Review_of_Differentiation,_Integration_and_Numerical_Integration

Review_of_Differentiation,_Integration_and_Numerical_Integration

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Unformatted text preview: Summary of Calculus Review of Dierentiation Denition of the Derivative f 0 (x) = lim h→0 f (x + h) − f (x) h Properties of Dierentiation d d d [f (x) ± g(x)] = [f (x)] ± [g(x)] dx dx dx d d [k · f (x)] = k · [f (x)] dx dx d [k] = 0 dx Power Rule Product Rule Quotient Rule Chain Rule d n [x ] = n · xn−1 dx d [f · g] = f 0 g + g 0 f dx   d f f 0g − g0f = dx g g2 dy dy du = · dx du dx 1 Exponential and Logarithmic Table Function Derivative dy dx y = ex dy dx y = ax = ax ln a dy dx y = ln x dy dx y = loga x 2 = ex = = 1 x 1 x ln a Trig Table Function Derivative dy dx y = sin x dy dx y = cos x dy dx y = sec x y = csc x = − sin x dy dx y = tan x dy dx y = cot x = cos x = sec2 x = sec x tan x = − csc x cot x dy dx = − csc2 x Inverse Trig Table function derivative y = sin−1 x y0 = y = cos−1 x √ 1 1−x2 1 y 0 = − √1−x 2 y = tan−1 x y0 = 3 1 1+x2 Review of Calculus - Applications of Dierentiation Rates of Change s(t) = v(t) = s0 (t) = a(t) = v 0 (t) = s00 (t) = distance function velocity function acceleration function Optimization (Maximization and Minimization) Obtain a function equate to f (x) in terms of one variable, then obtain f 0 (x), and lastly 0. Gradient Function and Tangent Lines f (x) = function f 0 (x) = gradient function Finding Relative and Absolute Extrema - First Derivative Test (FDT) To nd the critical points and to classify them as relative maximum or relative minimum, we use the First Derivative Test Steps: 1. 2. Find all the critical points of f (x) by equating f 0 (x) = 0 and solve for x. Then use the Sign Test using f 0 (x) to determine the nature of the critical point x0 . (a) (b) (c) If the values of f 0 (x) are positive on the left of x0 and negative on the right of x0 , then x0 is a relative maximum. If the values of f 0 (x) are negative on the left of x0 and positive on the right of x0 , then x0 is a relative minimum. If the values of f 0 (x) are positive (or negative, resp.) on the left of x0 and positive (or negative, resp.) on the right of x0 , then x0 is not a relative extrema. 4 Point of Inection and Concavity Given a function f (x), 1. 2. dierentiate f (x) twice [f 00 (x)] and equating it to 0 to nd point of inection x0 . Use Sign Test to nd concavity: (a) (b) For the region of positive f 00 (x) values, we have concave up (cup facing up). For the region of negative f 00 (x) values, we have concave down (cup facing down). 5 Review of Integration The Indenite Integral (Antiderivative) ˆ F (x) = f (x)dx The Denite Integral (Area under the Curve) ˆ b f (x)dx A= a Integration Table for Polynomials function Integration formula ´ 1 ´ x ´ x2 ´ x3 xn , n 6= −1 ´ 1dx = x + c xdx = 12 x2 + c x2 dx = 31 x3 + c x3 dx = 41 x4 + c xn dx = 6 1 xn+1 n+1 +c Integration Table for Trig, Exp, Log function integration − sin x − ´ sin x ´ sec2 x csc2 x 1 x ex ´ sin xdx = cos x + c sin xdx = − cos x + c ´ cos x csc x cot x ´ cos xdx = sin x + c sec2 x = tan x + c csc x cot xdx = − csc x + c ´ ´ csc2 xdx = − cot x + c 1 dx x ´ = ´ dx x = ln |x| + c ex dx = ex + c 7 Integration by u-substitution For ˆ f (g(x)) · g 0 (x)dx, let u = g(x), then du = g 0 (x)dx and ˆ ˆ 0 f (g(x)) · g (x)dx = Integration by Parts ˆ f (u)du. ˆ u dv = uv − 8 v du. Review of Numerical Integration Midpoint Approximation For Midpoint approximation, we have the following formulae to use: width of the subintervals, ∆x choice of midpoints of the subintervals, ˆ b−a n = 1 x∗k = a + (k − )∆x 2 b f (x)dx ≈ Mn = [f (x∗1 ) + f (x∗2 ) + . . . + f (x∗n )] · ∆x Area function, a Trapezoidal Approximation Also known as Trapeziodal Rule. The formula to use are: width of the subintervals, ˆ b f (x)dx ≈ Tn = Area function a ∆x = b−a n 1 [y0 + 2y1 + 2y2 + . . . + 2yn−1 + yn ] · ∆x 2 Simpsons Approximation Also known as Simpsons Rule. The formula to use is: ˆ b f (x)dx ≈ S2n = a 1 3  b−a 2n    y0 + 4y1 + 2y2 + 4y3 + 2y4 + . . . + 4y(2n)−1 + y2n . 9 ...
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