AP Calculus AB Unit 5: Analytical Applications of Differentiation Notes

Introduction

Derivative signs turn local slope information into interval-wide conclusions about increase, decrease, extrema, and concavity. The Mean Value and Extreme Value Theorems explain when key behaviors must occur.

First- and second-derivative tests classify candidates only under their stated conditions. A zero derivative is evidence to investigate, not a conclusion by itself.

Optimization applies the same analysis to a modeled objective on a feasible domain. Implicit relations require additional care because one input can lie on multiple branches.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 6 free-response questions. Calculator availability changes by part; the reader marks calculator decisions separately from mathematical justification.

5.1

Using the Mean Value Theorem

When must a curve contain a tangent parallel to its endpoint secant?

Short answer

If the function is continuous on the closed interval and differentiable inside it, the Mean Value Theorem guarantees at least one matching tangent slope.

Know firstContinuity · Differentiability · Average rate of change

Verify the bridge before finding the point

The endpoint secant on [a,b] has slope [f(b)−f(a)]/(b−a), an average rate across the interval. The Mean Value Theorem says a sufficiently smooth trip from one endpoint to the other must attain that same rate instantaneously somewhere inside.

The hypotheses do real work. Continuity prevents a jump across the interval, while differentiability prevents an interior corner or cusp from avoiding the required slope. Once both are stated, solve f′(c)=secant slope and retain only solutions in (a,b). The theorem guarantees existence, not uniqueness.

Formula

Formula and conditions

f'(c)=\frac{f(b)-f(a)}{b-a}

At some interior point, instantaneous and average rates agree.

Use when: f is continuous on [a,b] and differentiable on (a,b).
Do not assume: Solving the derivative equation without verifying both hypotheses does not invoke MVT.

The tangent matches the interval’s average slope

For f(x)=x² on [0,2], the secant slope is 2. The tangent at x=1 has the same slope, illustrating the point guaranteed by the Mean Value Theorem.

The interactive version of this figure is part of the full study guide.

The tangent matches the interval’s average slope: values

Case	Result
Continuity	on [0,2]
Differentiability	on $(0,2)$
Secant slope	2
Solve $f'(c)=2$	$c=1$

Use the evidence in order

State continuity on the closed interval.
State differentiability on the open interval.
Calculate the endpoint secant slope.
Solve f′(c) equal to that slope and keep interior solutions.

Keep these three ideas

MVT links an interval average with an interior instantaneous rate.
Both hypotheses must be named.
Only interior solutions count.

Keep reading Unit 5: Analytical Applications of Differentiation

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5.1 Using the Mean Value TheoremRead

5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

5.3 Determining Intervals on Which a Function Is Increasing or Decreasing

5.4 Using the First Derivative Test to Determine Relative (Local) Extrema

5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

5.6 Determining Concavity of Functions over Their Domains

5.7 Using the Second Derivative Test to Determine Extrema

5.8 Sketching Graphs of Functions and Their Derivatives

5.9 Connecting a Function, Its First Derivative, and Its Second Derivative

5.10 Introduction to Optimization Problems

5.11 Solving Optimization Problems

5.12 Exploring Behaviors of Implicit Relations

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