2.9.1 Compute the following derivatives using the chain rule. ...
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2.9.2 Compute the following derivatives using the chain rule. ...
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2.9.3 Compute the following derivatives using the chain rule. ...
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2.9.4 Compute the following derivatives using the chain rule. ...
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2.9.5 Compute the following derivatives using the chain rule. ...
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2.9.6 Compute the following derivatives using the chain rule. ...
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2.9.7 Compute the following derivatives using the chain rule. ...
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2.9.8 Compute the following derivatives using the chain rule. ...
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2.9.9 Compute the following derivatives using the chain rule. ...
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2.9.10 Compute the following derivatives using the chain rule. ...
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2.9.11 Compute the following derivatives using the chain rule. g ( y )=ln(1 + y).
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2.9.12 Compute the following derivatives using the chain rule. ...
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2.9.13 Compute the following derivatives using the chain rule. ...
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2.9.14 Compute the following derivatives using the chain rule. ...
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2.9.15 Compute the following derivatives using the chain rule. L ( x ) = ln(ln(x)).
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2.9.16 Compute the following derivatives using the chain rule. ... Hint: Rewrite with the exponential function.
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2.9.17
Compute the derivative of each of the following functions in the two
ways given. ...with a) the quotient rule and b) the chain rule.
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2.9.18
Compute the derivative of each of the following functions in the two
ways given. ...with a) the quotient rule and b) the chain rule.
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2.9.19 Compute the derivative of each of the following functions in the two ways given. g ( x ) = ln(3x) with a) a law of logs and b) the chain rule.
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2.9.20 Compute the derivative of each of the following functions in the two ways given. ... with a) a law of logs and b) the chain rule.
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2.9.21 Compute the derivative of each of the following functions in the two ways given. ... by a) expanding the binomial and taking the derivative of the polynomial and b) applying the chain rule.
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2.9.22
Compute the derivative of each of the following functions in the two
ways given. ... by a) expanding the binomial and taking the
derivative of the polynomial and b) applying the chain rule.
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2.9.23 Compute the derivative of each of the following functions in the two ways given. ... with a) the power rule and b) the chain rule after writing F(x) using the exponential function.
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2.9.24 Compute the derivative of each of the following functions in the two ways given. ... with a) the power rule and b) the chain rule after writing F(x) using the exponential function.
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2.9.25 Find
the derivatives of the inverses of the following functions in two ways:
first by finding the inverse and taking its derivative directly, and
then by using the formula for the derivative of the inverse (Theorem
2.13). f ( x )=3x + 1.
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2.9.26 Find
the derivatives of the inverses of the following functions in two ways:
first by finding the inverse and taking its derivative directly, and
then by using the formula for the derivative of the inverse (Theorem
2.13). g ( x )=−x + 3
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2.9.27 Find
the derivatives of the inverses of the following functions in two ways:
first by finding the inverse and taking its derivative directly, and
then by using the formula for the derivative of the inverse (Theorem
2.13). ...
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2.9.28 Find
the derivatives of the inverses of the following functions in two ways:
first by finding the inverse and taking its derivative directly, and
then by using the formula for the derivative of the inverse (Theorem
2.13). ...
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2.9.29 Find
the derivatives of the inverses of the following functions in two ways:
first by finding the inverse and taking its derivative directly, and
then by using the formula for the derivative of the inverse (Theorem
2.13). ...
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2.9.30 Find
the derivatives of the inverses of the following functions in two ways:
first by finding the inverse and taking its derivative directly, and
then by using the formula for the derivative of the inverse (Theorem
2.13). ...
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2.9.31 Check that implicit differentiation gives the same formula for dy/dx as does solving for y in terms of x and then finding the derivative in the ordinary way xy =1.
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2.9.32 Check that implicit differentiation gives the same formula for dy/dx as does solving for y in terms of x and then finding the derivative in the ordinary way ...
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2.9.33 We can use laws of exponents and the chain rule to check the power rule. Write ...using the exponential function.
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2.9.34
We can use laws of exponents and the chain rule to check the power
rule. Take the derivative and then rewrite as a power function.
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2.9.35 The equation for the top half of a circle is ... In
addition to using implicit differentiation (Example 2.9.10), we can
find the slope of the tangent with the chain rule, or find it
geometrically. Find the derivative of f (x) with the chain rule.
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2.9.36 The equation for the top half of a circle is ... In
addition to using implicit differentiation (Example 2.9.10), we can
find the slope of the tangent with the chain rule, or find it
geometrically. Find the slope of the ray connecting the center of the circle at (0, 0) to the point (x, f (x)) on
the circle. Then use the fact that the tangent to a circle is
perpendicular to the ray to find the slope of the tangent. Check that it
matches the result with the chain rule.
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2.9.37 Use
implicit differentiation to find the slope of the tangent to the
following relations at the given points. What do the shapes look like? ... at all points where x =−0.5, x = 0.5, x = −1, and x = 1. What happens at x = 0?
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2.9.38 Use
implicit differentiation to find the slope of the tangent to the
following relations at the given points. What do the shapes look like? ... at all points where x =−0.5, x = 0.5, x =−1, and x = 1. What happens at x = 0?
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2.9.39 The
following functional compositions describe connections between
measurements (as in Section 1.2, Exercises 53–56). Find the derivative
of the composition using the chain rule. The number of mosquitoes (M) that end up in a room is a function of how far the window is open (W , in ...) according to M(W ) = 5W + 2. The number of bites (B) depends on the number of mosquitoes according to B(M) = 0.5M. Find the derivative of B as a function of W .
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2.9.40 The
following functional compositions describe connections between
measurements (as in Section 1.2, Exercises 53–56). Find the derivative
of the composition using the chain rule. 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 )= 14 − T/5. Find the derivative of S as a function of W .
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2.9.41 The
following functional compositions describe connections between
measurements (as in Section 1.2, Exercises 53–56). Find the derivative
of the composition using the chain rule. The number of viruses (V , measured in trillions) that infect a person is a function of the degree of immunosuppression (I , the fraction of the immune system that is turned off by medication) according to .... 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 the derivative of F as a function of I .
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2.9.42 The
following functional compositions describe connections between
measurements (as in Section 1.2, Exercises 53–56). Find the derivative
of the composition using the chain rule. The length of an insect (L, in mm) is a function of the temperature during development (T , measured in ?C) according to L(T )= 10 + T/10. The volume of the bug (V, in cubic mm) is a function of the length according to .... The mass (M in milligrams) depends on volume according to M(V ) = 1.3V . Find the derivative of M as a function of T .
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2.9.43 The amount of carbon-14 ...left t years after the death of an organism is given by ... where ...is the amount left at the time of death. Suppose ...= ...atoms/g. Find the derivative of Q(t).
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2.9.44 The amount of carbon-14 ...left t years after the death of an organism is given by ... Evaluate the derivative at t = 0 and after one and two halflives. Find and explain their relationship.
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2.9.45 The method of implicit differentiation is often applied to related rates problems involving distances. Suppose
a cheetah is dashing due north at a rate of 30 m/s toward a popular
watering hole, and a misguided gazelle is running due east toward the
same spot at a rate of 20 m/s. The cheetah begins from a distance of 120
m, while the gazelle begins from a distance of 80 m. Find equations for
the position y(t) of the cheetah, x(t) for the position of gazelle, and the distance r (t) between them, and use it to find the rate of change of the distance between them at t = 0, t = 2, and t =4.
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2.9.46 The method of implicit differentiation is often applied to related rates problems involving distances. Suppose
a cheetah is dashing due south at a rate of 30 m/s toward a popular
watering hole, and a misguided gazelle is running due east toward the
same spot at a rate of 20 m/s. The cheetah begins from a distance of 100
m, while the gazelle begins from a distance of 80 m. Find equations for
the position y(t) of the cheetah, x(t) for the position of the gazelle, and the distance r (t) between them, and use it to find the rate of change of the distance between them at t = 0, t = 2, t = 3, and t = 4. What is the minimum distance between them? Does this occur before or after the first of them has reached the watering hole?
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2.9.47 We
can describe the energy use of an organism with two tissue types by
looking at the fraction of each type rather than the total mass of each
as in Example 2.9.12, by writing ... Here p is the fraction of tissue of the first type. Suppose ...= 0.01, ... = 0.022. Suppose that at some time M = 30.0, p = 0.3, dM/dt = 2.0, and dE/dt = 0.02. Find dp/dt. Why might this be a useful calculation?
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2.9.48 We
can describe the energy use of an organism with two tissue types by
looking at the fraction of each type rather than the total mass of each
as in Example 2.9.12, by writing ... Here p is the fraction of tissue of the first type. Suppose ... = 0.022. Suppose that at some time M =30.0, dM/dt =2.0 and dE/dt = 0.01 without setting a value for p. Find dp/dt for p =0.1, p =0.5, and p =0.9. Is there are value of p for which dp/dt = 0? Can you explain what is going on?
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2.9.49
Check the given solutions to the following differential equations.
Which solutions are increasing and which are decreasing? ...
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2.9.50
Check the given solutions to the following differential equations.
Which solutions are increasing and which are decreasing? ...
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2.9.51
Check the given solutions to the following differential equations.
Which solutions are increasing and which are decreasing? ...
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2.9.52
Check the given solutions to the following differential equations.
Which solutions are increasing and which are decreasing? ...
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2.9.53 Consider again energy use E by an organism with mass M, a fraction p of which is in a tissue with parameter ... and the remaining 1 − p in a tissue with parameter ..., ... Think of M, p, and E as functions of time. Suppose that M increases exponentially over time. Experiment with different equations for p as a function of time. When does E(t) grow exponentially?
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