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Interactive
CK-12 Foundation

Definition of Inverse Functions

For Students 11th - Higher Ed Standards
Investigate the definition of inverse functions graphically. Using the interactive, scholars create a graph of a function and view its resulting inverse. They then determine whether a group of functions have inverses that are also...
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Interactive
CK-12 Foundation

Differential Equations Representing Growth and Decay: Rice Legend

For Students 11th - Higher Ed
The legend of a wise man who asks a king for rice as a reward presents a context to study exponential solutions to differential equations. Pupils move quantities of rice to a chessboard and calculate the amount of rice for each day. To...
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Interactive
CK-12 Foundation

Length of a Plane Curve

For Students 11th - Higher Ed
Challenge your class to use straight lines when estimating the length of a curve. An engaging interactive allows individuals to place line segments one after another along the arc. Learners determine that the more lines used, the better...
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Interactive
CK-12 Foundation

Volume by Cross Section: Volume of the Cone

For Students 11th - Higher Ed
Discover another way to find the volume of a cone. Pupils explore how the area of a cross section changes as it moves through a cone. The interactive uses that knowledge to develop the integral to use to find the volume of the cone....
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Interactive
CK-12 Foundation

Method of Cylindrical Shells

For Students 11th - Higher Ed
Approximate the volume of a solid of revolution. Using a method similar to approximating the area under a curve, pupils investigate the volume of a solid of revolution. The learners use a given definite integral to find the volume of...
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Interactive
CK-12 Foundation

Area Sums: Estimation with Rectangles

For Students 11th - Higher Ed
The more rectangles, the better the estimate. Using the interactive, pupils explore estimating the area under a curve using left-hand sums. Learners respond to challenge questions on how to get better estimates using the same technique.
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Interactive
CK-12 Foundation

Analyzing the Graphs of Functions: Analyzing a Rational Function

For Students 11th - Higher Ed
Shift the function and transform the key features of the graph. By translating the graph of the rational function, class members find out how the key features alter. Pupils determine the domain, range, asymptotes, and intervals of...
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Interactive
CK-12 Foundation

Antiderivative: Piecing it Together

For Students 11th - Higher Ed
Build a function backwards. Given a graph of the derivative of a function, pupils piece together a graph of the original function, the antiderivative. Learners use their graphs and the graphs of the derivatives to answer questions about...
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Interactive
CK-12 Foundation

Newton's Method

For Students 11th - Higher Ed
Does the accuracy of the first guess make a difference down the line? Learners investigate the effects of the iterative process of finding roots, using Newton's Method. By moving the initial guess of a root on a graph, pupils observe the...
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Interactive
CK-12 Foundation

Absolute Extrema and Optimization: Building the Biggest Box

For Students 11th - Higher Ed
Optimally, you want the largest box. Given a square piece of box material, pupils determine the size of congruent squares to cut out of the corners to create a box with the greatest volume. Learners determine the equation of the volume...
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Interactive
CK-12 Foundation

Absolute Versus Local Extrema

For Students 11th - Higher Ed
Get the class to take an extreme look at functions. The interactive presents a function on a closed interval with a movable tangent line. Using the given function, pupils determine the extrema, critical points, and points of inflection. 
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Interactive
CK-12 Foundation

Derivatives with Mean Value Theorem and Rolle's Theorem: Maxima and Minima

For Students 11th - Higher Ed
Rolle with the mean values and derivatives. Scholars complete the statements of Rolle's Theorem and the Mean Value Theorem. Using the interactive to illustrate scenarios, pupils respond to questions concerning the theorems in terms of...
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Interactive
CK-12 Foundation

Related Rates of Car Speeds

For Students 11th - Higher Ed
Speed up your pupils' understanding of derivatives. Two cars travel in perpendicular distances to each other. With the aid of the interactive, learners visualize the situation. Pupils use the derivative to calculate the instantaneous...
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Interactive
CK-12 Foundation

Higher Order Derivatives—Acceleration and Jerk

For Students 11th - Higher Ed
Accelerate your class through finding the second derivative. Using a bank of equations, pupils determine the equations for distance, velocity, and acceleration and their associated function notations. With the equations, learners answer...
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Interactive
CK-12 Foundation

Chain Rule

For Students 11th - Higher Ed
Keep it on the chains. Pupils work through the steps in the chain rule using the interactive as a guide. Learners identify the two composed functions and their associated derivatives from a set of choices. They continue on to build the...
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Interactive
CK-12 Foundation

Constant, Identity, and Power Rules: Power Rule

For Students 11th - Higher Ed
Learn to gain power in finding derivatives. The resource introduces the power rule. Using the sliders in the interactive, pupils create polynomial functions and find their derivatives. Given the derivative of a polynomial, learners...
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Interactive
CK-12 Foundation

Derivatives of Trigonometric Functions: Derivative of sin(x)

For Students 11th - Higher Ed
Graphically find the derivative of sin(x). Using the interactive, pupils graph the slope of the tangent line to the sine function. Class members use the resulting graph to determine the derivative of the sine function. They verify their...
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Interactive
CK-12 Foundation

Tangent Line Approximation: Estimating Square Roots

For Students 11th - Higher Ed
Estimating a square root is as easy as evaluating a linear equation. Using the derivative of the square root function, pupils calculate an estimation of square roots. Class members determine the equation of the tangent line at the value...
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Interactive
CK-12 Foundation

Continuity of a Function: Continuity

For Students 10th - Higher Ed
Does the point continually move along the graph? Pupils drag a line across two functions to determine whether they are continuous or not. They answer questions about the properties of continuous and discontinuous functions. Using their...
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Interactive
CK-12 Foundation

Linear, Quadratic, and Cubic Models: The Box Model

For Students 10th - 12th Standards
Models make math manageable. Individuals investigate a cubic function that models the volume of a cube through the interactive.
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Interactive
CK-12 Foundation

Newton's Method

For Students 11th - Higher Ed
Use an iterative process to find an approximation of a square root. Pupils use the interactive to find an approximation to find the positive root of a quadratic function with Newton's method. With the graphs, learners position the...
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Interactive
CK-12 Foundation

Infinite Limit Type: Asymptotes and End Behavior Question

For Students 11th - Higher Ed Standards
There are an infinite number of reasons to use the resource. Scholars drag vertical and horizontal lines to the graph of a rational function to identify all asymptotes. They investigate the connection between asymptotes and limits to...
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Interactive
CK-12 Foundation

Infinite Limit Type: Properties of Limits

For Students 11th - Higher Ed Standards
Limits can provide some valuable information about graphs. A slider interactive lets learners see the behavior of a graph around asymptotes. They investigate relationships between limits and asymptotes.
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Interactive
CK-12 Foundation

One-Sided Limit Type: One-Sided Limits

For Students 11th - Higher Ed
Not everything that's one-sided is bad. A slider interactive aids learners in investigating one-sided limits from graphs. A set of challenge questions assesses their understanding of the relationship between one- and two-sided limits.