1. Collect homework.
2. Definitions:
Critical points occur when f'(x) = 0 or when f(x) is not differentiable. (Slopes are either zero or infinite - e.g., parabolas or cusps).
Stationary points are those where
f'(x) = 0.
The relative extrema of a function (if any) occur at critical points. But this does not mean that relative extrema occur at every critical point (they could be points of inflection, for example).
Relative maximums and minimums may occur when the first derivative is not zero (like a cusp).
3. To Find relative max/min:
(A) Take the first derivative and set equal to zero. Solve for x. These are critical points.
(B) 1st derivative test:
Test points on each side of the critical points found in (A) by substituting in the first derivative.
If the value of the derivative of the point to the left of the critical point is positive and the value of the derivative for the point to the right is negative, then the critical point is a relative maximum.
If the value of the derivative of the point to the left of the critical point is negative and the value of the derivative for the point to the right is positive, then the critical point is a relative minimum.
If the values of the derivative of the points to the left and the right of the critical point are the same (i.e., both positive or both negative), then the critical point is a point of inflection.
(C) 2nd derivative test:
Take the first derivative and set it equal to zero to solve for critical points.
Take the second derivative of the function.
Substitute the critical point in the second derivative.
If this value is negative, the critical point is a relative maximum.
If this value is positive, the critical point is a relative minimum.
If this value is zero, the critical point is a possible point of inflection. Test points on either side of the critical point by substituting them into the second derivative to verify that the concavity changed.
4. Example:
5. Look at the Top Ten List of Pick Up Lines for Math Chicks (in math humor) -
#4. My love for you is like the slope of a concave up function because it is always increasing.
Pick-Up Lines to use on Math Chicks
6. Distribute a copy of the Curve Sketching Techniques and the Curve Sketching Worksheet.
Curve Sketching Instructions