1.2.1 Identify
the variables and parameters in the following situations, give the
units they might be measured in, and choose an appropriate letter or
symbol to represent each. A scientist measures the mass of fish
over the course of 100 days, and repeats the experiment at three
different levels of salinity: 0%, 2% and 5%.
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1.2.2 Identify
the variables and parameters in the following situations, give the
units they might be measured in, and choose an appropriate letter or
symbol to represent each. A scientist measures the body
temperature of bandicoots every day during the winter, and does so at
three different altitudes: 500 m, 750 m, and 1000 m.
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1.2.3 Compute the values of the following functions at the points indicated and sketch a graph of the function. f ( x )= x + 5 at x =0, x =1, and x =4.
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1.2.4 Compute the values of the following functions at the points indicated and sketch a graph of the function. g ( y )=5y at y =0, y =1, and y =4.
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1.2.5 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.
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1.2.6 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.
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1.2.7 Graph the given points and say which point does not seem to fall on the graph of a simple function. (0, −1), (1, 1), (2, 1), (3, 5), (4, 7).
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1.2.8 Graph the given points and say which point does not seem to fall on the graph of a simple function. (0, 5), (1, 10), (2, 8), (3, 6), (4, 4).
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1.2.9 Graph the given points and say which point does not seem to fall on the graph of a simple function. (0, 2), (1, 3), (2, 6), (3, 11), (4, 10).
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1.2.10 Graph the given points and say which point does not seem to fall on the graph of a simple function. (0, 45), (1, 25), (2, 12), (3, 12.5), (4, 10).
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1.2.11 Evaluate the following functions at the given algebraic arguments. f ( x )= x + 5 at x =a, x =a + 1, and x =4a.
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1.2.12 Evaluate the following functions at the given algebraic arguments. g ( y )=5y at y = ..., y =2x + 1, and y =2 − x.
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1.2.13 Evaluate the following functions at the given algebraic arguments. ...
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1.2.14 Evaluate the following functions at the given algebraic arguments. ...
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1.2.15
Sketch graphs of the following relations. Is there a more convenient
order for the arguments? A function whose argument is the name of a
state and whose value is the highest altitude in that state. ...
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1.2.16
Sketch graphs of the following relations. Is there a more convenient
order for the arguments? A function whose argument is the name of a
bird and whose value is the length of that bird. ...
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1.2.17 For each of the following sums of functions, graph each component piece. Compute the values at x =−2, x =−1, x = 0, x = 1, and x = 2 and plot the sum. f ( x )=2x + 3 and g(x)=3x − 5.
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1.2.18 For each of the following sums of functions, graph each component piece. Compute the values at x =−2, x =−1, x = 0, x = 1, and x = 2 and plot the sum. f ( x )=2x + 3 and h(x)=−3x − 12.
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1.2.19 For each of the following sums of functions, graph each component piece. Compute the values at x =−2, x =−1, x = 0, x = 1, and x = 2 and plot the sum. F ( x )= ...+ 1 and G(x)= x + 1.
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1.2.20 For each of the following sums of functions, graph each component piece. Compute the values at x =−2, x =−1, x = 0, x = 1, and x = 2 and plot the sum. F ( x )= ... + 1 and H(x)=−x + 1.
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1.2.21 For each of the following products of functions, graph each component piece. Compute the value of the product at x =−2, x =−1, x = 0, x = 1, and x = 2 and graph the result. f ( x )=2x + 3 and g(x)=3x − 5.
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1.2.22 For each of the following products of functions, graph each component piece. Compute the value of the product at x =−2, x =−1, x = 0, x = 1, and x = 2 and graph the result. f ( x )=2x + 3 and h(x)=−3x − 12.
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1.2.23 For each of the following products of functions, graph each component piece. Compute the value of the product at x =−2, x =−1, x = 0, x = 1, and x = 2 and graph the result. F ( x )= ...+ 1 and G(x)= x + 1.
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1.2.24 For each of the following products of functions, graph each component piece. Compute the value of the product at x =−2, x =−1, x = 0, x = 1, and x = 2 and graph the result. F ( x )= ... + 1 and H(x)=−x + 1.
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1.2.25 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. f ( x )=2x + 3.
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1.2.26 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. g ( x )=3x − 5.
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1.2.27 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. G ( y )=1/(2 + y) for y ≥0.
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1.2.28 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. F ( y )= ... + 1 for y ≥0.
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1.2.29 Graph each of the following functions and its inverse. Mark the given point on the graph of each function. f ( x ) = 2x + 3. Mark the point (1, f (1)) on the graphs of f and ... (based on Exercise 25). Reference Exercise 25 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. f ( x )=2x + 3.
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1.2.30 Graph each of the following functions and its inverse. Mark the given point on the graph of each function. g ( x )= 3x − 5. Mark the point (1, g(1)) on the graphs of g and ... (based on Exercise 26). Reference Exercise 26 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. g ( x )=3x − 5.
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1.2.31 Graph each of the following functions and its inverse. Mark the given point on the graph of each function. G ( y ) = 1/(2 + y). Mark the point (1, G(1)) on the graphs of G and ... (based on Exercise 27). Reference Exercise 27 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. G ( y )=1/(2 + y) for y ≥0.
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1.2.32 F ( y )= ... + 1 for y ≥0. Mark the point (1, F(1)) on the graphs of F and ... (based on Exercise 28). Reference Exercise 28 Find
the inverses of each of the following functions. In each case, compute
the output of the original function at an input of 1.0, and show that
the inverse undoes the action of the function. F ( y )= ... + 1 for y ≥0.
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1.2.33 Find the compositions of the given functions. Which pairs of functions commute? f ( x )=2x + 3 and g(x)=3x − 5.
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1.2.34 Find the compositions of the given functions. Which pairs of functions commute? f ( x )=2x + 3 and h(x)=−3x − 12.
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1.2.35 Find the compositions of the given functions. Which pairs of functions commute? F ( x )= ... + 1 and G(x)= x + 1.
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1.2.36 Find the compositions of the given functions. Which pairs of functions commute? F ( x )= ...+ 1 and H(x)=−x + 1.
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1.2.37 Describe what is happening in the graphs shown. A plot of cell volume against time in days. ...
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1.2.38 Describe what is happening in the graphs shown. A plot of a Pacific salmon population against time in years. ...
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1.2.39
Describe what is happening in the graphs shown. A plot of the average
height of a population of trees plotted against age in years. ...
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1.2.40 Describe what is happening in the graphs shown. A plot of an Internet stock price against time. ...
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1.2.41
Draw graphs based on the following descriptions. A population of
birds begins at a large value, decreases to a tiny value, and then
increases again to an intermediate value.
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1.2.42 Draw graphs based on the following descriptions. The
amount of DNA in an experiment increases rapidly from a very small
value and then levels out at a large value before declining rapidly to
0.
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1.2.43
Draw graphs based on the following descriptions. Body temperature
oscillates between high values during the day and low values at night.
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1.2.44
Draw graphs based on the following descriptions. Soil is wet at
dawn, quickly dries out and stays dry during the day, and then becomes
gradually wetter again during the night.
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1.2.45
Evaluate the following functions over the suggested range, sketch a
graph of the function, and answer the biological question. The number
of bees b found on a plant is given by b = 2 f + 1 where f is the number of flowers, ranging from 0 to about 20. Explain what might be happening when f = 0.
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1.2.46
Evaluate the following functions over the suggested range, sketch a
graph of the function, and answer the biological question. The number
of cancerous cells c as a function of radiation dose r (measured in rads) is c =r − 4 for r greater than or equal to 5, and is zero for r less than 5. r ranges from 0 to 10. What is happening at r = 5 rads?
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1.2.47 Insect development time A (in days) obeys A = ...where T represents temperature in ?C for 10≤ T ≤ 40. Which temperature leads to the more rapid development?
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1.2.48 Tree height h (in meters) follows the formula ... where a represents the age of the tree in years for 0≤ a ≤ 1000. How tall would this tree get if it lived forever?
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1.2.49 Consider the following data describing the growth of a tadpole. ... Graph length as a function of age.
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1.2.50 Consider the following data describing the growth of a tadpole. ... Graph tail length as a function of age.
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1.2.51 Consider the following data describing the growth of a tadpole. ... Graph tail length as a function of length.
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1.2.52
Consider the following data describing the growth of a tadpole.
... Graph mass as a function of length and then graph length as a
function of mass. How do the two graphs compare?
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1.2.53 The following series of functional compositions describe connections between several measurements. The number of mosquitos (M) that end up in a room is a function of how far the window is open (W , in ...)according toM(W)= 5W + 2. The number of bites (B) depends on the number of mosquitos according to B(M)= 0.5M.
Find the number of bites as a function of how far the window is open.
How many bites would you get if the window was 10 ... open?
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1.2.54
The following series of functional compositions describe connections
between several measurements. The temperature of a room (T) is a function of how far the window is open (W) according to T (W)= 40 − 0.2W. How long you sleep (S, measured in hours) is a function of the temperature according to S(T )=
.... Find how long you sleep as a function of how far the window is
open. How long would you sleep if the window was 10 ...open?
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1.2.55 The following series of functional compositions describe connections between several measurements. The number of viruses (V , measured in trillions or ...) that infect a person is a function of the degree of immunosuppression (I , the fraction of the immune system that is turned off by stress) according to V (I ) = .... The fever (F, measured in ?C) associated with an infection is a function of the number of viruses according to F(V)= 37 + 0.4V. Find fever as a function of immunosuppression. How high will the fever be if immunosuppression is complete (I = 1)?
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1.2.56
The following series of functional compositions describe connections
between several measurements. The length of an insect (L, in mm) is a function of the temperature during development (T , measured in ?C) according to L(T )= .... The volume of the insect (V , in cubic mm) is a function of the length according to V (L)= .... The mass (M in milligrams) depends on volume according to M(V)= 1.3V. Find mass as a function of temperature. How much would an insect weigh that developed at 25?C?
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1.2.57 Each
of the following measurements is the sum of two components. Find the
formula for the sum. Sketch a graph of each component and the total as
functions of time for 0 ≤ t ≤ 3. Describe each component and the sum in words. A population of bacteria consists of two types a and b. The first follows a(t)=1 + ..., and the second follows b(t)= 1 − 2t + ... where populations are measured in millions and time is measured in hours. The total population is P(t) = a(t) + b(t).
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1.2.58 Each
of the following measurements is the sum of two components. Find the
formula for the sum. Sketch a graph of each component and the total as
functions of time for 0 ≤ t ≤ 3. Describe each component and the sum in words. ...
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1.2.59 Consider the following data describing a plant. ... Graph M as a function of a. Does this function have an inverse? Could we use mass to figure out the age of the plant?
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1.2.60 Consider the following data describing a plant. ... Graph V as a function of a. Does this function have an inverse? Could we use volume to figure out the age of the plant?
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1.2.61 Consider the following data describing a plant. ... Graph G as a function of a. Does this function have an inverse? Could we use glucose production to figure out the age of the plant?
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1.2.62 Consider the following data describing a plant. ... Graph G as a function of M.
Does this function have an inverse? What is strange about it? Could we
use glucose production to figure out the mass of the plant?
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1.2.63 The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the following exercises, find the formula for the total mass, sketch graphs of P(t), W(t), and the total mass as functions of time for 0≤ t ≤ 100, and describe the results in words. The population of people is P(t)= 2.0×... + 2.0×...t, and the mass per person W(t) (in kg) is W(t) = 80 − 0.5t.
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1.2.64 The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the following exercises, find the formula for the total mass, sketch graphs of P(t), W(t), and the total mass as functions of time for 0≤ t ≤ 100, and describe the results in words. The population is ..., and the mass per person W(t) is W(t) = 80 + 0.5t.
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1.2.65 The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the following exercises, find the formula for the total mass, sketch graphs of P(t), W(t), and the total mass as functions of time for 0≤ t ≤ 100, and describe the results in words. The population is ..., and the mass per person W(t) is W(t) = 80 − 0.5t.
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1.2.66 The total mass of a population (in kg) as a function of the time, t, is the product of the number of individuals, P(t), and the mass per person, W(t) (in kg). In each of the following exercises, find the formula for the total mass, sketch graphs of P(t), W(t), and the total mass as functions of time for 0≤ t ≤ 100, and describe the results in words. The population is ...and the mass per person W(t) is ...
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1.2.67 Have your graphics calculator or computer plot the following functions. How would you describe them in words?
a. f (x)= ...for 0≤ x ≤ 20.
b. g(x)=1.5 + ...sin(x) for 0 ≤ x ≤20.
c. h(x)=sin(5x) − cos(7x) for 0 ≤ x ≤5.
d. f (x) + h(x) for 0 ≤ x ≤ 20 (using the functions in parts a and c).
e. g(x) · h(x) for 0 ≤ x ≤ 20 (using the functions in parts b and c).
f. h(x) · h(x) for 0 ≤ x ≤ 20 (using the function in part c).
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1.2.68
Have your graphics calculator or computer plot the following
functions. How would you describe them in words? Have your computer
plot the function ... for values of x between −10 and 10.
a. How would you describe the result in words? b.
Blow up the graph by changing the range to find all points where the
value of the function is 0. For example, one such value is between 1 and
2. Plot the function again for x between 1 and 2 to zoom in.
c. If you only found 2 points where h(x)= 0, blow up the region between 0 and 1 to try to find two more.
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1.2.69
Have your graphics calculator or computer plot the following
functions. How would you describe them in words? Use your computer to
find and plot the following functional compositions.
a. ( f ? g)(x) and (g ? f )(x) if f (x)=sin(x) and g(x)= ...
b. ( f ? g)(x) and (g ? f )(x) if f (x)=ex and g(x)= ...
c. ( f ? g)(x) and (g ? f )(x) if f (x)=ex and g(x)= sin(x).
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1.2.70
Have your graphics calculator or computer plot the following
functions. How would you describe them in words? Have your graphics
calculator or computer plot the following functions for −2 ≤ x ≤ 2. Do they have inverses? ... Have
your computer try to find the formula for the inverses of these
functions and plot the results. Does the machine always succeed in
finding an inverse when there is one? Does it sometimes find an inverse
when there is none?
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