AP/Honors Calculus Chapter 4 Applications of Derivatives Chapter 4 Applications of Derivatives

Topics 4.1 Extreme Values 4.2 Mean Value Theorem 4.3 Connecting the Graph of f to f’ and f” 4.4 Modeling and Optimization 4.5 Linearization (and Newton’s Method) 4.6 Related Rates

The First Derivative Test Given f is a continuous function and c is a point on the open interval (a, b) and β is sufficiently small such that c – β and c + β are on (a, b), then: 1) c is a local max of f if f ‘ (c – β) > 0 and f ‘ (c + β) < 0 2) c is a local min of f if f ‘ (c – β) 0 In other words, if the function increases to the left of c and decreases to the right of c, then c is a local max or if the function decreases to the left of c and increases to the right of c, then c is a local min.

Concavity Concave up Concave down

Some Definitions and Theorems Concavity A differentiable function f is: 1) CONCAVE UP on an open interval R if f ‘ is increasing on R. 2) CONCAVE DOWN on an open interval R if f ‘ is decreasing on R.

Concavity Test A twice-differentiable function f is: 1) CONCAVE UP on any interval where f ‘’ > 0 2) CONCAVE DOWN on any interval where f ‘’ < 0

Inflection Points A point c is an INFLECTION POINT of a function f if the concavity changes on either side of c. MUST NOTE: Inflection points will occur where f ‘’ = 0 or where f ‘’ = dne. However, these will only provide POSSIBLE inflection points. The concavity on either side MUST be tested.

Second Derivative Test 1) If f ‘ (c) = 0 and f ‘’ (c) < 0, then f has a Lmax at c. 2) If f ‘ (c) = 0 and f ‘’ (c) > 0, then f has a Lmin at c. Note: The second derivative test does not work if f ‘’ = 0 or if f ‘’ = dne. One must then return to the first derivative test.

Using f’ and f’’ to graph f If f(x) = x 4 – 5x ) Find where the extrema of f occur. 3) Find the intervals where f is increasing or decreasing. 4) Find the intervals where f is concave up or concave down. 1) Find where the x and y intercepts of f occur. 5) Sketch a possible graph of f.

Assignment Begin 4.3