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Unformatted text preview: Review of Calculus
Definition of Derivative
In geometric terms, the derivative is the slope of a curve at a particular point. using an alternative definition, if x + h = c, then Definition of a partial derivative
This occurs when we hold all but one of the independent variables of a function
constant and is written by
or fx(x,y,z). Here x is allowed to change and y and z
are considered constant.
A function, f(x,y) is differentiable at (xo,yo) if fx(xo,yo) and fy(xo,yo) exist plus other
conditions that are related to being continuous. W e call f differentiable if it is
differentiable at every point in its domain.
1) if the partial derivatives fx and fy of a function f(x,y) are continuous throughout an
open region R, then f is differentiable at every point in R.
2) If a function f(x,y) is differentiable at (xo,yo) then f is continuous at (xo,yo) Integrals:
Definition of an integral:
In geometric terms, the integral is the area under a curve.
To approximate the area using the
Riemann sum, you add up all of the
rectangles 'in' the curve to get the area.
The diagram is a 'left-endpoint' sum. The
height of each rectangle is f(xk) and the
width is xk. Therefore, the area can be
approximated by: http://upload.wikimedia.org/wikipedia/commons/c/cc/Riemann_Sum_Left_Hand.png 1 Riemann Integral
This is the limit or the Riemann sum as xk gets smaller and smaller and just equals the
area under the curve. Fundamental Theorem of Calculus:
Let F(x) =
, then F'(x) = f(x) and That is the integral of the derivative of a function can be evaluated by taking the
antiderivative at the endpoints.
Another useful property of integrals is: You should be able to verify this by using the geometric definition of an integral.
Abbreviated table of integrals Do this by substitution: let u = x2 ==> du = 2xdx ==> xdx = We will mostly be using definite integrals so we use the Fundamental Theorem of
Calculus to evaluate them. If there is an indefinite integral, then remember to always
add in the 'C', the constant of integration. 2 ...
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This note was uploaded on 02/20/2012 for the course STAT 311 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.
- Spring '08