1 x 2 sqrt x 2

    • [PDF File]Techniques of Integration - Whitman College

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      cos2 x = 1−sin2 x sec2 x = 1+tan2 x tan2 x = sec2 x −1. If your function contains 1−x2, as in the example above, try x = sinu; if it contains 1+x2 try x = tanu; and if it contains x2 − 1, try x = secu. Sometimes you will need to try something a bit different to handle constants other than one. EXAMPLE10.2.2 Evaluate Z p 4− 9x2 dx. We ...

    • [PDF File]Techniques of Integration - Whitman College

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      Then since u = 1− x2: Z x3 p 1− x2 dx = 1 5 (1−x2)− 1 3 (1−x2)3/2 + C. To summarize: if we suspect that a given function is the derivative of another via the chain rule, we let u denote a likely candidate for the inner function, then translate the given function so that it is written entirely in terms of u, with no x remaining in the ...

    • [PDF File]Solution to Math 2433 Calculus III Term Exam. #3

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      x2 xdydx= Z 3 2 x x+ 6 x2 dx = Z 3 22 x2 + 6x x3 dx= 1 3 x3 + 3x2 1 4 x4 = 9 + 27 81 4 + 8 3 12 + 4 = 125 12 2. Let T be the solid bounded by the paraboloid z= 4 x2 y2 and below by the xy-plane. Find the volume of T. (Hint, use polar coordinates). Answer The intersection of z= 4 2x 22y and xyplane is 0 = 4 x2 y;i.e. x2 +y = 4: In polar ...

    • [PDF File]Integration by substitution

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      1+x2 dx = 2 3 (1+x2)3/2 +c We have completed the integration by substitution. Let us analyse this example a little further by comparing the integrand with the general case f(g(x))g′(x). Suppose we write g(x) = 1+x2 and f(u) = √ u Then we note that the composition1of the functions f and g is f(g(x)) = √ 1+x2. 1when finding the composition ...

    • [PDF File]Expected Value The expected value of a random variable ...

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      Calculate E(X2). E(X2) 2= 2sum_{i=1}^{6} i p(i) = 1 p(1) + 2 2 p(2) + 32 p(3) + 42 p(4) + 5 p(5) + 62 p(6) = 1/6*(1+4+9+16+25+36) = 91/6 E(X) is the expected value or 1st moment of X. E(Xn) is called the nth moment of X. Calculate E(sqrt(X)) = sum_{i=1}^{6} sqrt(i) p(i) Calculate E(eX) = sum_{i=1}^{6} ei p(i) (Do at home) Ex.

    • [PDF File]Table of Integrals

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      1 2 x2 1 2 a2 lnja2 + x2j (12) Z 1 ax2 + bx+ c dx= 2 p 4ac b2 tan 1 p 2ax+ b 4ac b2 (13) Z 1 (x+ a)(x+ b) dx= 1 b a ln a+ x b+ x; a6= b (14) Z x (x+ a)2 dx= a a+ x + lnja+ xj (15) Z x ax2 + bx+ c dx= 1 2a lnjax2 + bx+ cj b a p 4ac b2 tan 1 p 2ax+ b 4ac b2 (16) Integrals with Roots Z p x adx= 2 3 (x 2a)3=2 (17) Z 1 p x1a dx= 2 p x a (18) Z 1 p a ...

    • [PDF File]The Squeeze Theorem

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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]Homework Assignment 4 - Elizabethtown College

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      x2 + y2 sin 1 x y = lim (x;y)!(0;0) (x y) (x y) p x2 + y2 sin 1 x y = 0; because the rst factor, (x y), goes to 0, and everything else is bounded. This proves the di erentiability at the origin. On the other hand, the partial derivatives are not continuous. Ideed, for x6= ywe have f x(x;y) = 2(x y)sin 1 x y cos 1 x y and f y(x;y) = 2(x y)sin 1 ...

    • [PDF File]Chapter 2: Simple Linear Regression

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      1 The model The simple linear regression model for nobser- vations can be written as yi= β 0 +β 1xi+ei, i= 1,2,··· ,n. (1) The designation simple indicates that there is only one predictor variable x, and linear means that the model is linear in β 0 and β 1.The intercept β 0 and the slope β 1 are unknown constants, and

    • [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]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]Volumes by Cylindrical Shells: the Shell Method

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      Ex. Find the volume of the solid generated by revolving the region bounded by y = x2, y = 0, x = −1, and x = 1, about the line x = 2. The axis of rotation, x = 2, is a line parallel to the y-axis, therefore, the

    • [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 Inner products and norms - Princeton University

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      1 2! is inde nite. To see this, consider x= (1;0)T and x= (0;1)T: 2.2 Eigenvalues of positive semide nite matrices Theorem 2. The eigenvalues of a symmetric real-valued matrix Aare real. Proof: Let x 2Cn be a nonzero eigenvector of Aand let 2Cbe the corresponding eigenvalue; i.e., Ax = x. By multiplying either side of the equality by the conjugate

    • [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]Trig Substitution

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      Example 2. Compute Z 1 (x2 9)2 dx Soluion: This is almost identical to the rst example. Again, no u-substitution will work, and even though we have no square roots, we can use trig sub with x= 3sec and dx= 3sec tan d . Hence, the integral becomes: Z 1 (x 2 9) dx = Z 1 (9tan2 )2 (3sec tan d ) = Z sec 27tan3 ;d :

    • [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]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]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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