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f (x) Given the functions below, define u(x) = f(x) * g(x) and v(x) = g(x)' %3D a. State the points of x where f (x) is not differentiable b. State the points of x where g(x) is not differentiable c. Evaluate u'(1) . State the theorem used in solving this. d. Evaluate v'(5). State the theorem used in solving this.

f (x) Given the functions below, define u(x) = f(x) * g(x) and v(x) = g(x)' %3D a. State the points of x where f (x) is not differentiable b. State the points of x where g(x) is not differentiable c. Evaluate u'(1) . State the theorem used in solving this. d. Evaluate v'(5). State the theorem used in solving this.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Transcription of Educational Content** **Given Functions and Definitions** For the functions below, we define: - \( u(x) = f(x) \cdot g(x) \) - \( v(x) = \frac{f(x)}{g(x)} \) **Tasks:** a. Determine the points of \( x \) where \( f(x) \) is not differentiable. b. Determine the points of \( x \) where \( g(x) \) is not differentiable. c. Evaluate \( u'(1) \). State the theorem used in solving this problem. d. Evaluate \( v'(5) \). State the theorem used in solving this problem. **Graph Explanation:** The graph illustrates two piecewise linear functions, \( f(x) \) and \( g(x) \), plotted on a coordinate system with \( x \)-axis and \( y \)-axis both spanning from 0 to 5. - The function \( f(x) \) is shown in a pink line, while \( g(x) \) is depicted in a blue line. - The graph shows changes in slope at integer values of \( x \) such as 1 and 3, indicating these points may be of interest for differentiability analysis. - The lines cross the y-axis at \( f(0) = 1 \) and \( g(0) = -3 \), and the functions intersect each other at certain points, which could potentially affect differentiability or calculations involving \( v(x) \). This explanation is provided to aid in understanding the differentiability of functions and evaluating derivatives as per the given tasks.
Transcribed Image Text:**Transcription of Educational Content** **Given Functions and Definitions** For the functions below, we define: - \( u(x) = f(x) \cdot g(x) \) - \( v(x) = \frac{f(x)}{g(x)} \) **Tasks:** a. Determine the points of \( x \) where \( f(x) \) is not differentiable. b. Determine the points of \( x \) where \( g(x) \) is not differentiable. c. Evaluate \( u'(1) \). State the theorem used in solving this problem. d. Evaluate \( v'(5) \). State the theorem used in solving this problem. **Graph Explanation:** The graph illustrates two piecewise linear functions, \( f(x) \) and \( g(x) \), plotted on a coordinate system with \( x \)-axis and \( y \)-axis both spanning from 0 to 5. - The function \( f(x) \) is shown in a pink line, while \( g(x) \) is depicted in a blue line. - The graph shows changes in slope at integer values of \( x \) such as 1 and 3, indicating these points may be of interest for differentiability analysis. - The lines cross the y-axis at \( f(0) = 1 \) and \( g(0) = -3 \), and the functions intersect each other at certain points, which could potentially affect differentiability or calculations involving \( v(x) \). This explanation is provided to aid in understanding the differentiability of functions and evaluating derivatives as per the given tasks.
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as we can see the function f(x) has a very sharp turn at x=0 and x=2

so the function f(x) is not differentiable at 

{\color{Red} x=0, x=2}...............answer(a)

.

as we can see the function g(x) has a very sharp turn at x=2

so the function g(x) is not differentiable at 

{\color{Red} x =2}...............answer(b)

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