Analysis Day 3 - Differential Calculus Integral Calculus...

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Unformatted text preview: Differential Calculus Integral Calculus Multi-Variable 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 Multi-Variable 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 Well-known Taylor series Differential Calculus Integral Calculus Multi-Variable Calculus Integral Calculus and Multi-Variable calculus Integral Calculus Riemann Integral Some well-known summations Fundamental theorem of calculus Integrating rational functions, exponential, logarithmic, trigonometric and inverse trigonometric functions Tricks for performing integration Leibnizs rule Arc length Multi-Variable 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 Multi-Variable 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 Multi-Variable 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 Multi-Variable 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 Multi-Variable 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.

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Analysis Day 3 - Differential Calculus Integral Calculus...

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