Flipped Math
Calculus AB/BC - Determining Concavity of Functions over Their Domains
Time to take a second look at derivatives finding concavity. While watching the video, learners find out the definition of concavity. Individuals see how to determine whether an interval is concave up or concave down using graphs and the...
College Board
2012 AP® Calculus AB Free-Response Questions
What's on the test? Released items provide insight into the AP Calculus test. The six items are split into two groups, two calculator items and four non-calculator items. Resources allow teachers to see what kinds of answers test takers...
College Board
2010 AP® Calculus AB Free-Response Questions
Is this going to be on the test? The resource contains the AP Calculus AB free-response questions that were on the 2010 test. Pupils use the six questions to prepare for their tests. Teachers use the included response resources to...
Flipped Math
Calculus AB/BC Mid-Unit 6 Review: Integration and Accumulation of Change
Accumulate the content so far. Learners review the content from the first five lessons dealing with accumulation functions. The eight problems go over finding maximums, minimums, concavity, points of inflection, and intervals of increase...
Flipped Math
Calculus AB/BC - Interpreting the Behavior of Accumulation Functions Involving Area
Accumulation functions should behave like other functions. Using content from Unit 5, pupils analyze the first and second derivative to determine the behavior of an accumulation function. Learners find out how to calculate the relative...
Flipped Math
Calculus AB/BC Unit 5 Review: Analytical Applications of Differentiation
Time to put it all together in one sheet—well five. Pupils review all the topics presented in the resources for Analytical Applications of Differentiation. The five pages summarize the concepts with explanations and examples.
Flipped Math
Calculus AB/BC - Sketching Graphs of Functions and Their Derivatives
Find deeper meaning in graphs. Pupils use the knowledge gained from the previous sections in the unit to sketch graphs of a function's derivative. Learners also see how to sketch a graph of a function given the graph of its derivatives....
Flipped Math
Calculus AB/BC - Connecting a Function, Its First Derivative, and Its Second Derivative
Uncover what the first and second derivatives reveal about the function. Pupils move beyond sketching graphs to determine what derivatives tell about the function itself. The video reminds learners about particle motion and how to...
Houston Area Calculus Teachers
Cubic Polynomial
Birthdays are a time for celebration, and now they're a time for calculation! An AP calculus lesson uniquely explores relationships between a polynomial function and its derivative. Learners create a cubic function based on their...
Curated OER
Analyze This Function
In this derivatives worksheet, learners solve 11 short answer and graphing problems. Students find inflection points, increasing and decreasing intervals, maximums and minimums, and graph an exponential derivative. Learners use the...
Curated OER
From F to F
In this derivatives worksheet, students solve 16 short answer problems. Students use a graph of a derivative to determine local maximums and minimums, concavity, inflection points, etc.
Curated OER
Quartic Inflection
Learners use their TI-89 calculator to find properties of the quartic polynomials. The instructional activity includes detailed instructions along with screen shots of the graphing calculator display.
Texas Instruments
Quartic Regions
Young scholars explore quartic functions in this calculus lesson. They investigate an application of derivatives and definite integrals, then explore the question of which of the "three bumps" of a quartic is the largest.
Curated OER
Intervals
In this intervals worksheet, students solve and complete 8 various types of problems. First, they find the intervals of x for each curve. Then, students sketch a graph of each showing all maximum and minimum points and all inflection...
Curated OER
Worksheet 11: Derivative
For this derivative worksheet, students compute the relative maximum, relative minimum and inflection points of a function. They determine the derivative of a function and find the circumference of inscribed circles. This two-page...
Curated OER
Taylor Polynomials, Minima, Maxima and Inflection Points
In this Taylor polynomial worksheet, learners answer 3 problems regarding quadratic polynomials, 2nd degree Taylor polynomials, and cubic polynomials.
Curated OER
Rate Equations: Equilibrium and Inflection Values
In this rate equation, learners are given 7 short-answer problems regarding functions, maximum and minimum values, equilibrium values, and inflection values.
Harvey Mudd College
Harvey Mudd College: Concavity and the Second Derivative Test
An excellent tutorial explaining concavity and the second derivative test, including a labeled diagram, examples, and links to a quiz.
Texas Instruments
Texas Instruments: Using Traffic Data to Reinforce Inflection Points in Calculus
Real data from the morning rush on the Chicago expressways is the basis for a piece-wise regression. The students' success depends on their understanding of inflection points. Students need to use the Select command to split the data...
Mathematics Archives
Math Archives: Visual Calculus: Graph a Function, Its Second Derivative
This online tutorial site shows that the concavity of a function is related to the second derivative of the function by using an online graphing system. Inflection points are also briefly discussed.
University of California
U of Ca, Davis: Graphing Functions Using Derivatives
Learn about the graphing of functions through useage of the first and second derivatives.
![Houghton Mifflin: Math Graphs: Derivative and Second Derivative [Pdf] Unknown Type](https://d15y2dacu3jp90.cloudfront.net/images/attachment_defaults/resource/large/FPO-knovation.png "Houghton Mifflin: Math Graphs: Derivative and Second Derivative [Pdf] Unknown Type")
Math Graphs
Houghton Mifflin: Math Graphs: Derivative and Second Derivative [Pdf]
Learners use the graph of the derivative to approximate the slope, relative minimum, concavity, and inflection points. The exercise is available in PDF format.