AP Calculus BC 5.3: Increasing and Decreasing
Practice AP Calculus BC questions on using the sign of the first derivative to justify where a function increases or decreases.
- Syllabus
- Effective Fall 2025
- Course
- AP Calculus BC
FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives question 1
[Maximum number: 2]
Let f be a differentiable function with f(4)=3. On the interval 0≤x≤7, the graph of f′, the derivative of f, consists of a semicircle and two line segments, as shown in the figure above.
Let g be the function defined by g(x)=f(x)-x. On what intervals, if any, is g decreasing for 0≤x≤7 ? Show the analysis that leads to your answer.
g′(x)=f′(x)−1g′(x)=f′(x)−1
1 point
f′(x)−1≤0⇒f′(x)≤1
The graph of g is decreasing on the interval 0≤x≤5 because g′(x)≤0 on this interval.
Interval with reason
1 point
Scoring notes:
- The first point can be earned for f′(x)≤1 or the equivalent, in words or symbols.
- Endpoints do not need to be included in the interval to be eligible for the second point.
Total for part (c) 2 points
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