November 19, 2015
AP Calculus
Section 3.1
Day 1
Objective:
1. Understand the definition of extrema
of a function on an interval.
2. Understand the definition of relative
extrema of a function on an open interval.
November 19, 2015
Extrema of a Function
Not a
Maximum
Maximum
Minimum
(a) f is continuous
[-1,2] is closed
Maximum
Not a
Minimum
Minimum
(b) f is continuous
(-1,2) is open
(c) g is not continuous, [-1, 2] is closed.
These are absolute max and min called
extreme values (or extrema).
Extrema can occur at interior points or endpoints of an
interval (as long as it is a closed interval). Extrema that
occur at the endpoints are calledendpoint extrema.
November 19, 2015
Consider the graph of
3
y = x - 2x
2
relative max at (0, 0)
relative min at (2, -4)
***We say relative because there is an interval
for which it is a max/min
November 19, 2015
Relative maximums and minimums can occur at
hills/valleys or at sharp turns.
hill/valley: derivative is zero
sharp turn: function is not
differentiable
November 19, 2015
Critical Number
Definition of a Critical Number
Let f be defined at c. Critical numbers are c such that
f’(c ) = 0 or
f is not differentiable at c,
***Maximums and minimums occur at critical numbers!!
BUT......critical numbers are not
necessarily maximums or minimums!!
November 19, 2015
Ex 1:Find critical numbers, state the max/min and whether it is relative or absolute.
November 19, 2015
Ex 2: Find the critical #s.
2
2
f(x) = x (x - 4)
November 19, 2015
Ex 2: Find the critical #s.
f(x) = (3x2 - 2) 1/2
November 19, 2015
Ex 3: Find the critical #s.
f(x) = 2 sec x + tan x
0 < x < 2π
November 19, 2015
Homework
p. 204-205: 3-6 all, 11, 17, 27, 33,
34, 37, 47, 51, 53
(2 days)
November 19, 2015
AP Calculus
Section 3.1
Day 2
Objective: Find extrema on a closed interval.
November 19, 2015
The Extreme Value Theroem
If f is continuous on a closed interval [ a,b], then f has
both an absolute minimum and an absolute maximum
at some numbers c and d on the interval [a,b].
Extrema of a Function
Not a
Maximum
Maximum
Minimum
(a) f is continuous
[-1,2] is closed
Minimum
(b) f is continuous
(-1,2) is open
Maximum
Not a
Minimum
(c) g is not continuous, [-1, 2] is closed.
November 19, 2015
Steps for Finding Extrema on a Closed Interval
1. Find the critical numbers (set f ' = 0).
2. Evaluate f at each critical number in the interval.
3. Evaluate f at each endpoint of the interval.
4. The least of these values is the minimum.
The greatest is the maximum.
November 19, 2015
Ex 1: Find the extrema of f(x) = 3x 4 - 4x3 on the interval [-1,2].
Maximum
(2,16)
(-1,7)
(0,0)
(1,-1)
Minimum
November 19, 2015
Ex 2: Find the extrema of f(x) = 2x -3x 2/3 on the interval [-1,3].
Maximum
(0,0)
(1,-1)
(-1,-5)
Minimum
(3,6-3∛9)
November 19, 2015
Critical Number
Definition of a Critical Number
Let f be defined at c. Critical numbers are c such that
f’(c ) = 0 or
f is not differentiable at c,
***Maximums and minimums occur at critical numbers!!