What is the difference between a relative extrema and absolute extrema?
Relative extremas are one of many minimas or maximas a graph may have but absolute is the overall maxima or minima with the greatest value
100
What are the three things you need to have to prove Rolle's theorem?
F(x) needs to be continuous on the closed interval [a,b], have a derivative on the open interval (a,b) and F(a)=F(b)
200
f(x) = (3x - 2x2)(5 + 4x)
f'(x) = -24x2 + 4x + 15
200
State Rolle's Theorem.
Given a function continuous on [a, b] and differentiable on (a, b), f'(c) = 0 for some value of c in (a, b) if f(a) = f(b).
200
Identify the two justification methods commonly used in optimization problems to determine whether or not a relative extremum actually exists.
1) First Derivative Test
2) Concavity (Sign of Second Derivative)
200
Identify the steps in determining an ABSOLUTE maximum. Be as specific as possible!
1) Find the first derivative of the function.
2) Identify the critical values (first derivative = 0 or DNE).
3) Determine function values of critical values AND endpoints.
4) Identify ABSOLUTE maximum = greatest y-value.
200
Why do we use Rolle's Theorem?
Rolle's theorem essentially states that a differentiable function, which attains equal values at two points, must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero.
300
f(x) = sin x tan x
f'(x) = cos x tan x + sin x sec2x
300
Find a value of c such that the conclusion of the Mean Value Theorem is satisfied for f(x) = -2x3 + 6x - 2 on the interval [-2 , 2].
c = +2 sqrt(1/3) and c = -2 sqrt(1/3)
300
Find two positive numbers whose sum is 47 and whose product is as large as possible.
47/2 and 47/2
This is true for any question like this!
300
Find the absolute extrema for the interval
f(x)=x^3-3x^2 [0,4]
What is
F(0)=0
F(2)=-4
F(4)=16
300
Apply rolle's theorem to this function: f(x)= x^2-5x+6.
This follows Rolle's theorem x= 5/2
400
The derivative of sin23x
6sin3xcos3x
400
The sum of the first number and twice the second number is 108. The product of the two numbers is a maximum. Find both numbers.
54 and 27
400
Find the absolute extremas for the interval
f(x)=x+1/x [1,5]
What is
F(1)=2
F(5)=5.2
400
Why does this function not follow Rolle's theorem?
f(x)=x^2 - 1/x
It is not continuous on x=0 so Rolle's doesn't apply.
500
f(x) = ln(x/(x2 + 1))
f'(x) = (1/x) - (2x/(x2 + 1))
500
A girl is building a cardboard house without a roof. She has a sheet of cardboard that is 10 ft by 10 ft. She is going to cut four squares out of the four corners of the sheet of cardboard. What are the dimensions of the squares she must cut to make a roofless house with the largest possible volume?
5/3 ft by 5/3 ft
500
Find the x-coordinates of the relative extrema of the function f(x) = .5x - sin x in the interval (0, 2π). Identify any relative extremum as a relative maximum or relative minimum.
relative minimum at x = π/3
relative maximum at x = 5π/3
500
Does this problem follow Rolle's Theorem
F(x)= 1-| x |, [-1,1]
No, Rolle's does not apply. Is not differentiable.