Analysis of Functions using Derivatives
Lessons in this chapter
About Analysis of Functions using Derivatives
Analysis of Functions using Derivatives shows how the derivative reveals a function’s complete shape, from extrema to concavity.
Finding Extrema and Extreme Value and Mean Value Theorems cover locating a function’s maximum and minimum points and the theorems that guarantee they exist.
First Derivative Test and Concavity and Second Derivative Test give two systematic tests, using the first and second derivatives, for classifying extrema and describing a graph’s concavity.
Graphing Functions Using Calculus combines every derivative-based tool to sketch a complete, accurate graph, and Optimization closes the chapter by using derivatives to find the best possible value in a real situation, like minimum cost or maximum area.
Six lessons, each a worksheet worked in the page, run from Finding Extrema through First Derivative Test and Concavity and Second Derivative Test to Optimization. Problems are scored as you answer them, so a sign chart read the wrong way round shows up at that question. Free to practice online.
Frequently asked questions
Finding Extrema locates critical points and the maximum and minimum values a function reaches. Extreme Value and Mean Value Theorems give the guarantees behind that, First Derivative Test uses the sign of the derivative to classify each critical point, and Concavity and Second Derivative Test adds inflection points and a second way to classify. Graphing Functions Using Calculus assembles the full shape of a curve from that information, and Optimization applies it to real problems such as maximum area or minimum cost. This is Unit 5 of the AP Calculus AB course framework, Analytical Applications of Differentiation, past anything in the high school Common Core.
A function’s extrema (its maximum and minimum values) occur either at a critical point, where the derivative is zero or undefined, or at an endpoint of the interval.
The Extreme Value Theorem guarantees a continuous function on a closed interval has both a maximum and minimum value, and the Mean Value Theorem guarantees a point where the instantaneous rate of change matches the average rate of change over that interval.
The First Derivative Test classifies a critical point as a maximum, minimum, or neither by checking whether the derivative changes sign from positive to negative, negative to positive, or not at all.
Concavity describes whether a graph curves upward or downward, and the Second Derivative Test uses the sign of the second derivative at a critical point to classify it as a maximum or minimum instead.
Combine every tool: intercepts, extrema, concavity, and end behavior, to sketch a graph that’s accurate in every key feature, not just roughly shaped.