Summary AP Calculus AB Unit 5: Analytical applications of Differentiation - Study Guide

AP Calculus AB Unit 5: Analytical Applications



Analytical Applications
Mean Value Theorem and Curve Sketching



1. Existence Theorems

Mean Value Theorem (MVT)

If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b)
such that:
f (b) − f (a)
f ′ (c) =
b−a
Interpretation: There is a point where the instantaneous rate of change (slope of tangent) equals
the average rate of change (slope of secant).

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an
absolute minimum on that interval.



2. First Derivative Analysis

Critical Points
Critical Points (c): Values where f ′ (c) = 0 OR f ′ (c) is undefined.

• Candidates for relative extrema.

• Always check endpoints on closed intervals!


• f ′ (x) > 0 =⇒ f (x) is Increasing.
• f ′ (x) < 0 =⇒ f (x) is Decreasing.

Derivative Test
First Derivative Test (Relative Extrema) At a critical point c:

1. If f ′ changes from + to –, then f (c) is a Relative Maximum.

2. If f ′ changes from – to +, then f (c) is a Relative Minimum.

3. If signs do not change, there is no extremum.



3. Second Derivative Analysis

The second derivative tells us about the shape (curvature) of the graph.

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