Recall: The derivative does not exist because......

1. Recall: The derivative does not exist because...... Corner What feature is common to the functions in each row, but is not shared by the functions in the other row?

2. Suppose a function is continuous but not differentiable at x=a. What analysis can we do to determine if the function has a cusp, a corner, or a vertical tangent at x=a?

3. Cusp : A point is called a cusp for f(x) if f(x) is continuous at x = a and f '(x) tends to as x approaches a from one side and tends to as x approaches a from the other side. f(x) always has a local maximum or local minimum at any point where it has a cusp. and

4. Vertical Tangent f(x) has a vertical tangent at if f(x) is continuous at x = a and

5. Corner A corner is similar to a cusp, but the function need not become infinitely steep at a corner. A point is called a corner for f(x) if f(x) is continuous at x = a and f '(x) tends to any two different values as x approaches a from the left and the ride side. f(x) always has a local maximum or local minimum at any point where it has a corner.

6. Ex.1. i) Determine all critical points. ii) Describe the type of local extrema iii) Sketch.

7. Ex. 2 i) Determine all critical points ii) Determine any local extrema. iii) Sketch.

8. Ex. 3 i) Determine all critical points. ii) Determine any local extrema iii) Any cusps, turning points or inflection points. iv) Sketch.

9. i) Sketch. ii) Determine all critical points. iii) Determine any local extrema iv) Any cusps, turning points or inflection points.

10. Practice

12. Attachments