Miss Battaglia AP Calculus AB/BC.  Min & max are the largest and smallest value that the function takes at a point Let f be defined as an interval I.

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Presentation transcript:

Miss Battaglia AP Calculus AB/BC

 Min & max are the largest and smallest value that the function takes at a point Let f be defined as an interval I containing c.  f(c) is the min of f on I if f(c)<f(x) for all x in I  f(c) is the max of f on I if f(c)>f(x) for all x in I

f is continuous [-1,2] is closed f is continuous (-1,2) is open g is not continuous [-1,2] is closed min Not a min max Not a max

If f is continuous on a closed interval [a,b] then f has both a minimum and a maximum on the interval.

 Think of a relative max as occurring on a “hill” on the graph and a relative min as occurring on a “valley” of a graph. If there is an open interval containing c on which f(c) is a max, then f(c) is called a relative max of f, or you can say f has a relative max at (c,f(c)) If there is an open interval containing c on which f(c) is a min, then f(c) is called a relative min of f, or you can say f has a relative min at (c,f(c)) AKA local max and local min

f’(c) = o or undefined

What is the value of the derivative at the relative max (3,2)?

Find the value of the derivative at the relative min (0,0).

 Let f be define at c. If f’(c)=0 or if f is not differentiable at c, then c is a critical number of f. c has to be in the domain of f, but does not have to be in the domain of f’.

 If f has a relative min or a relative max at x=c, then c is a critical number of f.  Is the converse true? Think about y=x 3.. Is 0 a critical value? Is it a relative min or max?

To find the extrema of a continuous function f on a closed interval [a,b], use the following steps: 1. Find the critical numbers of f in (a,b) 2. Evaluate f at each critical number in (a,b) 3. Evaluate f at each endpoint of [a,b] 4. The least of these values is the minimum. The greatest is the maximum.

Find the extrema of f(x)=3x 4 -4x 3 on the interval [-1,2]

Find the extrema of f(x)=2x-3x 2/3 on the interval [-1,3]

Find the extrema of f(x)=2sinx – cos2x on the interval [0,2π]

 Read 3.1 Page 169 #11-27 odd, 39