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f(z + h) – f(z) can be written in the form (VBz + Ch) + (/E)' where A, B, and C are constants. (Note: It's possible for one or more of these constants to be 0.) Find the constants. A = B = C = f(z + h) - f(x) Use your answer from above to find lim h0 h f(r + h) - f(x) lim h0 h Finally, find each of the following: f'(1) = S'(2) = s'(3) =

f(z + h) – f(z) can be written in the form (VBz + Ch) + (/E)' where A, B, and C are constants. (Note: It's possible for one or more of these constants to be 0.) Find the constants. A = B = C = f(z + h) - f(x) Use your answer from above to find lim h0 h f(r + h) - f(x) lim h0 h Finally, find each of the following: f'(1) = S'(2) = s'(3) =

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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### Understanding Limits and Derivatives The expression \[ \frac{f(x + h) - f(x)}{h} \] can be written in the form \[ \frac{A}{(\sqrt{Bx + Ch}) + (\sqrt{x})}, \] where \( A \), \( B \), and \( C \) are constants. (Note: It's possible for one or more of these constants to be 0.) Follow the instructions below to find the constants and proceed with the calculations. First, determine the constants: - \( A \) = [textbox for input] - \( B \) = [textbox for input] - \( C \) = [textbox for input] Next, use your result to find the limit: \[ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = [textbox for input] \] Finally, find each of the following derivatives: 1. \( f'(1) = \) [textbox for input] 2. \( f'(2) = \) [textbox for input] 3. \( f'(3) = \) [textbox for input] #### Additional Instructions for the Students: - Substitute \( h \) with approaches zero to evaluate the limit. - Use your calculated constants \( A \), \( B \), and \( C \) to simplify the expression before taking the limit. - Compute \( f'(x) \) using the definition of the derivative and then find the specific values for \( f'(1) \), \( f'(2) \), and \( f'(3) \).
Transcribed Image Text:### Understanding Limits and Derivatives The expression \[ \frac{f(x + h) - f(x)}{h} \] can be written in the form \[ \frac{A}{(\sqrt{Bx + Ch}) + (\sqrt{x})}, \] where \( A \), \( B \), and \( C \) are constants. (Note: It's possible for one or more of these constants to be 0.) Follow the instructions below to find the constants and proceed with the calculations. First, determine the constants: - \( A \) = [textbox for input] - \( B \) = [textbox for input] - \( C \) = [textbox for input] Next, use your result to find the limit: \[ \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = [textbox for input] \] Finally, find each of the following derivatives: 1. \( f'(1) = \) [textbox for input] 2. \( f'(2) = \) [textbox for input] 3. \( f'(3) = \) [textbox for input] #### Additional Instructions for the Students: - Substitute \( h \) with approaches zero to evaluate the limit. - Use your calculated constants \( A \), \( B \), and \( C \) to simplify the expression before taking the limit. - Compute \( f'(x) \) using the definition of the derivative and then find the specific values for \( f'(1) \), \( f'(2) \), and \( f'(3) \).
**Transcription of Image Text:** "Let \( f(x) = 4 + 5\sqrt{x} \). Then the expression" This snippet demonstrates the beginning of a mathematical problem involving a function defined as \( f(x) = 4 + 5 \sqrt{x} \). The function \( f(x) \) consists of a constant term (4) and a term that involves the square root of \( x \) multiplied by 5.
Transcribed Image Text:**Transcription of Image Text:** "Let \( f(x) = 4 + 5\sqrt{x} \). Then the expression" This snippet demonstrates the beginning of a mathematical problem involving a function defined as \( f(x) = 4 + 5 \sqrt{x} \). The function \( f(x) \) consists of a constant term (4) and a term that involves the square root of \( x \) multiplied by 5.
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