Answered: (B) Let {fn(x)}1 = {r ln x. cos(x –…

(B) Let {fn(x)}1 = {r ln x. cos(x – 1)"} be a sequence of meaurable functions defined over [1,2]. Show that: a) {fn(x)}=1 converges to a function f(x) to be determined. 100

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.

Chapter4: Calculating The Derivative

Section4.4: Derivatives Of Exponential Functions

Problem 20E

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

(B) Let {fn(x)}1 = {rlnx. cos(x – 1)"}1
be a sequence of meaurable functions defined over [1,2]. Show that:
(a) {fn(x)}1 converges to a function f(x) to be determined.
200
n=1
n=1

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This exercise shows another way to derive the formula for the derivative of the natural logarithm function using the definition of the derivative. a. Using the definition of the derivative, show that d(lnx)dx=limh0ln(1+hx)1h. b. Eliminate h from the result in part a using the substitution h=x/m to show that d(lnx)dx=limmln[(1+1m)m]1/x c. What property should the function g have that would yield limmg(h(m))=g(limmh(m))? Assuming that the natural logarithm has this property, and using the result about (1+1/m)m from Section 2.1, show that d(lnx)dx=lne1/x=1x.