12. Now, recall our discussions with respect to indefinite and definite integrals, and do the following: a) State the most general antiderivative of f(z) = =², i.e. compute the indefinite integral

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.CR: Chapter 3 Review
Problem 12CR: Determine whether each of the following statements is true or false and explain why. The derivative...
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Question 12a
10. Let f(z) be an always negative function with the property that f'(x) <0 for any x.
a) Let g(z) = (f(z))?. For what values of z will g(z) be increasing? Think about the
connection between increasing/decreasing functions and the algebraic sign of their
corresponding derivatives.
b) Let h(z) = f(f(z)). For what values of z will h(z) be decreasing?
11. Does the natural exponential function f(z) =e" admit any inflection points? Explain why
or why not.
12. Now, recall our discussions with respect to indefinite and definite integrals, and do the
following:
a) State the most general antiderivative of f(z) = z, ie. compute the indefinite integral
b) Even more generally, state
c) Now, express Va by means of a rational exponent and compute
d) What do you think happens when n=-1, in other words what is
e) Apply the Fundamental Theorem of Calculus in order to evaluate the definite integral:
You must show your work step by step in order to receive credit.
13. Finally, recall from class that the so-called hyperbolic sine and cosine functions are defined
in terms of exponential functions, ie.
e-e
sin h(2) =
and
cos A(2) =
Prove that
(cos h(z))² – (sin h(z))² = 1
As always, you must explain your work each and every step in order to receive credit.
Transcribed Image Text:10. Let f(z) be an always negative function with the property that f'(x) <0 for any x. a) Let g(z) = (f(z))?. For what values of z will g(z) be increasing? Think about the connection between increasing/decreasing functions and the algebraic sign of their corresponding derivatives. b) Let h(z) = f(f(z)). For what values of z will h(z) be decreasing? 11. Does the natural exponential function f(z) =e" admit any inflection points? Explain why or why not. 12. Now, recall our discussions with respect to indefinite and definite integrals, and do the following: a) State the most general antiderivative of f(z) = z, ie. compute the indefinite integral b) Even more generally, state c) Now, express Va by means of a rational exponent and compute d) What do you think happens when n=-1, in other words what is e) Apply the Fundamental Theorem of Calculus in order to evaluate the definite integral: You must show your work step by step in order to receive credit. 13. Finally, recall from class that the so-called hyperbolic sine and cosine functions are defined in terms of exponential functions, ie. e-e sin h(2) = and cos A(2) = Prove that (cos h(z))² – (sin h(z))² = 1 As always, you must explain your work each and every step in order to receive credit.
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ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,