Model 1: Finding the Equation of f '(x) from a Graph of f(x) a. C. f(x)=2 -6 -5 -4 -3 -2 -1 f'(x)=0 y=-10x-25 -5 -4 -3 -2 -1 ya-4x-4 a. 4 y 3 1 -1 -2 -3 a. 24 18 12 1 -6 -12 6y=2x-1 2 3 4 1 2 5 6 7 h (x)=x² 3 y=8x-16 4 5 2. Now consider Graph b in Model 1. 6 $ 3. Now consider Graph c in Model 1. b. Construct Your Understanding Questions 1. Consider Graph a in Model 1. d. a. What is the slope of the line f(x) = 2? b. Is your answer consistent with the graph of its derivative, f'(x) = 0? -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 On Graph b, sketch the derivative of the function g(x) = 2x, shown there. b. Determine the equation of this derivative: g'(x) = -2 1 -1 -2 -3 1 ܂ 1 g(x) = 2x 2 3 4 5 p(x)=x² 6 12 3 4 5 6 7 3 2 1 -5-4-3-2-1₁ -2 +3 -4 -5 -6 -7 -8 -9 -10- 7 * $ 1 2 3 4 5 6 On the axes (above, right) sketch the derivative of the function shown in Graph c. To help, the equations of tangent lines to the graph at x= -5, -2, 1, and 4 are given. b. From your graph, determine the equation of this derivative: h'(x) = C. f in Graph a is the derivative of the function g in Graph b. Does the same relationship hold for g and h in Graph b and Graph c?
Model 1: Finding the Equation of f '(x) from a Graph of f(x) a. C. f(x)=2 -6 -5 -4 -3 -2 -1 f'(x)=0 y=-10x-25 -5 -4 -3 -2 -1 ya-4x-4 a. 4 y 3 1 -1 -2 -3 a. 24 18 12 1 -6 -12 6y=2x-1 2 3 4 1 2 5 6 7 h (x)=x² 3 y=8x-16 4 5 2. Now consider Graph b in Model 1. 6 $ 3. Now consider Graph c in Model 1. b. Construct Your Understanding Questions 1. Consider Graph a in Model 1. d. a. What is the slope of the line f(x) = 2? b. Is your answer consistent with the graph of its derivative, f'(x) = 0? -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 On Graph b, sketch the derivative of the function g(x) = 2x, shown there. b. Determine the equation of this derivative: g'(x) = -2 1 -1 -2 -3 1 ܂ 1 g(x) = 2x 2 3 4 5 p(x)=x² 6 12 3 4 5 6 7 3 2 1 -5-4-3-2-1₁ -2 +3 -4 -5 -6 -7 -8 -9 -10- 7 * $ 1 2 3 4 5 6 On the axes (above, right) sketch the derivative of the function shown in Graph c. To help, the equations of tangent lines to the graph at x= -5, -2, 1, and 4 are given. b. From your graph, determine the equation of this derivative: h'(x) = C. f in Graph a is the derivative of the function g in Graph b. Does the same relationship hold for g and h in Graph b and Graph c?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:**Model 1: Finding the Equation of \( f'(x) \) from a Graph of \( f(x) \)**
This illustration presents four graphs labeled a, b, c, and d. Each graph represents a different function. The aim is to determine the derivative of these functions using their respective graphs.
- **Graph a:**
- The graph displays a horizontal line where \( f(x) = 2 \).
- The derivative, \( f'(x) = 0 \), is represented by a horizontal line along the x-axis.
- **Graph b:**
- The graph represents the line \( g(x) = 2x \), which is linear and passes through the origin with a slope of 2.
- **Graph c:**
- The graph features two intersecting lines with equations \( y = 10x - 25 \) and \( g = 4x + 2 \).
- It also includes a quadratic function \( h(x) = x^2 \) and two linear functions \( y = 8x - 16 \) and \( g = 2x - 1 \).
- **Graph d:**
- The graph depicts the function \( p(x) = x^3 \), suggesting a cubic relationship.
**Construct Your Understanding Questions**
1. **Consider Graph a in Model 1.**
- a. What is the slope of the line \( f(x) = 2 \)?
- b. Is your answer consistent with the graph of its derivative, \( f'(x) = 0 \)?
2. **Now consider Graph b in Model 1.**
- a. On Graph b, sketch the derivative of the function \( g(x) = 2x \), shown there.
- b. Determine the equation of this derivative: \( g'(x) = \).
3. **Now consider Graph c in Model 1.**
- a. On the axes (above, right) sketch the derivative of the function shown in Graph c. To help, the equations of tangent lines to the graph at \( x = -5, -2, 1, \) and \( 4 \) are given.
- b. From your graph, determine the equation of this derivative: \( h'(x) = \).
- c. If \( f \) in Graph a is the derivative of the function \(
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