Math Problem Statement

. Use the following graph to evaluate a. f′(1) and b. f′(6). This graph shows two connected line segments: one going from (1, 0) to (4, 6) and the other going from (4, 6) to (8, 8).

Solution

To find the derivatives f(1)f'(1) and f(6)f'(6) based on the graph of the function, we'll follow these steps:

Step 1: Understanding the Graph

You mentioned that the graph consists of two connected line segments:

  1. The first line segment goes from (1,0)(1, 0) to (4,6)(4, 6).
  2. The second line segment goes from (4,6)(4, 6) to (8,8)(8, 8).

Since the function consists of line segments, its derivative will be the slope of the line at the given points.

Step 2: Finding the Slope of the Line Segments

The slope of a line segment can be calculated using the formula for the slope of a line: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points on the line.

For f(1)f'(1):

Since the point (1,0)(1, 0) lies on the first line segment, and the line segment goes from (1,0)(1, 0) to (4,6)(4, 6), we can calculate the slope of this line: f(1)=6041=63=2f'(1) = \frac{6 - 0}{4 - 1} = \frac{6}{3} = 2 Thus, f(1)=2f'(1) = 2.

For f(6)f'(6):

The point (6,y)(6, y) lies on the second line segment, which goes from (4,6)(4, 6) to (8,8)(8, 8). We calculate the slope of this line: f(6)=8684=24=0.5f'(6) = \frac{8 - 6}{8 - 4} = \frac{2}{4} = 0.5 Thus, f(6)=0.5f'(6) = 0.5.

Final Answers:

  • f(1)=2f'(1) = 2
  • f(6)=0.5f'(6) = 0.5

Would you like further details on how the slope affects the graph, or do you have any other questions? Here are some related questions to expand on the topic:

  1. How does the concept of the slope of a line relate to the derivative of a function?
  2. What happens to the slope of the function at points where the function is not differentiable?
  3. How can you determine whether a function is increasing or decreasing from its derivative?
  4. Can the derivative of a function be negative? If so, what does it signify about the graph?
  5. How would you calculate the derivative at a point where the graph has a corner or cusp?

Tip: When working with piecewise linear graphs, you can calculate the derivative by simply finding the slope of each segment of the graph.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Slopes of Line Segments

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Derivative as the slope of the tangent line

Suitable Grade Level

Grades 10-12