Math Problem Statement
Match the first and second derivative of each function below with its sign or value.
Solution
To match the first and second derivatives of each function with their signs or values, let's analyze each graph and determine their behaviors:
Graph Analysis:
-
Graph of :
- This is a linear function, so the slope (first derivative) is constant and negative.
- The second derivative, since this is a linear function, is zero.
First derivative (): Negative (constant).
Second derivative (): Zero. -
Graph of :
- This is an exponential growth function, so the slope (first derivative) is always positive and increasing.
- The second derivative, which represents the rate of change of the slope, is also positive.
First derivative (): Positive (increasing).
Second derivative (): Positive. -
Graph of :
- This resembles a logistic or asymptotic growth curve. The slope (first derivative) starts positive and decreases over time, approaching zero as the graph levels out.
- The second derivative is initially positive (as the slope is increasing), but it becomes negative (as the slope starts to decrease).
First derivative (): Positive (decreasing).
Second derivative (): Changes sign: Positive initially, then becomes negative.
Matching Signs/Values:
- For :
- : Negative
- : Zero
- For :
- : Positive
- : Positive
- For :
- : Positive (decreasing)
- : Positive initially, then negative
Would you like a table summarizing these or further clarifications? Let me know!
Here are 5 related questions for deeper exploration:
- What is the general form of a logistic function, and how does it relate to ?
- How can we calculate the first and second derivatives for an exponential function like ?
- Why is the second derivative zero for linear functions like ?
- What real-world scenarios might each of these functions model?
- How can you interpret the change in sign of the second derivative in ?
Tip: For understanding second derivatives, think of them as measuring "acceleration" or "curvature" of the graph!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Derivatives
Graph Analysis
Exponential Functions
Linear Functions
Logistic Growth
Formulas
First derivative f'(x)
Second derivative f''(x)
Theorems
Derivative of a linear function is constant
Derivative of an exponential function is proportional to itself
Second derivative test for concavity
Suitable Grade Level
Grades 11-12