calculus

# calculus - Essential techniques of Calculus J R Chasnov...

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Unformatted text preview: Essential techniques of Calculus J. R. Chasnov Department of Mathematics Hong Kong University of Science and Technology Fall 2007 1 Differentiation 1.1 Definition of derivative The derivative of the function y = f ( x ), denoted as f ′ ( x ) or dy/dx , is defined as the slope of the tangent line to the curve y = f ( x ) at the point ( x, y ). This slope is obtained by a limit. First, a line is drawn which passes through the points ( x, f ( x )) and ( x + h, f ( x + h )) (see Fig. 1). The slope of this line is the rise: f ( x + h ) − f ( x ) divided by the run: ( x + h ) − x = h . As h → 0, the drawn line approaches the tangent line. The derivative is thus defined as the limit f ′ ( x ) = lim h → f ( x + h ) − f ( x ) h . (1) Note that taking the limit is not the same as directly substituting h = 0, which gives f ′ ( x ) = 0 / 0, an undefined value that is not of much use. Commonly, we will need to cancel the h in the denominator against a factor of h in the numerator before setting h = 0 and evaluating the derivative. x x+h f(x) f(x+h) y=f(x) slope = [f(x+h)-f(x)]/h Figure 1: The derivative as the limit of a slope. 1 The notation dy/dx for the derivative of y = f ( x ) comes from denoting the change h in the dependent variable x by Δ x , pronounced “delta x,” and the resulting change in the dependent variable y by Δ y , where Δ y = f ( x + Δ x ) − f ( x ) . Therefore, (1) can be rewritten in this new notation as f ′ ( x ) = lim Δ x → Δ y Δ x = dy dx , the second line defining dy/dx = f ′ ( x ). Note that strictly speaking dy/dx is defined after taking the limit and is a single entity. Sometimes, however, it is expedient to treat dy/dx as a fraction. Next we derive some general rules that will be useful in evaluating the deriva- tives of complicated functions. Then we will determine specific rules differentiat- ing some simple functions that can be used to construct complicated functions. 1.2 Derivative of the sum or difference of two functions Let h ( x ) = f ( x ) + g ( x ). We have h ′ ( x ) = lim Δ x → h ( x + Δ x ) − h ( x ) Δ x = lim Δ x → f ( x + Δ x ) + g ( x + Δ x ) − ( f ( x ) + g ( x )) Δ x = lim Δ x → f ( x + Δ x ) − f ( x ) Δ x + lim Δ x → g ( x + Δ x ) − g ( x ) Δ x = f ′ ( x ) + g ′ ( x ) . The result can be written as ( f + g ) ′ = f ′ + g ′ . In a similar fashion, ( f − g ) ′ = f ′ − g ′ . These simple rules generalize to the sum or difference of an arbitrary number of functions. 1.3 Derivative of the product of two functions If h ( x ) = f ( x ) g ( x ), then h ′ ( x ) = lim Δ x → h ( x + Δ x ) − h ( x ) Δ x 2 = lim Δ x → f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = lim Δ x → f ( x + Δ x ) g ( x + Δ x ) − f ( x ) g ( x + Δ x ) + f ( x ) g ( x + Δ x ) − f ( x ) g ( x ) Δ x = lim Δ x → f ( x + Δ x ) − f ( x ) Δ x g ( x + Δ x ) + lim Δ x → f ( x ) g ( x + Δ x ) − g ( x ) Δ x = f ′ ( x ) g ( x ) +...
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