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Unformatted text preview: Differential Calculus Integral Calculus MultiVariable Calculus Day 3 Review of Basic Calculus Sivaram sivaambi@stanford.edu Institute of Computational and Mathematical Engineering Stanford University September 21, 2011 Differential Calculus Integral Calculus MultiVariable Calculus Differential Calculus Product rule Quotient rule Chain rule Exponential and logarithmic function Trigonometric and Inverse trigonometric function LHospital rule Maximum and Minimum of a function Rolles theorem Mean Value theorem Implicit differentiation Taylor Series Wellknown Taylor series Differential Calculus Integral Calculus MultiVariable Calculus Integral Calculus and MultiVariable calculus Integral Calculus Riemann Integral Some wellknown summations Fundamental theorem of calculus Integrating rational functions, exponential, logarithmic, trigonometric and inverse trigonometric functions Tricks for performing integration Leibnizs rule Arc length MultiVariable Calculus Partial derivatives Integral to compute surface areas and volumes Change of coordinates Cylindrical polar coordinates Spherical coordinates Z  exp( x 2 ) dx = Differential Calculus Integral Calculus MultiVariable Calculus Derivative The left derivative of a function f at a point a is given by f ( a ) = lim h  f ( a + h ) f ( a ) h . The right derivative of a function f at a point a is given by f ( a ) + = lim h + f ( a + h ) f ( a ) h . The function f is said to be differentiable at the point a if we have f ( a ) = f ( a ) + . The derivative at the point a is then denoted by f ( a ) (or) df dz a . When the left and right derivatives match, the derivative can be directly obtained as f ( a ) = lim h f ( a + h ) f ( a ) h . Differential Calculus Integral Calculus MultiVariable Calculus Differentiation The geometrical interpretation of the the derivative at the point a is the slope of the tangent at the point a . Figure: Geometrical interpretation of the the derivative at the point x Differential Calculus Integral Calculus MultiVariable Calculus Properties of derivatives The derivative of a constant is zero i.e. if f ( x ) = c , x S , then f ( x ) = 0 , x S . f ( x ) = lim h f ( x + h ) f ( x ) h = lim h c c h = lim h 0 = 0 . ( f ( x ) + g ( x )) = f ( x ) + g ( x ) wherever f ( x ) and g ( x ) are defined. Proof: (?) d ( x n ) dx = nx n 1 where n R Proof: (?) The derivative of a polynomial of degree n is a polynomial of degree at most n 1. Proof: (?) Differential Calculus Integral Calculus MultiVariable Calculus Properties of derivatives Product rule: If f and g are two functions whose derivative exists at x = a , then ( f ( x ) g ( x ))  x = a = f ( a ) g ( a ) + f ( a ) g ( a ) Proof: (?) Quotient rule: If f and g are two functions whose derivative exists at...
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This note was uploaded on 10/01/2011 for the course EE 221 taught by Professor Ee221a during the Spring '08 term at University of California, Berkeley.
 Spring '08
 ee221a

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