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This animation helps explain One-Dimentional Motion

A Review of Tangent Lines

Tangent Line Rule: Tangent lines lie above a graph when the graph is concave down and vice versa...


Inverse Functions

Definition. Let f be a function with domain D and range R. A function g with domain R and range D is an inverse function for f if, for all x in D,

y = f(x) if and only if x = g(y).
Max's and Min's

Related Rates:

Outline of steps in a related rate problem.

  1.  Recognition: Each related rate problem has a "character" of its own. The diversity of situations to be modeled is an issue that tends to inhibit students from seeing a pattern for the solution process. This is made more abstract by the fact that most related rate problems are posed as (dreaded) word problems. So the algebraic portions of the problem are imbedded within verbal descriptions that connect various components of the problem. Students need some guide posts to assist with recognition other than the fact that the title of the section is Related Rates. Several suggested guide posts that can useful are:
  • Related rate problems often involve a situation in which you are asked to calculate the rate at which one quantity changes with respect to time from the rate at which a second quantity changes with respect to time.
  • Related rate problems can be recognized because the rate of change of one or more quantities with respect to time is given and the rate of change with respect to time of another quantity is required.

Certainly the recognition process depends on "reading the problem", which is often given as step 1 in text books.

2. Read the problem: The reading of course must be accompanied by understanding. For beginning students one reading is rarely sufficient. The first reading can be used to get familiar with the general situation (the "character") of the problem. 

  • What is the physical process involved? 
  • Is a geometric figure mentioned? 
  • What are the features of the process and/or figure? 

For this reading identification of what is going on at every instant in time is a primary goal. 

A second (or later) reading can be used to focus on a geometric model of the general situation. Here is where an accompanying animation as part of a lecture can provide practice with the visualization of components changing. It is at this point that we usually tell the student to draw a diagram that is a geometric embodiment of the process described in the problem statement. This is a key interpretive step and we need to devise ways for students to practice this step. (See the gallery of animations below.)

3. Label the diagram: Here the components of the diagram are to be identified as described in the verbal description. It important that the student interpret correctly which parts of the figure are changing with respect to time and any parts that remain fixed throughout. Here again using an animation as part of an example can aid in distinguishing between these two aspects of the problem. Basically there will be two (or more) varying quantities, but additional information in the problem concerning sizes of quantities at a particular instant of time can lead to difficulty. We must emphasize that no numerical values should be assigned to any quantity varying with respect to time until after the derivative is taken. (See Step 5.)

4. Find an equation linking the varying quantities: The result of this step is completely dependent on the "character" of the problem. It is possible that two or more equations must be combined to get a single equation that relates the quantities that vary with respect to time. Certain fundamentals arise repeatedly involving right triangles, rectangles, circles, cones, and spheres.  

5. Differentiation: Before differentiating, the student should be strongly encouraged to write down the quantity to be determined. Generally this is the rate of change of one of the quantities with respect to time so students should learn that they expect to solve an equation for a derivative expression. Take the derivative implicitly with respect to time of both sides of the equation constructed in Step 4. That is,

Solve for the rate identified as the one requested in the statement of the problem. Recall that there will be other rates in the expression that result from the implicit differentiation.

6. Insert specific information: Substitute specific numerical values for quantities that varied with respect to time and any rate information that was specified. Simplify to obtain a specific expression for the rate requested in the statement of the problem.

Animation