Calculus - Review of Calculus Derivatives: Definition of...

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Unformatted text preview: Review of Calculus Derivatives: 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. Differentiable 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. Note: 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. Riemann Sum 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: 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.

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