2.6.1 Find the derivatives of the following functions using the product rule. f ( x )=(2x + 3)(−3x + 2).
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2.6.2 Find the derivatives of the following functions using the product rule. g ( z )=(5z − 3)(z + 2).
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2.6.3 Find the derivatives of the following functions using the product rule. ...
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2.6.4 Find the derivatives of the following functions using the product rule. ...
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2.6.5 Find the derivatives of the following functions using the product rule. h ( x )=(x + 2)(2x + 3)(−3x + 2) (apply the product rule`twice).
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2.6.6 Find the derivatives of the following functions using the product rule. F ( w )= (w − 1)(2w − 1)(3w − 1) (apply the product rule twice).
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2.6.7 Find the derivatives of the following functions using the quotient rule. ...
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2.6.8 Find the derivatives of the following functions using the quotient rule. ...
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2.6.9 Find the derivatives of the following functions using the quotient rule. ...
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2.6.10 Find the derivatives of the following functions using the quotient rule. ...
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2.6.11 Find the derivatives of the following functions using the quotient rule. ...
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2.6.12 Find the derivatives of the following functions using the quotient rule. ...
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2.6.13 ... f ( x )=2x + 3 and g(x)=−3x + 2.
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2.6.14 ... ...
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2.6.15 Suppose p(x) = f (x)g(x). Test out the incorrect formula...on the following functions. f ( x )= x, g(x)= ...
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2.6.16 Suppose p(x) = f (x)g(x). Test out the incorrect formula...on the following functions. f ( x )=1, g(x)= ...
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2.6.17 Suppose that f (x ) is a positive increasing function defined for all x. Use the product rule to show that ... is also increasing.
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2.6.18 Suppose that f (x ) is a positive increasing function defined for all x. Use the quotient rule to show that ... is decreasing.
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2.6.19
For positive integer powers, it is possible to derive the power rule
with mathematical induction. The idea is to show that a formula is true
for n = 1, and then that if it is true for some particular n, it must then also be true for n + 1. Check that the power rule is true for n = 1.
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2.6.20
For positive integer powers, it is possible to derive the power rule
with mathematical induction. The idea is to show that a formula is true
for n = 1, and then that if it is true for some particular n, it must then also be true for n + 1. Use the product rule on ...to check the power rule for n = 2 using only the power rule with n = 1.
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2.6.21
For positive integer powers, it is possible to derive the power rule
with mathematical induction. The idea is to show that a formula is true
for n = 1, and then that if it is true for some particular n, it must then also be true for n + 1. Use the product rule on ...to check the power rule for n = 3 using only the power rule with n = 1 and n = 2.
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2.6.22
For positive integer powers, it is possible to derive the power rule
with mathematical induction. The idea is to show that a formula is true
for n = 1, and then that if it is true for some particular n, it must then also be true for n + 1. Assuming that the power rule is true for n, find ... using the product rule, and check that it too satisfies the power rule.
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2.6.23 The
total mass of a population is the product of the number of individuals
and the mass of each individual. In each case, time is measured in years
and mass is measured in kilograms.
a. Find the total mass as a function of time.
b. Compute the derivative.
c. Find the population, the mass of each individual, and the total mass at the time when the derivative is equal to zero.
d. Sketch a graph of the total mass over the next 100 years. The population P is ...and the weight per person W(t) is W(t)=80 − 0.5t (as in Section 1.2, Exercise 63).
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2.6.23 The
total mass of a population is the product of the number of individuals
and the mass of each individual. In each case, time is measured in years
and mass is measured in kilograms.
a. Find the total mass as a function of time.
b. Compute the derivative.
c. Find the population, the mass of each individual, and the total mass at the time when the derivative is equal to zero.
d. Sketch a graph of the total mass over the next 100 years. The population P is ...and the weight per person W(t) is W(t)=80 − 0.5t (as in Section 1.2, Exercise 63).
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2.6.25 The
total mass of a population is the product of the number of individuals
and the mass of each individual. In each case, time is measured in years
and mass is measured in kilograms.
a. Find the total mass as a function of time.
b. Compute the derivative.
c. Find the population, the mass of each individual, and the total mass at the time when the derivative is equal to zero.
d. Sketch a graph of the total mass over the next 100 years. The population P is ...and the weight per person W(t) is W(t)=80 − 0.5t (as in Section 1.2, Exercise 65).
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2.6.26 The
total mass of a population is the product of the number of individuals
and the mass of each individual. In each case, time is measured in years
and mass is measured in kilograms.
a. Find the total mass as a function of time.
b. Compute the derivative.
c. Find the population, the mass of each individual, and the total mass at the time when the derivative is equal to zero.
d. Sketch a graph of the total mass over the next 100 years. The population P is ...and the weight per person W(t) is W(t)=80 − 0.005t2 (as in Section 1.2, Exercise 66).
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2.6.27 In
each of the following situations (extending Exercise 38 in Section
2.5), the mass is the product of the density and the volume. In each
case, time is measured in days and density is measured in grams per ....
a. Find the mass as a function of time.
b. Compute the derivative.
c. Sketch a graph of the mass over the next 30 days. The above-ground volume is ...=3.0t + 20.0 and the above-ground density is ...= 1.2 − 0.01t.
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2.6.28 In
each of the following situations (extending Exercise 38 in Section
2.5), the mass is the product of the density and the volume. In each
case, time is measured in days and density is measured in grams per ....
a. Find the mass as a function of time.
b. Compute the derivative.
c. Sketch a graph of the mass over the next 30 days. The below-ground volume is ...=−1.0t + 40.0 and the below-ground density is ...= 1.8 + 0.02t.
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2.6.29 Suppose that the fraction of chicks that survive, P(N), as a function of the number N of
eggs laid, is given by the following forms (variants of the model
studied in Example 2.5.14). The total number of offspring that survive
is S(N)= N · P(N). Find the number of surviving offspring when the bird lays 1, 5, or 10 eggs. Find S ′ (N). Sketch a graph of S(N). What do you think is the best strategy for each bird? P ( N )=1 − 0.08N
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2.6.30 Suppose that the fraction of chicks that survive, P(N), as a function of the number N of
eggs laid, is given by the following forms (variants of the model
studied in Example 2.5.14). The total number of offspring that survive
is S(N)= N · P(N). Find the number of surviving offspring when the bird lays 1, 5, or 10 eggs. Find S ′ (N). Sketch a graph of S(N). What do you think is the best strategy for each bird? P ( N )=1 − 0.16N.
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2.6.31 Suppose that the fraction of chicks that survive, P(N), as a function of the number N of
eggs laid, is given by the following forms (variants of the model
studied in Example 2.5.14). The total number of offspring that survive
is S(N)= N · P(N). Find the number of surviving offspring when the bird lays 1, 5, or 10 eggs. Find S ′ (N). Sketch a graph of S(N). What do you think is the best strategy for each bird? ...
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2.6.32 Suppose that the fraction of chicks that survive, P(N), as a function of the number N of
eggs laid, is given by the following forms (variants of the model
studied in Example 2.5.14). The total number of offspring that survive
is S(N)= N · P(N). Find the number of surviving offspring when the bird lays 1, 5, or 10 eggs. Find S ′ (N). Sketch a graph of S(N). What do you think is the best strategy for each bird? ...
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2.6.33 Find the derivative of the updating function from Equation 1.10.7 ...with the following values of the parameters s and r . s = 1.2, r = 2.0.
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2.6.34 Find the derivative of the updating function from Equation 1.10.7 ...with the following values of the parameters s and r . s = 1.8, r = 0.8.
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2.6.35 Suppose that the mass M(t) of an insect (in grams) and the volume V (t) (in ...) are known functions of time (in days).
a. Find the density ρ(t) as a function of time.
b. Find the derivative of the density.
c. At what times is the density increasing?
d. Sketch a graph of the density over the first 5 days. ...
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2.6.36 Suppose that the mass M(t) of an insect (in grams) and the volume V (t) (in ...) are known functions of time (in days).
a. Find the density ρ(t) as a function of time.
b. Find the derivative of the density.
c. At what times is the density increasing?
d. Sketch a graph of the density over the first 5 days. ...
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2.6.37 In
a discrete-time dynamical system describing the growth of a population
in the absence of immigration and emigration, the final population is
the product of the initial population and the per capita production.
Represent the initial population by ... . In each case, find the final population as a function f (... ) of the initial population, find the derivative, and sketch the function. ...
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2.6.38 In
a discrete-time dynamical system describing the growth of a population
in the absence of immigration and emigration, the final population is
the product of the initial population and the per capita production.
Represent the initial population by ... . In each case, find the final population as a function f (... ) of the initial population, find the derivative, and sketch the function. ...
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2.6.39 The following steps should help you to figure out what happens to the Hill function ... for large values of n.
a. Compute the value of the function at x = 0, x = 1, and x =2.
b. Compute the derivative and evaluate at x = 0, x = 1, and x =2.
c. Sketch a graph.
d. ...can be thought of as representing a response to a stimulus of strength x.
Would the response work as a good filter, giving a small output for
inputs less than 1 and a large output for inputs greater than 1? With n = 3
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2.6.40 The following steps should help you to figure out what happens to the Hill function ... for large values of n.
a. Compute the value of the function at x = 0, x = 1, and x =2.
b. Compute the derivative and evaluate at x = 0, x = 1, and x =2.
c. Sketch a graph.
d. ...can be thought of as representing a response to a stimulus of strength x.
Would the response work as a good filter, giving a small output for
inputs less than 1 and a large output for inputs greater than 1? With n = 10
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2.6.41 Consider the functions ... for various values of n.We will compare these functions with ...
a. Plot ...and g(x) the intervals 0≤ x ≤0.5 and 0≤ x ≤0.9.
b. Take the derivative of g(x). Can you see how the derivative is related to g(x) itself? In other words, what function could you apply to the formula for g(x) to get the formula for g ′ (x)?
c. Apply this same function to .... Can you see why these functions are good approximations to g(x)? Can you see why these approximations are best for small values of x? What happens to the approximations for x near 1?
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2.6.42 Consider the function ...
a. Make one graph of u(x) and v(x) for 0 ≤ x ≤ 1, and another of r (x). Could you have guessed the shape of r (x) from looking at the graphs of u(x) and v(x)?
b. What happens at the critical point?
c. Find the exact location of the critical point ...
d. Compare ... Why are they equal?
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