AP/Honors Calculus Chapter 4 Applications of Derivatives
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