Unit IX. Applications of Derivatives
Activities- 1 Motion Application Activities
This link includes a series of exercises to teach students about graphs, distance, velocity, and acceleration. A motion sensor device is needed for some activities.
eTools- 2 Drawing Curve on Slope Field
This applet draws a slope field and allows several choices of the function, and allows the user to draw the curve on the slope field. - 3 Optimization
This applet allows the user to try to maximize the area of a rectangle with fixed perimeter.
- 4 Related Rates Problem
This applet allows the user to explore a related rates problem by controlling the position of a runner. The applet then displays the length of the runner's shadow s as a function of the runner's position x. - 5 Slope Fields Interactive Applet
This is a quick applet for graphing slope fields for differential equations. Clicking on the field will produce the solution curve through the selected point. Tutorials- 6 Derivative of an Inverse (Polynomial Function)
This website shows how to find the derivative of the inverse function of a given polynomial.
- 7 Derivative of an Inverse (Power Function)
This website tutorial shows how to find the derivative of the inverse function of f(x) = x^n when n is even.
- 8 Derivative of an Inverse Formula Proof
This website proves a formula for the derivative of an inverse function.
- 9 L'Hopital's Rule Problem
This tutorial uses l'Hopital's rule to find the limit as x goes to infinity of (1+1/x)^x. - 10 Optimization Problems
This website contains a collection of maximization and minimization problems with demonstrations: maximize the areas of various regions, and investigate the behavior of a falling ball. - 11 Related Rates Demo
This page includes a description of solution techniques for (time-dependent) related rates problems, and a gallery of ten "movies" to illustrate different problems.
- 12 Related Rates Problem (Shortest Ladder)
This applet allows the user to find the length of the shortest ladder that can reach from the ground over the fence to the wall of the building. - 13 Slope Fields
This website defines a slope field. The Slope Fields Calculator draws integral curves in a particular slope field. Videocasts- 14 Section G: Motion (DE streaming)
The Candy-Coated World of Calculus, Program 8: Practical Applications. Section G discusses derivatives and distance, motion, velocity, and acceleration (Student Login: hcpss_student, Password: student). - 15 Velocity and Acceleration
This link contains RealMedia lessons of worked problems on velocity and acceleration.
Unit VIII. The Derivative as a Function
Activities- 1 Increasing, Decreasing, & Concavity
This activity requires students to use Geometer’s Sketchpad to analyze the graph of a function, determining its absolute and local extrema, intervals of increasing and decreasing, concavity, and inflection points. eTools- 2 Graphs of f and f"
This link allows the user to graph selected functions, their tangent at a particular point, and their derivatives.
Quizzes and Tests- 3 Graphs of f and f' Matching
This link allows the user to match graphs of functions with the graphs of their derivatives.
- 4 Graphs of f, f', and f'' Puzzles
The user matches graphs of polynomial functions with the graphs of their derivatives. Tutorials- 5 Maxima and Minima
This link contains a discussion of methods for finding the maxima and minima of differentiable functions.
- 6 Proof of Mean Value Theorem
This link contains a proof of the Mean Value Theorem. - 7 Rolle's Theorem
This link contains a proof of Rolle's Theorem. Videocasts- 8 Mean Value Theorem
This link contains RealMedia lessons of worked problems on the Mean Value Theorem. - 9 Sections A,B, AAAA C: Applications (DE streaming)
The Candy-Coated World of Calculus, Program 6: Derivative Applications. Part 2 Sections A-C discuss critical points, increasing/decreasing, and the first derivative test (Student Login: hcpss_student, Password: student). - 10 Sections D AAAA E: Concavity/Extrema Tests (DE streaming)
The Candy-Coated World of Calculus, Program 7: Figuring Out Curves. Part 2 Sections D AAAA E discuss the Second Derivative Test and Concavity (Student Login: hcpss_student, Password: student).
