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Analysis Day 3

# Analysis Day 3 - Dierential Calculus Integral Calculus Day...

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Differential Calculus Integral Calculus Multi-Variable Calculus Day 3 Review of Basic Calculus Sivaram [email protected] Institute of Computational and Mathematical Engineering Stanford University September 21, 2011

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Differential Calculus Integral Calculus Multi-Variable Calculus Differential Calculus Product rule Quotient rule Chain rule Exponential and logarithmic function Trigonometric and Inverse trigonometric function L’Hospital rule Maximum and Minimum of a function Rolle’s 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 Leibniz’s 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 = π

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Differential Calculus Integral Calculus Multi-Variable Calculus Derivative The left derivative of a function f at a point a is given by f 0 ( a ) - = lim h 0 - f ( a + h ) - f ( a ) h . The right derivative of a function f at a point a is given by f 0 ( a ) + = lim h 0 + f ( a + h ) - f ( a ) h . The function f is said to be differentiable at the point a if we have f 0 ( a ) - = f 0 ( a ) + . The derivative at the point a is then denoted by f 0 ( a ) (or) df dz a . When the left and right derivatives match, the derivative can be directly obtained as f 0 ( a ) = lim h 0 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 0

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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 0 ( x ) = 0 , x S . f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 c - c h = lim h 0 0 = 0 . ( f ( x ) + g ( x )) 0 = f 0 ( x ) + g 0 ( x ) wherever f 0 ( x ) and g 0 ( 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 )) 0 | x = a = f 0 ( a ) g ( a ) + f ( a ) g 0 ( a ) Proof: (?) Quotient rule: If f and g are two functions whose derivative exists at ‘ a ’ and we have g ( a ) 6 = 0, then f ( x ) g ( x ) 0 x = a = g ( a ) f 0 ( a ) - f ( a ) g 0 ( a ) ( g ( a )) 2 Proof: (?) Chain rule: If f and g are two functions whose derivative exists at ‘ a ’ and g ( a ) respectively, then df ( g ( x )) dx = df ( y ) dy y = g ( a ) × dg ( x ) dx x = a Proof: (?)

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Differential Calculus
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Analysis Day 3 - Dierential Calculus Integral Calculus Day...

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