Math Problem Statement
How do you determine from just an interval where it is concave up or down for sinusoidal functions
Solution
To determine where a sinusoidal function (like or ) is concave up or concave down over an interval, you need to analyze the second derivative of the function. Here's how you can do it step-by-step:
1. Find the Second Derivative:
- For , the first derivative is , and the second derivative is:
- For , the first derivative is , and the second derivative is:
2. Determine Concavity:
- A function is concave up where its second derivative is positive: .
- A function is concave down where its second derivative is negative: .
3. Apply This to Sinusoidal Functions:
-
For :
- Since , concavity depends on the sign of .
- Concave Up: ⟶ , which occurs on intervals like , , etc.
- Concave Down: ⟶ , which occurs on intervals like , , etc.
-
For :
- Since , concavity depends on the sign of .
- Concave Up: ⟶ , which occurs on intervals like , , etc.
- Concave Down: ⟶ , which occurs on intervals like , , etc.
4. Analyze the Interval:
- Given a specific interval, check where or is positive or negative within that interval to determine concavity.
Example:
For on the interval :
- From to , , so is concave down.
- From to , , so is concave up.
Would you like more detailed steps on any part of this?
Here are 5 questions to explore the topic further:
- What is the concavity of on the interval ?
- How would you determine the concavity of ?
- What are the points of inflection for in the interval ?
- How do concavity and points of inflection relate to the graph's curvature?
- How does changing the amplitude of affect its concavity?
Tip: The second derivative test is a powerful tool to determine not just concavity but also to test for local maxima or minima of functions.
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Math Problem Analysis
Mathematical Concepts
Calculus
Sinusoidal Functions
Concavity
Second Derivative
Formulas
y'' = -sin(x) for y = sin(x)
y'' = -cos(x) for y = cos(x)
Theorems
Second Derivative Test
Suitable Grade Level
Grades 11-12 or Introductory College Level