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-5 10 ↑y 1 f(x) g(x) 1. Given f (x) and g(x) are both piecewise functions as shown in the graph above, if h (x) = f (x) · g(x) find h' (1). 710

-5 10 ↑y 1 f(x) g(x) 1. Given f (x) and g(x) are both piecewise functions as shown in the graph above, if h (x) = f (x) · g(x) find h' (1). 710

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need an explanation. On how they found the points on the graph, and also why they found h'(-1) even though h'(1) was asked to be found. 

### Problem: Given \( f(x) \) and \( g(x) \) are both piecewise functions as shown in the graph above, if \( h(x) = f(x) \cdot g(x) \), find \( h'(1) \). ### Solution: 1. Evaluate \( f(-1) \): \[ f(-1) = -8 \] 2. Find \( f'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)): \[ f'(-1) = 3 \] 3. Evaluate \( g(-1) \): \[ g(-1) = 4 \] 4. Find \( g'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)): \[ g'(-1) = -1 \] 5. Use the product rule to find \( h'(-1) \): \[ h'(-1) = f(-1) \cdot g'(-1) + g(-1) \cdot f'(-1) \] 6. Substitute the known values: \[ h'(-1) = (-8) \cdot (-1) + (4) \cdot (3) \] 7. Calculate \( h'(-1) \): \[ h'(-1) = 20 \] ### Graph Explanation: The graph displays two piecewise functions, \( f(x) \) in red and \( g(x) \) in blue, plotted on a Cartesian plane. The x-axis ranges from -7 to 10, and the y-axis ranges from -10 to 10. Points of intersection and changes in slopes are critical for evaluating values and derivatives at specific points.
Transcribed Image Text:### Problem: Given \( f(x) \) and \( g(x) \) are both piecewise functions as shown in the graph above, if \( h(x) = f(x) \cdot g(x) \), find \( h'(1) \). ### Solution: 1. Evaluate \( f(-1) \): \[ f(-1) = -8 \] 2. Find \( f'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)): \[ f'(-1) = 3 \] 3. Evaluate \( g(-1) \): \[ g(-1) = 4 \] 4. Find \( g'(-1) \) (the derivative is the slope of the tangent line at \( x = -1 \)): \[ g'(-1) = -1 \] 5. Use the product rule to find \( h'(-1) \): \[ h'(-1) = f(-1) \cdot g'(-1) + g(-1) \cdot f'(-1) \] 6. Substitute the known values: \[ h'(-1) = (-8) \cdot (-1) + (4) \cdot (3) \] 7. Calculate \( h'(-1) \): \[ h'(-1) = 20 \] ### Graph Explanation: The graph displays two piecewise functions, \( f(x) \) in red and \( g(x) \) in blue, plotted on a Cartesian plane. The x-axis ranges from -7 to 10, and the y-axis ranges from -10 to 10. Points of intersection and changes in slopes are critical for evaluating values and derivatives at specific points.
Expert Solution
Step 1: Given the information

The graph of the functions are given.

The aim is to find the derivative value h apostrophe open parentheses 1 close parentheses where h open parentheses x close parentheses equals f open parentheses x close parentheses times g open parentheses x close parentheses.

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