in words?
Get solution
1.4.34 Not
very many functions commute with each other. The following problems ask
you to find all linear functions that commute with the given linear
function. Find all functions of the form g(x)=mx + b that commute with the function f (x)= 2x. Can you explain your answer in words?
Get solution
1.4.35 Many
fundamental relations express a proportional relation between two
measurements with different dimensions. Find the slopes and the
equations of the relations between the following quantities. Volume = area × thickness. Find the volume V as a function of the area A if the thickness is 1.0 cm.
Get solution
1.4.36 Many
fundamental relations express a proportional relation between two
measurements with different dimensions. Find the slopes and the
equations of the relations between the following quantities. Volume = area × thickness. Find the volume V as a function of thickness T if the area is 7.0 ....
Get solution
1.4.37
Many
fundamental relations express a proportional relation between two
measurements with different dimensions. Find the slopes and the
equations of the relations between the following quantities. Total mass
= mass per bacterium × number of bacteria. Find total mass M as a function of the number of bacteria b if the mass per bacterium is 5.0 ×...g.
Get solution
1.4.38 Total mass = mass per bacterium × number of bacteria. Find total mass M as a function of mass per bacterium m if the total number is ....
Get solution
1.4.39 A ski slope has a slope of −0.2. You start at an altitude of 10,000 ft. Write the equation giving altitude a as a function of horizontal distance moved d.
Get solution
1.4.40 A ski slope has a slope of −0.2. You start at an altitude of 10,000 ft. Write the equation of the line in meters.
Get solution
1.4.41
A ski slope has a slope of −0.2. You start at an altitude of 10,000
ft. What will be your altitude when you have gone 2000 ft horizontally?
Get solution
1.4.42
A ski slope has a slope of −0.2. You start at an altitude of 10,000
ft. The ski run ends at an altitude of 8000 ft. How far will you have
gone horizontally?
Get solution
1.4.43 The following data give the elevation of the surface of the Great Salt Lake in Utah. ... Graph these data.
Get solution
1.4.45 The following data give the elevation of the surface of the Great Salt Lake in Utah. ... What
was the slope between 1965 and 1975? What would the surface elevation
have been in 1990 if things had continued as they began? How different
is this from the actual depth?
Get solution
1.4.45 The following data give the elevation of the surface of the Great Salt Lake in Utah. ... What
was the slope between 1965 and 1975? What would the surface elevation
have been in 1990 if things had continued as they began? How different
is this from the actual depth?
Get solution
1.4.46 The following data give the elevation of the surface of the Great Salt Lake in Utah. ... What
was the slope during between 1985 and 1995? What would the surface
elevation have been in 1965 if things had always followed this trend?
How different is this from the actual depth?
Get solution
1.4.47 Graph
the following relations between measurements of a growing plant,
checking that the points lie on a line. Find the equations in both
point-slope and slope-intercept form. ... Mass as a function of age. Find the mass on day 1.75.
Get solution
1.4.48 Graph
the following relations between measurements of a growing plant,
checking that the points lie on a line. Find the equations in both
point-slope and slope-intercept form. ... Volume as a function of age. Find the volume on day 2.75.
Get solution
1.4.49
Graph
the following relations between measurements of a growing plant,
checking that the points lie on a line. Find the equations in both
point-slope and slope-intercept form. ... Glucose production as a
function of mass. Estimate glucose production when the mass reaches
20.0 g.
Get solution
1.4.50 Graph
the following relations between measurements of a growing plant,
checking that the points lie on a line. Find the equations in both
point-slope and slope-intercept form. ... Volume
as a function of mass. Estimate the volume when the mass reaches 30.0
g. How will the density at that time compare with the density when a = 0.5?
Get solution
1.4.51 Consider the data in the following table (adapted from Parasitoids by
H. C. F. Godfray), describing the number of wasps that can develop
inside caterpillars of different weights. ... Graph these data.
Which point does not lie on the line?
Get solution
1.4.52 Consider the data in the following table (adapted from Parasitoids by
H. C. F. Godfray), describing the number of wasps that can develop
inside caterpillars of different weights. ... Find the equation
of the line connecting the first two points.
Get solution
1.4.53 Consider the data in the following table (adapted from Parasitoids by
H. C. F. Godfray), describing the number of wasps that can develop
inside caterpillars of different weights. ... How many wasps
does the function predict would develop in a caterpillar weighing 0.72
g?
Get solution
1.4.54 Consider the data in the following table (adapted from Parasitoids by H. C. F. Godfray), describing the number of wasps that can develop inside caterpillars of different weights. ... How
many wasps does the function predict would develop in a caterpillar
weighing 0.0 g? Does this make sense? How many would you really expect?
Get solution
1.4.55 The world record times for various races are decreasing at roughly linear rates (adapted from Guinness Book of Records, 1990). The
men’s Olympic record for the 1500 meters was 3:36.8 in 1972 and 3:35.9
in 1988. Find and graph the line connecting these. (Don’t forget to
convert everything into seconds.)
Get solution
1.4.56 The world record times for various races are decreasing at roughly linear rates (adapted from Guinness Book of Records,
1990). The women’s Olympic record for the 1500 meters was 4:01.4 in
1972 and 3:53.9 in 1988. Find and graph the line connecting these.
Get solution
1.4.57 The world record times for various races are decreasing at roughly linear rates (adapted from Guinness Book of Records,
1990). If things continue at this rate, when will women finish the
race in exactly no time? What might happen before that date?
Get solution
1.4.58 The world record times for various races are decreasing at roughly linear rates (adapted from Guinness Book of Records, 1990). If things continue at this rate, when will women be running this race faster than men?
Get solution
1.4.59 Try
Exercise 58 on the computer. Compute the year when the times will reach
0. Give your best guess of the times in the year 1900.
Get solution
1.4.60 Graph the ratio of temperature measured in Fahrenheit to temperature measured in Celsius for −273 ≤ ?C< 200.
What happens near ?C= 0? What happens for large and small values of ?C?
How would the results differ if the zero for Fahrenheit were changed to
match that of Celsius?
Get solution
