AP/Honors CalculusAP/Honors CalculusChapter 4

Applications of DerivativesChapter 4

Applications of Derivatives

Topics4.1 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 TestGiven 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) < 02) c is a local min of f if f ‘ (c – β) < 0 and f ‘ (c + β) > 0In 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.

ConcavityConcave up

Concave down

Some Definitions and Theorems

ConcavityA 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 TestA twice-differentiable function f is:

1) CONCAVE UP on any interval where f ‘’ > 0

2) CONCAVE DOWN on any interval where f ‘’ < 0

Inflection PointsA point c is an INFLECTION POINT of a function f if the concavity changes on either side of c.

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 MUSTMUST 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 Note: The second derivative test does not work if ff ‘’‘’ = 0 or if = 0 or if ff ‘’‘’ = dne. One must then return to the = dne. One must then return to the first derivative test.first derivative test.

Using f’ and f’’ to graph f

If f(x) = x4 – 5x2 + 4

2) 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 4.3Begin 4.3