1.7.1 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.2 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.3 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.4 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.5 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.6 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.7 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.8 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.9 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.10 Use the laws of exponents to rewrite the following (if possible). If no law of exponents applies, say so. ...
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1.7.11
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ln(1)
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1.7.12
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ln(−6.5)
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1.7.13
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.14
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.15
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.16
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.17
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.18
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.19
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.20
Use the laws of logs to rewrite the following if possible. If no law
of logs applies or the quantity is not defined, say so. ...
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1.7.21 Using the fact that ...
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1.7.22 Using the fact that ...
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1.7.23 Solve the following equations for x and check your answer. ...
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1.7.24 Solve the following equations for x and check your answer. ...
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1.7.25 Solve the following equations for x and check your answer. ...
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1.7.26 Solve the following equations for x and check your answer. ...
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1.7.27 Sketch graphs of the following exponential functions. For each, find the value of x where
it is equal to 7.0. For the increasing functions, find the doubling
time, and for the decreasing functions, find the half-life. For what
value of x is the value of the function 3.5? For what value of x is the value of the function 14.0? ...
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1.7.28 Sketch graphs of the following exponential functions. For each, find the value of x where
it is equal to 7.0. For the increasing functions, find the doubling
time, and for the decreasing functions, find the half-life. For what
value of x is the value of the function 3.5? For what value of x is the value of the function 14.0? ...
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1.7.29 Sketch graphs of the following exponential functions. For each, find the value of x where
it is equal to 7.0. For the increasing functions, find the doubling
time, and for the decreasing functions, find the half-life. For what
value of x is the value of the function 3.5? For what value of x is the value of the function 14.0? ...
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1.7.30 Sketch graphs of the following exponential functions. For each, find the value of x where
it is equal to 7.0. For the increasing functions, find the doubling
time, and for the decreasing functions, find the half-life. For what
value of x is the value of the function 3.5? For what value of x is the value of the function 14.0? ...
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1.7.31 Sketch graphs of the following updating functions over the given range and mark the equilibria. ...
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1.7.32 Sketch graphs of the following updating functions over the given range and mark the equilibria. F ( x )= ln(x) + 1 for 0 ≤ x ≤2. (Although this cannot be solved algebraically, you can guess the answer.)
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1.7.33 Find the equations of the lines after transforming the variables to create semilog or double-log plots. Suppose M(t)= ... Find the slope and intercept of ln(M(t)).
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1.7.34 Find the equations of the lines after transforming the variables to create semilog or double-log plots. Suppose L(t)= .... Find the slope and intercept of ln(L(t)).
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1.7.35 Find the equations of the lines after transforming the variables to create semilog or double-log plots. Suppose ... Find the slope and intercept of ln(M(t)) as a function of ln(S(t)).
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1.7.36 Find the equations of the lines after transforming the variables to create semilog or double-log plots. Suppose ...Find the slope and intercept of ln(L(t)) as a function of ln(K(t)).
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1.7.37 Find
the solution of each discrete-time dynamical system, express it in
exponential notation, and solve for the time when the value reaches the
given target. Sketch a graph of the solution. ...
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1.7.38 Find
the solution of each discrete-time dynamical system, express it in
exponential notation, and solve for the time when the value reaches the
given target. Sketch a graph of the solution. ...
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1.7.39 Find
the solution of each discrete-time dynamical system, express it in
exponential notation, and solve for the time when the value reaches the
given target. Sketch a graph of the solution. ...
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1.7.40 Find
the solution of each discrete-time dynamical system, express it in
exponential notation, and solve for the time when the value reaches the
given target. Sketch a graph of the solution. ...
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1.7.41 Suppose the size of an organism at time t is given by ... where ...
is the initial size. Find the time it takes for the organism to double
or quadruple in size in the following circumstances. ... = 1.0 cm and α = 1.0/day.
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1.7.42 Suppose the size of an organism at time t is given by ... where ...
is the initial size. Find the time it takes for the organism to double
or quadruple in size in the following circumstances. ... = 2.0 cm and α = 1.0/day.
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1.7.43 Suppose the size of an organism at time t is given by ... where ...
is the initial size. Find the time it takes for the organism to double
or quadruple in size in the following circumstances. ... = 2.0 cm and α = 0.1/hour.
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1.7.44 Suppose the size of an organism at time t is given by ... where ...
is the initial size. Find the time it takes for the organism to double
or quadruple in size in the following circumstances. ... = 2.0 cm and α = 0.0/hour
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1.7.45 How much is left after 50,000 years? What fraction is this of the original amount?
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1.7.46 How much is left after 100,000 years? What fraction is this of the original amount?
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1.7.47 Find the half-life of ...
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1.7.48
About
how many half-lives will occur in 50,000 years? Roughly what fraction
will be left? How does this compare with the answer of Exercise 45?
Exercise 45 How much is left after 50,000 years? What fraction is this
of the original amount?
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1.7.49 Suppose a population has a doubling time of 24 years and an initial size of 500. What is the population in 48 years?
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1.7.50 Suppose a population has a doubling time of 24 years and an initial size of 500. What is the population in 12 years?
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1.7.51 Suppose a population has a doubling time of 24 years and an initial size of 500. Find the equation for population size P(t) as a function of time.
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1.7.52 Suppose a population has a doubling time of 24 years and an initial size of 500. Find
the one-year discrete-time dynamical system for this population (figure
out the factor multiplying the population in one year).
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1.7.53 Suppose a population is dying with a half-life of 43 years. The initial size is 1600. How long will it take to reach 200?
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1.7.54 Suppose a population is dying with a half-life of 43 years. The initial size is 1600. Find the population in 86 years.
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1.7.55
Suppose a population is dying with a half-life of 43 years. The
initial size is 1600. Find the equation for population size P(t) as a function oftime.
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1.7.56 Suppose a population is dying with a half-life of 43 years. The initial size is 1600. Find
the one year discrete-time dynamical system for this population (figure
out the factor multiplying the population in one year).
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1.7.57 Plot semilog graphs of the values. The growing organism in Exercise 41 for 0≤t ≤10. Mark where the organism has doubled in size and when it has quadrupled in size.
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1.7.58 Plot semilog graphs of the values. The carbon-14 in Exercise 45 for 0 ≤ t ≤20, 000. Mark where the amount of carbon has gone down by half.
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1.7.60 Plot semilog graphs of the values. The population in Exercise 53 for 0≤t ≤100. Mark where the population has gone down by half.
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1.7.60 Plot semilog graphs of the values. The population in Exercise 53 for 0≤t ≤100. Mark where the population has gone down by half.
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1.7.61 The following pairs of measurements can be described by ordinary, semilog, and double-log graphs.
a. Graph each measurement as a function of time on both ordinary and semilog graphs.
b. Graph the second measurement as a function of the first on both ordinary and double-log graphs. The antler size A(t) in centimeters of an elk increases with age t in years according to ...and its shoulder height L(t) increases according to ....
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1.7.62 The following pairs of measurements can be described by ordinary, semilog, and double-log graphs.
a. Graph each measurement as a function of time on both ordinary and semilog graphs.
b.
Graph the second measurement as a function of the first on both
ordinary and double-log graphs. Suppose a population of viruses in an
infected person grows according to ...and that the immune response (described by the number of antibodies) increases according to ...during the first week of an infection. When will the number of antibodies equal the number of viruses?
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1.7.63 The following pairs of measurements can be described by ordinary, semilog, and double-log graphs.
a. Graph each measurement as a function of time on both ordinary and semilog graphs.
b.
Graph the second measurement as a function of the first on both
ordinary and double-log graphs. The growth of a fly in an egg can be
described allometrically (see H. F. Nijhout and D. E. Wheeler, 1996).
During growth, two imaginal disks (the first later becomes the wing and
the second becomes the haltere) expand according to ...where size is measured in ... and time is measured in days. Development takes about 5 days.
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1.7.64 The following pairs of measurements can be described by ordinary, semilog, and double-log graphs.
a. Graph each measurement as a function of time on both ordinary and semilog graphs.
b.
Graph the second measurement as a function of the first on both
ordinary and double-log graphs. While the imaginal disks are growing
(Exercise 63), the yolk of the egg is shrinking according to Y(t)= ...and Y (t).
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1.7.65 For each of the given shapes, find the constant c in the power relationship ...between the surface area S and volume V . By how much is does c exceed the value ...=4.836 for the sphere (which is in fact the minimum for any shape). For a cube with side length w.
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1.7.66 For each of the given shapes, find the constant c in the power relationship ...=4.836 for the sphere (which is in fact the minimum for any shape). For a cylinder with radius r and height 3r
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1.7.67 Many
measurements in biology are related by power functions. For each of the
following, graph the second measurement as a function of the first on
both ordinary and double-log graphs. The −3/2 law of self-thinning in plants argues that the mean weight W of surviving trees in a stand increases while their number N decreases, related by ... Suppose ...trees
start out with mass of 0.001 kg. Graph the relationship, and find how
heavy the trees would be when only 100 remain alive, and again when only
1 remains alive. Is the total mass larger or smaller than when it
started?
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1.7.68 Many
measurements in biology are related by power functions. For each of the
following, graph the second measurement as a function of the first on
both ordinary and double-log graphs. Suppose that the population density D of a species of mammal is a decreasing function of its body mass M according to the relationship ... Suppose that an unlikely 1 g mammal would have a density of ...
per hectare. What is the predicted density of species with mass of 1000
g? A species with a mass of 100 kg? According to the metabolic scaling
law (Example 1.7.24), which species will use the most energy?
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1.7.69 Use your computer to find the following. Plot the graphs to check. ...
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1.7.70 Solve for the times when the following hold. Plot the graphs to check your answer. ... ...
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1.7.71 Plot the following functions. ...
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1.7.72 Compute the following. Does this give you any idea why e is special? ... ...
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1.7.73 Suppose that the antler size A(t) in
centimeters of an elk increases with age in years during the first five
years of growth according to the exponential function ...
a. Plot A and L as functions of t on ordinary and on semilog graphs.
b. Plot A as a function of L with an ordinary and with a semilog graph.
c. Find when the antler size would exceed the shoulder height of the elk.
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