1. 2.3 Curve Sketching (Introduction)

2. We have four main steps for sketching curves: 1.Starting with f(x), compute f’(x) and f’’(x). 2.Locate all relative maximum and minimum points and make a partial sketch. 3.Examine concavity of f(x) and locate inflection points. 4.Consider other properties of the graph such as intercepts and complete the sketch.

3. Locating Relative Extreme Points

4. The tangent line has a slope of zero at relative maximum and relative minimum points. So, to find relative extreme points, we find values of x so that f’(x) = 0. Look for possible relative extreme points of f(x) by setting f’(x) = 0 and solving for x.

5. Is the point a relative maximum point or a relative minimum point? How can we tell? Check concavity at relative extreme point using second derivative. Examine slope of nearby points on either side using the first derivative.

6. Locating Inflection Points

7. An inflection point can only occur at a value of x for which f’’(x) = 0 because the curve is concave up when f’’(x) is positive and concave down when f’’(x) is negative. Look for possible points of inflection by setting f’’(x) = 0 and solving for x.

8. 1. Find relative extreme points…find x where f’(x) = 0 Relative extreme point at (-1, f(-1)=-8)

9. 2. Check concavity at relative extreme point, x = -1. So, the second derivative is 6 (concave up) for all x, including x = -1.

10. (-1, -8) Concave up

11. 1. Find relative extreme points…find x where f’(x) = 0

12. Relative extreme point at (3, f(3)= 3) Relative extreme point at (2, f(2)= 4)

13. 2. Check concavity at relative extreme points, x = 2, 3.

14. 3. Find inflection points, f’’(x) = 0 Inflection point at (2.5, f(2.5)=3.5)

15. (2, 4) Concave down (3, 3) Concave up (2.5, 3.5) Inflection point