Theorems
What Theorem is this?
If f(x) is continuous on [a,b] AND differential on the interval (a,b), then there is at least one number x=c in (a,b) S.T f'(c)= f(b)-f(a)/b-a
Mean Value Theorem
A farmer has 500 feet of fencing and uses his barn to enclose a rectangular pen. if x is the width and y is the length of the pen, what is the equation for the enclosure?
2x+y=500
if f'(3)=0 and f"(3) = 12 what do we know about f(3)
because f"(3) > 0 the graph of f(x) is concave up and therefor x=3 is a relative minimum.
What is Mr Woods Favourite Math concept?
Proof by contradiction (assume something is true and show that that assumption results in a contradiction to prove that your starting assumption must be false)
Does EVT guarantee absolute extrema for f(x) = 1/x on (0,2)? if so, find them.
No, because the interval is open EVT cannot guarantee any extrema if the endpoints aren't included.
A farmer uses 200 feet of fencing to build a rectangular pen against a barn. Find the dimensions that maximize the area.
X=50, Y=100
Find all of the relative extrema of f(x)= x4-4x3 along [-1,4]
x=0 is neither a maximum or minimum
x=3 is a relative minimum
where did Mr. Woods attend high school?
Roosevelt High school in Seattle
A runners position can modelled by s(t) and is differentiable & continuous along [16,26].
the runner starts at s(16)= 260 & ends at s(26)=400
is it possible that at some point the runner was moving at a speed greater than 9m/s at some point during their run
Yes because of MVT, (400-260)/10 = 14m/s, this means that there is a moment where the runners speed is 14m/s.
14>9, T.F there is a point where the runner is faster than 9m/s.
a rectangle is bounded by the line y=-x+13, what length and width should the rectangle have to maximize its area?
To maximize the area, X & Y= 6.5
Total Area = 42.25
Find the absolute Max & Min of f(x)= x4-8x2 across [-3,3]
Abs Max f(3/-3)=9
Abs Min f(2/-2) = -16
Frank