Theorems
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c) = 0
Rolle's theorem
This is another term for a sharp point on a graph that makes it undifferentiable
cusp
Derivative of e^x
e^x
limit as x goes to infinity of f(x) = 2x
This many people are credited with discovering calculus
2
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a)) / (b-a)
The mean value theorem
This function is made up of several "stairs" of disconnected lines
step function
Derivative of pi
zero
limit as x goes to infinity of f(x) = x^2
infinity
When trying to describe how an object falls, this person found that the speed of the object increased every split second and that no mathematics currently used could describe the object at any moment in time.
If (c, f(c)) is a point of inflection of the graph of f, then either f"(c) = 0 or f" does not exist at x=c
Points of inflection
This function includes several different functions
Piecewise function
Derivative of ln(x)
1/x
limit as x goes to infinity of f(x) = 1/x
0
In this century calculus was first discovered
Mid 17th century
If a function f is continuous on the closed interval[a, b] and F is an antiderivative of f on the interval [a, b]. then Sf(x)dx = F(b) - F(a)
The fundamental theorem of calculus
This function goes towards negative infinity on both ends faster and faster as it goes away from x=0
negative quadratic function
Derivative of tan(x)
sec^2(x)
limit as x goes to infinity of f(x) = -x^1/2
negative infinity
The German who is credited as one of the people to independently develop the foundation of calculus
If f is a continuous on the closed interval [a, b] then there exists a number c in the closed interval [a, b] such that Sf(x)d = f(c)(b-a)
Mean value theorem for integrals
This function approaches infinity towards the right at a slower and slower rate the farther away it goes away from x=0
square root function
Derivative of sin(2x)
2cos(2x)
limit as x goes to infinity of f(x) = (1/3)^x
0
Is widely regarded by scholars and historians as the greatest mathematician in history
Ms. LA