Although Corollary 2 does not apply to the function 1 / (z — z_0) in the plane punctured at z_0, Theorem 6 can be used as follows to show that
(see attached 1)
for any circle C traversed once in the positive direction surrounding the point z_0. Introduce a horizontal branch cut from z_0 to infinity as in Fig. 4.25. In the resulting "slit plane" the function 1 / (z — z_0) has the anti-derivative Log (z — z_0). Apply Theorem 6 to compute the integral along the portion of C from alpha to beta as indicated in Fig. 4.25. Now let alpha and beta approach the point t on the cut to evaluate the given integral over all of C.
(see attached 1)
Theorem 6. Suppose that the function f (z) is continuous in a domain D and has an anti-derivative F(z) throughout D; i.e., dF(z)Idz = f (z) for each z in D. Then for any contour gamma lying in D, with initial point z_I and terminal point z_T, we have
(see attached 2).

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3)
(2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2
(3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i
(4) Evaluate the integral between 2pi and 0 of
e^-i@ . e ^e^i@ d@

Let f(x) be a continuous function of one variable.
a) Give the definition of the derivative.
b) Use this definition to find the derivative of f(x)=x^2+2x-5
c) Evaluate f'(2)

1. (L'Hopital's Rule) Show that if f,g:X->R, x0 belonging to X is a limit point of X such that f(x0) = g(x0) = 0, f,g are differentiable at x0, and g'(x0) != 0, then there is some delta > 0 such that g(x) != 0 for all x belonging to (X INTERSECTION (x0 - delta, x0 + delta) and
lim x->x0 [f(x)/g(x)] = f'(x0) / g'(x0) .
Hint

An elliptic curve can be written as y^2=x^3+ax+b. I need a proof for why x^3+ax+b either have 3 real roots or 1 real root and 2 complex roots. I don't have anything that I know about it prior to asking for help here at Brainmass.

Let f(x) = x2 + 4x.
(a) Find the derivative f 'of f.
(b) Find the point on the graph of f where the tangent line to the curve is horizontal.
Hint: Find the value of x for which f '(x) = 0.
(c) Sketch the graph of f and the tangent line to the curve at the point found in part (b).
Find the slope m of the tangent line to

Find the derivative;
G(v)= (v^3-1)/(v^3+1)
Find the limit;
lim(sin3x)/(sin5x)
x->0
Find the derivative;
R(w)= (cosw)/(1-sinw)
H(o)=(1+seco)/(1-seco)
Find the derivative;
F(x)= cos(3x^2)+{cos^2}3x
N(x)=(sin5x-cos5x)^5
"Assume that the equation determines a differentiable function f such that y=f(x),

Let be a compact interval and let A be a collection of continuous functions on which satisfy the properties of the Stone-Weierstrass Theorem
[Stone-Weierstrass Theorem: Let K be a compact subset of and let A be a collection of continuous functions on K to R with the properties:
a) The constant function belongs to A.
b) I

Calc Proofs
1) Using the following two functions X and X^2 develop their derivatives using the Definition of a Derivative
for three values of "h"
h = .1
h = .01
h= .001
and then repeat the calculation
in the limit as h->0
2) Using the two functions above, show that the Finite Sum approximation of the area