Math Problem Statement

The graph of a function f is given. Use it to sketch the graph of the derivative f'.

Solution

The image you provided illustrates a function f(x)f(x) and its derivative f(x)f'(x). Let's break it down based on what’s given:

Top Graph:

  • The graph at the top represents the function f(x)f(x).
  • Points AA, BB, CC, and PP are labeled.
    • At points AA and CC, the tangents are horizontal, meaning the slope m=0m = 0, so f(A)=0f'(A) = 0 and f(C)=0f'(C) = 0.
    • At point PP, the slope is estimated as m=32m = \frac{3}{2}, so f(P)=32f'(P) = \frac{3}{2}.

Bottom Graph:

  • The lower graph shows the derivative f(x)f'(x).
    • It highlights how f(x)f'(x) corresponds to the slope of f(x)f(x) at different points.
    • At points AA', BB', and CC', which correspond to points AA, BB, and CC on f(x)f(x), the derivative crosses the x-axis because the slope is zero at these points.

Solution Breakdown:

  • The goal is to plot f(x)f'(x) using the slopes of f(x)f(x) at several points.
  • We are instructed to estimate the derivative at specific points and then connect these points to sketch the graph of f(x)f'(x).
  • Specifically:
    1. For x=5x = 5, f(5)=32f'(5) = \frac{3}{2}.
    2. At points AA, BB, and CC, f(x)=0f'(x) = 0 (because the slopes are horizontal).
    3. Between points AA and BB, the slope of f(x)f(x) is positive, meaning f(x)>0f'(x) > 0 there.
    4. Between points BB and CC, the slope of f(x)f(x) is negative, so f(x)<0f'(x) < 0 in that region.

Let me know if you would like more details on any part or if you want me to solve the specific blanks!

Here are some relative questions:

  1. How do we interpret horizontal tangents when sketching the derivative graph?
  2. What is the relationship between the slope of f(x)f(x) and the sign of f(x)f'(x)?
  3. Why does the derivative f(x)f'(x) cross the x-axis at points AA, BB, and CC?
  4. How does the concavity of f(x)f(x) affect the behavior of f(x)f'(x)?
  5. What can we deduce about f(x)f'(x) at points where f(x)f(x) has maximum or minimum values?

Tip: Remember, the derivative graph f(x)f'(x) tells us the slope of the original function f(x)f(x) at every point, so analyzing the slope at key points helps in sketching the derivative correctly!

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Graphing

Formulas

f'(x) represents the slope of f(x)
Tangent line slopes give f'(x) values

Theorems

The Derivative Theorem
Critical Points and Inflection Points

Suitable Grade Level

Grades 10-12