Có 99+ tài liệu thuộc chủ đề "khóa học tính toán"
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Use substitution to get rid of the √ x.. 802 CHAPTER 25 Finding Antiderivatives—An Introduction to Indefinite Integration Either we can express this as. If we could get the sum in the numerator as opposed to the denominator, we’d be happier. We’ll accomplish this with another substitution.. We can’t let u = x 2 or 1 − x 2 because...
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we chop [1, 5] into 50 equal pieces each of length x . we chop [1, 5] into 400 equal pieces each of length x . Using T 400 as an upper bound and M 400 as a lower bound, we’ve nailed down the value of this integral to 4 decimal places.. the endpoints of integration, and the number of...
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Simpson’s rule generally gives a substantially better numerical estimate than either the midpoint or the trapezoidal sums. Below we do this for the integral 5. EXAMPLE 26.3 Compare M n , T n , and Simpson’s rule (the weighted average) for 5 1 1. Simpson’s rule = 2M 4 3 + T . Simpson’s rule = 2M 8 3 +...
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The limits of integration, while not explicit in the summation notation of the Riemann sum, are implicitly there. The limits of integration are always determined by the endpoints of the interval being chopped up.. 1+ 1 r 2 kilograms per square meter, where r is the number of meters from the center of the meteorite’s impact. What is the mass...
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The number of calories per cubic millimeter varies with x , where x is the depth from the top of the mold. The calorie density is given by δ(x) calories/mm 3 . Top of the mold. A mineral suspended in the liquid is setting. Its density at a depth of h meters from the top is given by 5h milligrams...
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27.2 Slicing to Find the Area Between Two Curves 851. Write an integral (or the sum or difference of integrals) giving the area of the region shaded below. Which of the expressions below give the area of the shaded region? (Select all such expressions.). 1 (arctan x − π 4 ) dx. Find the area between the curve y =...
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EXAMPLE 28.5 Let’s model a bagel by revolving a disk of radius 1 centered at the origin about the vertical line x = 2. What is the volume of the bagel?. Figure 28.13. SOLUTION We can slice the disk either along the x-axis or along the y-axis. We’ll opt for slicing along the x -axis.. We can revolve the half-disk...
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lift up the ith slice ≈ (weight of ith slice. (distance to the top) The volume of the ith slice = π(4) 2 x = 16π x m 3 . the mass of the ith slice π x kg.. We calculate the weight of the slice by multiplying by g ≈ 9.8 m/sec 2 . The weight of the ith...
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We do integration by parts again.. 3 sin 3x. e 2x sin 3x dx. 3 e 2x sin 3x − 1 3 sin 3x · 2e 2x dx. 9 e 2x sin 3x − 4 9 e 2x sin 3x dx. 13 9 e 2x sin 3x dx. 9 e 2x sin 3x (29.1) e 2x sin 3x dx. 13...
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Method 2: We’d love to get rid of that square root in the denominator. EXAMPLE 29.14 Find 1 0. cos 2 θ can be replaced by cos θ because θ has been restricted to. In both of the previous examples we’ve said. It is because we’ve restricted θ to. Notice that once we’ve established that θ. Generalizing the Method of...
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EXAMPLE 29.19 Find − 2x − 1. Do not attempt partial fractions until the integral is in the form. 29.4 IMPROPER INTEGRALS. In this section we’ll push the boundaries of the definition of the definite integral.. The definition of the definite integral b. The integral. x 2 dx is improper because the interval of integration is unbounded. 0 x −...
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1 Figure 29.13. Figure 29.14 REMARK. c xe − x 2 dx. xe − x 2 dx = 1. 1 Figure 29.15. EXAMPLE 29.29 Is. Figure 29.16. See Figure 29.17.. Figure 29.17. We can argue that the harmonic series . We can argue that the infinite series. Figure 29.18. Similarly, after completing Exercise 29.11 we can argue that. We will...
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Figure 30.1. Refining the tangent line approximation: Any polynomial approximation, P k (x), of sin x about x = 0 certainly ought to be as good a local approximation as is the tangent line approximation, P 1 (x. Therefore, P k (x) is of the form x + a 2 x 2 + a 3 x 3. Note that sin...
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EXAMPLE 30.5 Approximate √ 5. 34 using the appropriate second degree Taylor polynomial.. We must center the Taylor polynomial at a point near 34 at which the values of f , f , and f can be readily computed.. Center the Taylor polynomial at x = 32.. If you study closely the numerical data in this section you can start...
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30.3 TAYLOR SERIES. Defining Taylor Series. In many examples in this chapter we’ve observed that the higher the degree of the Taylor polynomial generated by f at x = b, the better it approximates f (x) for x near b. For functions such as sin x and cos x , the higher the degree of the Taylor polynomial the longer...
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Find the Maclaurin series for 1 1. What is the radius of convergence?. Use the binomial series to find the Maclaurin series for √ 1. In Problems 25 through 34, use any method to find the Maclaurin series for f (x).. (A graphing instrument can be used.). 0 for all k. Conclude that the Maclaurin series for f (x) converges...
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EXERCISE 30.6 Verify that f (x. y has a power series representation, then that solution must be of the form C 0 cos x + C 1 sin x, where C 0 and C 1 are constants.. A graph of the partial sum J 0 (x. Not only are we able to show that the series for e x ,...
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Let x = x(t ) be the number of hundreds of animals of species A at time t.. For each system of differential equations, describe the nature of the interaction between the two species. What happens to each species in the absence of the other?. dt = 0.02x − 0.001x 2 − 0.002xy. dt = 0.008y − 0.004y 2 −...
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PH ƯƠ NG PHÁP TR I HÌNH TRÊN M T PH NG Ả Ặ Ẳ Ng ườ i so n :Tr n Th Hi n ạ ầ ị ề. T toán tr ổ ườ ng THPT Chuyên H Long ạ. Khi gi i m t bài toán v t di n mà các d ki n c a nó liên...