2.8.1 Compute the first and second derivatives of the following functions. ...
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2.8.2 Compute the first and second derivatives of the following functions. ...
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2.8.3 Compute the first and second derivatives of the following functions. ...
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2.8.4 Compute the first and second derivatives of the following functions. ...
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2.8.5 Compute the first and second derivatives of the following functions. ...
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2.8.6 Compute the first and second derivatives of the following functions. ...
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2.8.7 Compute the first and second derivatives of the following functions. ...
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2.8.8 Compute the first and second derivatives of the following functions. ...
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2.8.9 Compute the first and second derivatives of the following functions. f ( x )= x + 4 ln(x).
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2.8.10 Compute the first and second derivatives of the following functions. ...
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2.8.11 Compute the first and second derivatives of the following functions. g ( z )=(z + 4) ln(z).
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2.8.12 Compute the first and second derivatives of the following functions. ...
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2.8.13 Compute the first and second derivatives of the following functions. ...
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2.8.14 Compute the first and second derivatives of the following functions. ...
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2.8.15 Compute the first and second derivatives of the following functions. ...
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2.8.16 Compute the first and second derivatives of the following functions. ...
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2.8.17 Compute the first and second derivatives of the following functions. ...
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2.8.18 Compute the first and second derivatives of the following functions. ...
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2.8.19 Use
the first and second derivatives to sketch graphs of the following
functions on the given domains. Identify regions where the function is
increasing, and where it is concave up. ...
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2.8.20 Use
the first and second derivatives to sketch graphs of the following
functions on the given domains. Identify regions where the function is
increasing, and where it is concave up. ...
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2.8.20 Use
the first and second derivatives to sketch graphs of the following
functions on the given domains. Identify regions where the function is
increasing, and where it is concave up. ...
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2.8.22 Use
the first and second derivatives to sketch graphs of the following
functions on the given domains. Identify regions where the function is
increasing, and where it is concave up. ...
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2.8.23 Use
the first and second derivatives to sketch graphs of the following
functions on the given domains. Identify regions where the function is
increasing, and where it is concave up. ...
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2.8.24 Use
the first and second derivatives to sketch graphs of the following
functions on the given domains. Identify regions where the function is
increasing, and where it is concave up. M ( x )=(x + 2) ln(x) for 1≤ x ≤3.
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2.8.25 We can return to the definition to find derivatives of other exponential functions. For each of the following
a. Write down the definition of the derivative of this function.
b. Simplify with a law of exponents and factor.
c. Estimate the limit by plugging in small values of h.
d. Exponentiate the limit to figure out what it is. f ( x ) = 5x .
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2.8.26 We can return to the definition to find derivatives of other exponential functions. For each of the following
a. Write down the definition of the derivative of this function.
b. Simplify with a law of exponents and factor.
c. Estimate the limit by plugging in small values of h.
d. Exponentiate the limit to figure out what it is. ... After following the steps, use the fact that g(x) = ...to find the derivative with the pro duct rule.
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2.8.27
Polynomials form a useful set of functions in part because the
derivative of a polynomial is another polynomial. Another set of
functions with this useful property is called the generalized
polynomials,
formed as products of polynomials and exponential functions. One simple
group of generalized polynomials are the products of linear functions
with the exponential function, taking the form ... for various
values of a and b. We will call these generalized firstorder polynomials. Set a = 1 and b = 1. Find the first and second derivatives of h(x ).
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2.8.28
Polynomials form a useful set of functions in part because the
derivative of a polynomial is another polynomial. Another set of
functions with this useful property is called the generalized
polynomials,
formed as products of polynomials and exponential functions. One simple
group of generalized polynomials are the products of linear functions
with the exponential function, taking the form ... for various
values of a and b. We will call these generalized firstorder polynomials. Use the results of Exercise 27 to guess the tenth derivative of h(x) when a = 1 and b = 1.
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2.8.29
Polynomials form a useful set of functions in part because the
derivative of a polynomial is another polynomial. Another set of
functions with this useful property is called the generalized
polynomials,
formed as products of polynomials and exponential functions. One simple
group of generalized polynomials are the products of linear functions
with the exponential function, taking the form ... for various
values of a and b. We will call these generalized firstorder polynomials. Find a generalized first-order polynomial such that h(1) = 0. Where is the critical point? Where is the point of inflection?
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2.8.30 Let ...be the solution of the equation .... Show that the critical point of a generalized first-order polynomial is x∗ − 1 and the point of inflection is ....
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2.8.31 We
can return to the definition to figure out the derivative of the
natural log. We will first find the derivative at different values of x. For each value of x,
a. Write down the definition of the derivative for ln(x).
b. Plug in some small values of h to guess the limit.
c. Check that your answer matches the value of the derivative according to the formula. x =1
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2.8.32 We
can return to the definition to figure out the derivative of the
natural log. We will first find the derivative at different values of x. For each value of x,
a. Write down the definition of the derivative for ln(x).
b. Plug in some small values of h to guess the limit.
c. Check that your answer matches the value of the derivative according to the formula. x =2
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2.8.33 Instead of plugging different values of x into the definition of the derivative for ln(x),
as in Exercises 31 and 32, we can use a law of logs to find the
derivative in general. Write down the definition of the derivative at x = 2, and use a law of logs to try to convert the limit into something that looks like the limit at x = 1, as in Exercise 31. Hint: Substitute a new variable for h/2.
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2.8.34 Instead of plugging different values of x into the definition of the derivative for ln(x),
as in Exercises 31 and 32, we can use a law of logs to find the
derivative in general. Write down the definition of the derivative for
general x, and use a law of logs to try to convert the limit into something that looks like the limit at x = 1, as in Exercise 31. Hint: Substitute a new variable for h/x.
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2.8.35 Suppose a population of bacteria grows according to .... Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero? ...
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2.8.36 Suppose a population of bacteria grows according to .... Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero? m ( t )=1 − t.
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2.8.37 Suppose a population of bacteria grows according to .... Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero? ...
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2.8.38 Suppose a population of bacteria grows according to .... Find the first and second derivative to graph the total mass when the mass per individual m(t) has the following forms. Is the total mass ever greater than it is at t = 0? When does the total mass reach zero? ...
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2.8.39
Find the first and second derivatives of the following functions
(related to the gamma distribution) and sketch graphs for 0≤ x ≤ 2. ...
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2.8.40
Find the first and second derivatives of the following functions
(related to the gamma distribution) and sketch graphs for 0≤ x ≤ 2. ...
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2.8.41 The
following are differential equations that could describe a bacterial
population. For each, describe in words what the equation says and check
that the given solution works. Say whether the solution is an
increasing or decreasing function. ...
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2.8.41 The
following are differential equations that could describe a bacterial
population. For each, describe in words what the equation says and check
that the given solution works. Say whether the solution is an
increasing or decreasing function. ...
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2.8.43 The
following are differential equations that could describe a bacterial
population. For each, describe in words what the equation says and check
that the given solution works. Say whether the solution is an
increasing or decreasing function. ...
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2.8.44 The
following are differential equations that could describe a bacterial
population. For each, describe in words what the equation says and check
that the given solution works. Say whether the solution is an
increasing or decreasing function. ...
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2.8.45 Consider again the function ... describing the gamma distribution.
a. Find the critical point in general.
b. Find the point or points of inflection. What happens when n <1?
c. Graph this function for n = 0.5, n = 2, and n = 5. What happens for very large values of n?
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2.8.46
Consider the generalized polynomial ... As defined in
Exercise 27, a generalized polynomial is formed by multiplying and
adding polynomials and exponential functions.
a. Find all critical points and points of inflection of R(x) for −1≤ x ≤ 1.
b. Find the fifth derivative of R(x). Does it look any simpler than R(x) itself? Compare with what happens when you take many derivatives of a polynomial or of the exponential function.
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