2
if F is a continuous function on the closed interval [a,b], then F must attain a maximum and a minimum, each at least once, on [a,b].
4
If n is a positive integer, then d/dx (x^n) = nx^(n-1)
5
The point or place where something begins, arises, or is derived.
7
the function F is said to have a relative maximum at the point c if f(c) > f(x) for all x in an open interval containing c.
9
a function that is a reverse of another function. Ex: f-1(x)
10
If f(x)<g(x)<h(x) when x is near a (except possibly at a) and lim x_a f(x)= lim x_a h(x) = L then lim x_a g(x)=L
12
Implied though not plainly expressed.
13
The action or process of differentiating.
14
If f and g are both differentiable and F= f*g is the composite function defined by F(x)= f(g(x)), then F is differentiable and F’ is given by the product
15
Condition 1: f is continuous on [a,b] Condition 2: f is differentiable on (a,b) Conclusion: f’(x)= f(b)-f(a) / (b-a)
16
on the curve y=f(x) at the point P(a,f(a)) is the line through P with the slope m=lim x_a (f(x)-f(a))/ (x-a) provided that this limit exists.
17
Condition 1: f is continuous on the closed interval [a,b] Condition 2: f is differentiable on the open interval (a,b) Conclusion: if f(a)=f(b), then there is at least one number c in (a,b) such that f’(c)=0
18
If f and g are differentiable, then d/dx[f(x)/g(x)] = (g(x)(d/dx)[f(x)] - f(x)(d/dx)[g(x)]) / [g(x)]^2
20
implied though not plainly expressed.
22
the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions.