1 x 2 sqrt x

    • [PDF File]TRAPEZOIDAL METHOD Let f x) have two continuous derivatives on

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      INTEGRATING sqrt(x) Consider the numerical approximation of Z 1 0 sqrt(x)dx= 2 3 In the following table, we give the errors when using both the trapezoidal and Simpson rules. n ET n Ratio EnS Ratio 2 6.311E−2 2.860E−2 4 2.338E−22.70 1.012E−22.82 8 8.536E−32.74 3.587E−32.83 16 3.085E−32.77 1.268E−32.83 32 1.108E−32.78 4.485E− ...

    • [PDF File]Maxima by Example: Ch.7: Symbolic Integration

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      x(b2 ¡x2)¡1=2 dx: (%i3) integrate (x/ sqrt (bˆ2 - xˆ2), x); 2 2 (%o3) - sqrt(b - x ) (%i4) diff(%,x); x (%o4) -----2 2 sqrt(b - x ) Example 3 The definite integral can be related to the ”area under a curve” and is the more accessible concept, while the integral is simply a function whose first derivative is the original integrand.

    • [PDF File]Maxima by Example: Ch.4: Solving Equations

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      (%i2) allroots(x 2 - 2*x + 1); (%o2) [x = 1.0, x = 1.0] (%i3) multiplicities; (%o3) not_set_yet As we expected, allroots does not affect multiplicities; only solve and realroots set its value. 4.1.3 General Quadratic Equation or Function To get our feet wet, lets turn on the machinery with a general quadratic equation or expression. There are

    • [PDF File]Table of Integrals - UMD

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      1 2 secxtanx+ 1 2 ln|secxtanx| (76)!secxtanxdx=secx (77)!sec2xtanxdx= 1 2 sec2x (78)!secnxtanxdx= 1 n secnx, n!0 2 (79)!cscxdx=ln|cscx"cotx| (80)!csc2xdx="cotx (81)!csc3xdx=" 1 2 cotxcscx+ 1 2 ln|cscx"cotx| (82)!cscnxcotxdx=" 1 n cscnx, n!0 (83)!secxcscxdx=lntanx TRIGONOMETRIC FUNCTIONS WITH xn (84)!xcosxdx=cosx+xsinx (85)!xcos(ax)dx= 1 a2 ...

    • [PDF File]NUMERICAL INTEGRATION: ANOTHER APPROACH

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      The case n=2.Wewantaformula w1f(x1)+w2f(x2) ≈ Z 1 −1 f(x)dx The weights w1,w2 and the nodes x1,x2 aretobeso chosen that the formula is exact for polynomials of as large a degree as possible. We substitute and force

    • [PDF File]Working a difference quotient involving a square root

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      x+h x h p p x+h+ x p x+: The key idea is that the numerators multiply in a nice way. Note that the two numerators together have the form (A B)(A+B) which is equal to A2 B2 (you might recall the phrase difference of squares). The squaring eliminates the square roots from the numerator. As a result, our expression above becomes p x+h x h p p x+h+ ...

    • [PDF File]7.2 Finding Volume using the Washer Method

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      3 7.2 Finding Volume using the Washer Method Example 1) Find the volume of the solid formed by revolving the region bounded by the graphs y = √x and y = x2 about the x-axis.

    • [PDF File]Square Roots via Newton’s Method - MIT Mathematics

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      x n+1 = 1 2 x n + a x n : The intuition is very simple: if x n is too big (> p a), then a=x n will be too small (< p a), and so their arithmetic mean x n+1 will be closer to p a. It turns out that this algorithm is very old, dating at least to the ancient Babylonians circa 1000 BCE.1 In modern times, this was seen to

    • [PDF File]Finding Square Roots Using Newton’s Method

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      0 = f(xk)+f′(xk)(xk+1 −xk), that is, xk+1 = xk − f(xk) f′(xk). Applied to compute square roots, so f(x) := x2 −A, this gives xk+1 = 1 2 xk + A xk . (1) From this, by simple algebra we find that x k+1 −xk = 1 2xk (A−x2). (2) Pick some x0 so that x2 0 > A. then equation (2) above shows that subsequent approxi-mations x1, x2 ...

    • [PDF File]{1, (ln(x) + y)}/sqrt(1 + (ln(x) + y)^2)

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      variable 1 • lower limit 1. • upper limit 1 • variable 2: • lower limit 2 • upper limit 2. Input: {1, (In(x) + y)}/sqrt(l O 2 2 Il, log(x) + y} -2 21] VectorPlot 1 + (log(x) + Result log(x) is the natural logarithm 0.0 0.5

    • [PDF File]INTERPOLATION - University of Iowa

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      x0 =0,x1 = π 4,x2 = π 2 and yi=cosxi,i=0,1,2 This gives us the three points (0,1), µ π 4, 1 sqrt(2) ¶, ³ π 2,0 ´ Now find a quadratic polynomial p(x)=a0 + a1x+ a2x2 for which p(xi)=yi,i=0,1,2 The graph of this polynomial is shown on the accom-panying graph. We later give an explicit formula.

    • [PDF File]Disciplined Convex Programming and CVX - Stanford University

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      sqrt(x) √ x (x ≥ 0) ccv, nondecr ... x2, |x| ≤ 1 2|x|−1, |x| > 1 cvx Convex Optimization, Boyd & Vandenberghe 11. Composition rules • can combine atoms using valid composition rules, e.g.: – a convex function of an affine function is convex – the negative of a convex function is concave

    • [PDF File]Does it converge or diverge? If it converges, find its ...

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      x3 +1 x+1 = x2 −x+1 so that x3 + 1 = (x + 1)(x2 − x + 1) (NOTE: On the exam, you will be able to factor the polynomial easier than this!) By Partial Fractions, x3 x3 +1 = 1− 1 (x+1)(x2 −x+1) = 1+ 1 3 · 1 x+1 + 1 3 · 2−x x2 −x+1 Now you have to complete the square to finish things off, and after some long algebra, x+ 1 6 ln(x2 ...

    • [PDF File]18.06 Problem Set 6 - Solutions - MIT

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      (3) Apply the Gram-Schmidt algorithm to the set {1,x,x2} to obtain an orthonormal basis {f 0,f 1,f 2} of all degree-2 polynomials. Solution Denote g 0 = 1,g 1 = x and g 2 = x2. We begin by letting G 0 = g 0 = 1. For G 1: G 1 = g 1 − hG 0,g 1i hG 0,G 0i G 0 = x− R 1 0 xdx R 1 0

    • [PDF File]Math 241 Homework 12 Solutions - University of Hawaiʻi

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      semicircle y =-2 1 - x 2 to the semicircle y = 2 1 - x . 5. The base of a solid is the region between the curve y = 22 sin x and the interval 30, p 4 on the x-axis. The cross-sections perpen-dicular to the x-axis are a. equilateral triangles with bases running from the x-axis to the

    • [PDF File]The Squeeze Theorem - UCLA Mathematics

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      x!1 8x5 + 3x2 4 4 9x5 = lim x!1 8 + 3x 3 4x 5 4x 5 59 = lim x!1(8 + 3x 3 4x 5) lim x!1(4x 9) = 8 9 = 8 9: This technique of writing the denominator as a constant term plus terms with negative exponents is a good general strategy for determining the end behavior of rational functions. 2.Consider f(x) = sin(2x+ 7)cos(x2) + cos2(4 x3) x. Find lim ...

    • [PDF File]Trigonometric Substitutions Math 121 Calculus II

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      Since the derivative of f(x) = 1 2 x2 is x, the length is L= Z 1 0 p 1 + x2 dx: We’ll use the trig sub of the second kind with x= tan , dx= sec2 d , and p 1 + x2 = sec . Then the integral becomes L= Z ˇ=4 0 sec3 d : It takes an application of integration by parts to nd that an antiderivative of sec3 is 1 2 sec tan + 1 2 ln j+ tan . Given ...

    • [PDF File]PLOTTING AND GRAPHICS OPTIONS IN MATHEMATICA

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      Plot Sqrt 1-x^2 ,-Sqrt 1-x^2 , x,-1, 1 , AspectRatio ÆAutomatic -1.0 -0.5 0.5 1.0-1.0-0.5 0.5 1.0 voila. Or, we can use the AspectRatio command to make an even more oblate shape (but the figure is still a circle):

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