Theorems I Have Known
Everyday Derivatives
Commonplace Integrals
It's All About Graphs
That's Some Sum!
100
This theorem helps you find the limit of a function f(x) with
g(x) ≤ f(x) ≤ h(x)
and
limx → a g(x)  =  limx → a h(x)  =  L
What is the Squeeze Theorem?
100
This function is the derivative of e2x
What is 2 e2x?
100
This function is the antiderivative of (1+x2)-1.
What is arctan(x)+C?
100
The graph of f is this on an interval where f'(x) is negative.
What is decreasing?
100
This number is ∑i=14 2i.
What is 20?
200
This theorem says that, if f is a continuous function on [a,b] and M is a number between f(a) and f(b), then f( c )  =  M for some c between a and b.
What is the Intermediate Value Theorem?
200
This function is the derivative of x ln(x)  -  x.
What is ln(x)?
200
This function is given by
∫1x   t-1   dt.
What is ln(x)?
200
This is a point where the derivative of f either does not exist or equals zero.
What is a critical point?
200
This formula gives the left Riemann sum for
\∫ab  f(x)   dx
if [a,b] is partitioned into N equal intervals.
What is
&sumi=1N  f(xi-1)   Δ x?
300
This theorem says that if f is continuous on [a,b] and differentiable on (a,b), then there is a point c between a and b so that f'( c ) has the same slope as a line between (a,f(a)) and (b,f(b))
What is the Mean Value Theorem?
300
This function is the derivative of arcsin(x).
What is (1-x2)1/2?
300
This function is the indefinite integral of x exp(x2).
What is (1/2) exp(x2)+C?
300
This sign change of f'(x) guarantees that a critical point c is a local maximum.
What is positive to negative?
300
This integral is the limit of the Riemann sums ∑i=1N   (i/N)2   (1/N).
What is ∫01  x  dx?
400
This theorem says that, to compute ∫ab  f(x)   dx, you need only find an antiderivative F and compute F(b)-F(a)
What is the Fundamental Theorem of Calculus, Part I?
400
If f is a continuous function, this function is the derivative of
A(x) = ∫ax   f(t)   dt.
What is f(x)?
400
This integral gives the area to the right of x = h(y) and to the left of x = g(y) between y = c and y = d.
What is
∫cd   (g(y)-h(y))   dy?
400
These are the points where an absolute maximum or minimum of a function on a closed interval may occur.
What are critical points and endpoints?
400
In a Riemann sum R(f,P,C), this is what C is.
What is a collection of points ci with xi-1 ≤ ci ≤ xi?
500
This Theorem says that the derivative of the "area function"
A(x) = ∫ax f(t)   dt
is exactly f(x).
What is the Fundamental Theorem of Calculus, Part II?
500
This number is the derivative of f(g(x)) at x=3 if g'(3) = 2, g(3)=5, and f'(5)=7.
What is 14?
500
This number is the value of the integral ∫-11 (1-x2)1/2.
What is π/2?
500
This equation gives the horizontal asymptote for the function
f(x) = (3x2+5x+4)/(5x2+x +17).
What is y = 3/5?
500
In a Riemann sum R(f,P,C), this is what measures the "size" of P.
What is the norm of a partition?
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