Value Theorems & Variation
If f is continuous on [a,b] and k is between f(a) and f(b), then there exists a c in [a,b] such that f(c)=k.
IVT - Intermediate Value Theorem
f(x) will have a min/max where f'(x) is _________
0 or undefined
d/dx (5x2)
10x
Int (cos(x)) dx
sin(x)+c
How does a penguin build its house?
Igloos it together.
If f is continuous on [a,b], then f has a maximum and a minimum value in the interval [a,b].
EVT - extreme value theorem
f(x) is increasing when f'(x) is ________
positive
d/dx sin(2x)
2cos(x)
Int (sec2(x)) dx
tan(x)+c
What's a frogs favorite part of calculus?
de-ribbit-ives
If f is continuous on [a,b] and differentiable on (a,b), then for some c in (a,b), f'(c)=[f(b)-f(a)]/[b-a].
MVT - mean value theorem
f(x) will be _______________, when f''(x)>0.
concave up
d/dx ln(x)
1/x
Int (12x3) dx
3x4+c
What's the integral of (1/cabin)d(cabin)?
a natural log cabin
y=kx has this type of variation
direct variation
f(x) will have inflection points where ________ = 0 or undefined.
f''(x)
d/dx (3x+2)5
5(3x+2)4.3 = 15(3x+2)4
Int (1/x) dx from 1 to 5
ln(5)
2 things in calculus we never want to lose or forget...
our lims and +Cs
y=k/x has this type of variation
inverse variation
If f(x) is strictly increasing, then the left riemann sum will be an OVER/UNDER estimate for the area.
under
d/dx (tan-1(x))
1/(1+x2)
Int (x2+2) dx from 0 to 3
15
If dy/dx=ky, then y=________
cekt