Consider the function y = 2- 8x + 6 on the interval 8x² a) Find the slope of the secant line on this interval. m= - 3 5 2 2 b) Find the value(s) for c that satisfy the Rolle's Theorem on the given interval. C=
Consider the function y = 2- 8x + 6 on the interval 8x² a) Find the slope of the secant line on this interval. m= - 3 5 2 2 b) Find the value(s) for c that satisfy the Rolle's Theorem on the given interval. C=
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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Question
![**Function Analysis and Application of Rolle's Theorem**
Consider the function \( y = \sqrt{8x^2 - 8x + 6} \) on the interval \(\left[-\frac{3}{2}, \frac{5}{2}\right]\).
a) **Finding the Slope of the Secant Line:**
To find the slope of the secant line on this interval, calculate:
\[ m = \frac{y\left(\frac{5}{2}\right) - y\left(-\frac{3}{2}\right)}{\frac{5}{2} + \frac{3}{2}} \]
b) **Application of Rolle's Theorem:**
Find the value(s) for \( c \) that satisfy Rolle's Theorem on the given interval. According to Rolle’s Theorem, if a function is continuous on the interval \([a, b]\), differentiable on \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).
\[ c = \text{{value needed}} \]
This exercise involves calculating derivatives and verifying the conditions of continuity and differentiability for the given function within the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03532cdf-3ae3-42ec-b6da-b6f111ab5ca1%2F375191ef-f70a-47da-85dc-8f5258c7e04f%2Ftwnmsbb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Function Analysis and Application of Rolle's Theorem**
Consider the function \( y = \sqrt{8x^2 - 8x + 6} \) on the interval \(\left[-\frac{3}{2}, \frac{5}{2}\right]\).
a) **Finding the Slope of the Secant Line:**
To find the slope of the secant line on this interval, calculate:
\[ m = \frac{y\left(\frac{5}{2}\right) - y\left(-\frac{3}{2}\right)}{\frac{5}{2} + \frac{3}{2}} \]
b) **Application of Rolle's Theorem:**
Find the value(s) for \( c \) that satisfy Rolle's Theorem on the given interval. According to Rolle’s Theorem, if a function is continuous on the interval \([a, b]\), differentiable on \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).
\[ c = \text{{value needed}} \]
This exercise involves calculating derivatives and verifying the conditions of continuity and differentiability for the given function within the specified interval.
Expert Solution
Step 1
The given function is;
y=sqrt(8x2-8x+6), [-3/2,5/2].
Step by step
Solved in 3 steps with 2 images
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