Derivatives and Graphs Contest

Misc

Exam 2 is coming up next Thursday (April 11).

Covers material since first exam (e.g., implicit differentiation, related rates, initial value problems, linear approximation, extreme values, shapes of graphs — roughly the material on problem sets 5 and 6).

Rules and format otherwise similar to first exam, especially open-references, open-calculator/computer, rules.

Practice exam forthcoming.

Shapes of Graphs

A Contest

In small groups, pick 2 or more derivative/2nd derivative values below and attach them to 2 or more intervals/values below, in such a manner that what you have is consistent with a differentiable (or twice-differentiable if you include 2nd derivatives) function. Then describe that function’s graph in as much detail as you can.

Derivative/2nd derivative values:

• f′(x) < 0
• f′(x) = 0
• f′(x) > 0
• f′′(x) < 0
• f′′(x) = 0
• f′′(x) > 0

Intervals/values:

• -2 < x < -1
• x = -1
• -1 < x < 0
• x = 0
• 0 < x < 1
• x = 1
• 1 < x < 2
• Or pick your own intervals/values

(Very modest) prizes for the most original and precise use of derivatives to describe a graph.

Examples:

1. Orion’s Belt
2. Kite
3. Airplane
4. “Heart of Sadness”
5. Heart

Next

Look at any other graphs people want to offer.

Then the behavior of the “ends” of graphs, i.e., behavior of functions as x approaches ±∞.

Read section 4.6.