1. Chapter 4
AP Calculus BC

2. 4.1 Extreme Values of Functions
Absolute Maximums/Minimums
Local Maximums/Minimums
Graphs
Theorem 1 – If f is cts. on a closed interval [a,b], then f has both a max. and a min. on the interval.
Theorem 2 – Local Extreme values – If a function, f, has a local max. or local min. at an interior pt., c, of its domain, and if f’ exists at c, then f’(c) = 0.
Critical Points are where f’ = 0 or DNE
Examples:

3. 4.2 Mean Value Theorem Examples: Examples:
Y=f(x) is cts. on[a,b] and differentiable on (a,b) then
Examples:
Corollary 1: f is cts. on [a,b] and differentiable on (a,b)
f’>0 on (a,b) then f increases [a,b]
f’<0 on (a,b) then f decreases [a,b]
Examples:
Corollary 2: If f’(x)=0 at each pt in interval then f(x)=C.
Corollary 3: Functions with the same derivative differ by a constant.

4. 4.2 cont’d Do your homework!!!!!!!! Examples: Antiderivatives –
Reverse of derivatives
Position, velocity, and acceleration
Do your homework!!!!!!!!

5. 4.3 Connecting f’ and f” with the graph of f
Thm. 4 – 1st derivative test
Critical points where f’=0 or DNE
f” = 0 or DNE possible points of inflection
“concavity changes”
1. f’ goes + to - Local Max
2. f’ goes – to + Local Min
3. Left end pt + Local Min
- Local Max
Concavity: y= f(x)
Concave up if f”>0
Concave down if f”<0
Right end pt - Local Min
+ Local Max
Find extreme values
Find concavity:
Examples:

6. 4.3 cont’d. Graph examples Given f’ Given f WHY????
Theorem 5 – 2nd derivative test for local extrema
If f’(c)=0 and f”(c)<0 then f has a local max at c.
If f’(c)=0 and f”(c)>0 then f has a local min at c.
WHY????

7. 4.4 Modeling and Optimization
Strategy p. 219…….
Examples:
1. Two numbers sum is 20. Find the product to be as large as possible.
2. A rectangle inscribed under one arch of the sine curve, largest area ?
3. Open top box out of 20 by 25 foot sheet, cut squares out of corners, largest volume?
Thm. 6 – Maximum Profit is where R’ = C’
Thm. 7 – Min. Avg. Cost is where avg. cost = marginal cost.

8. 4.5 Linearization/Newton’s(Euler’s)
If f is differentiable at x = a, then the equation of the tangent line:
L(x)=f(a) + f’(a)(x-a) defines the linearization of f at a.
The approx. f(x)~L(x) is the standard linear approx. of f at a.
The point x = a is the center of the approximation.
Examples:
Differentials: dy = f’ dx
Separation of variables
Examples:

9. 4.5 cont’d (Newton/Euler)
CHART:
Orig. Pt. dx dy/dx dy New pt.
0.1
given
Continue………
Do an example with the chart……..

10. 4.6 Related Rates EXAMPLES:
Multiple Variables changing with respect to time, t.
Derivatives of each individual variable with respect to t.
Take a derivative with respect to t for each.
Hot Air Balloon
EXAMPLES:
Highway Chase
Ladder Problem