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 and 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:
- The first line segment goes from to .
- The second line segment goes from to .
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: where and are the coordinates of the two points on the line.
For :
Since the point lies on the first line segment, and the line segment goes from to , we can calculate the slope of this line: Thus, .
For :
The point lies on the second line segment, which goes from to . We calculate the slope of this line: Thus, .
Final Answers:
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:
- How does the concept of the slope of a line relate to the derivative of a function?
- What happens to the slope of the function at points where the function is not differentiable?
- How can you determine whether a function is increasing or decreasing from its derivative?
- Can the derivative of a function be negative? If so, what does it signify about the graph?
- 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