Math Problem Statement
The function graphed above has:
Positive derivative on the interval(s) Incorrect
Negative derivative on the interval(s)
Solution
Let's analyze the graph to determine where the derivative of the function is positive or negative.
Positive Derivative
The derivative of a function is positive when the graph is increasing, meaning that the curve is sloping upwards. From the graph, the function is increasing on the interval:
- Approximately from to .
Negative Derivative
The derivative is negative when the function is decreasing, or the curve is sloping downwards. Based on the graph:
- The function is decreasing on two intervals: from to and from to .
Summary:
- Positive derivative: .
- Negative derivative: .
Would you like more details, or have any questions?
Here are some related questions:
- How do you calculate the derivative from a graph?
- What does the second derivative tell us about the concavity of a function?
- How can you identify critical points on this graph?
- What are inflection points, and can we find any here?
- How would the graph change if the function were shifted vertically?
Tip: The intervals of increasing or decreasing behavior correspond to where the first derivative is positive or negative, respectively.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Graph Interpretation
Formulas
First derivative test
Theorems
First derivative theorem
Increasing and decreasing functions
Suitable Grade Level
Grades 10-12
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