Math Problem Statement
Solution
The image shows the graph of and asks to sketch the graph of under two conditions. Let's break it down step by step for both parts.
(a) Suppose is continuous on and . Sketch the graph of .
Since you're given , which represents the derivative of , we can reconstruct by integrating over the intervals.
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From to , . This means the slope of is 2, so increases linearly with a slope of 2 over this interval. Starting at , by , will have increased by , so .
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From to , , which means the slope is zero, implying is constant over this interval. Therefore, for .
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From to , , meaning the slope of is -1. The function will decrease linearly with a slope of -1 over this interval. From , by , decreases by , so .
The graph of should reflect this behavior:
- A line with slope 2 from to , starting at .
- A flat, constant section from to .
- A line with slope -1 from to , ending at .
(b) Suppose but is NOT continuous. Sketch a possible graph of . How many possibilities are there?
If is not continuous, we can introduce discontinuities, such as jumps or breaks, anywhere within the domain . However, is still given, so the behavior of between any jumps would still need to follow the slope dictated by .
One possibility:
- Keep the same slopes for the continuous parts as in part (a).
- Introduce a jump discontinuity at some point, for example, at . Instead of smoothly transitioning from the increasing part to the flat part, have a sudden jump in the value of .
There are many possibilities, depending on where you choose to introduce discontinuities. For instance:
- A jump at .
- A jump at .
- Multiple jumps at different points.
Each choice represents a different possibility.
Would you like more details on any part or a sketch of the graphs?
Here are 5 relative questions to deepen your understanding:
- What is the significance of the derivative in shaping the graph of ?
- How does the continuity of constrain the possibilities in part (b)?
- What type of discontinuities are possible when is not continuous?
- How would the graph of change if had more intervals with different slopes?
- How does the value of affect the overall shape of the graph?
Tip: When interpreting , remember that positive values indicate increasing slopes, negative values indicate decreasing slopes, and zero means the function is constant.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Continuity
Formulas
f(x) = ∫ f'(x) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grade 12 - College Level
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