Approximating definite
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Calculus: Early Transcendentals (3rd Edition)
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- (a) Evaluate the integral: Hint: = Your answer should be in the form kn, where k is an integer. What is the value of k? d dx —arctan(r) a₁ = a2 = 2 16 x² + 4 · 6²³ a3 = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. 16 f(x) = x² + 4 What are the first few terms of S? ao= a4 = dr 1 I²+1 (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of in terms of an infinite series. Approximate the value of by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.)arrow_forwardCan you show me how to solve this?arrow_forwardPlease solution speedarrow_forward
- Use series to approximate the definite integral to within the indicated accuracy: sin(x) dx, with an error < 10 4 Note: The answer you derive here should be the partial sum of an appropriate series (the number of terms determined by an error estimate). This number is not necessarily the correct value of the integral truncated to the correct number of decimal places. 0.234arrow_forward16 dz 2 + 4 (a) Evaluate the integral: Your answer should be in the form kr, where k is an integer. What is the value of k? Hint: arctan(z) | r2 +1 (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function 16 f(=) Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. r2 + 4 What are the first few terms of S? a, = 32 a2 = 20 128 az = 112 512 a4 = 576 of of ofarrow_forwardUse a power series to approximate the value of the integral with an error of less than 0.0001. (Round your answer to four decimal places.) e-x3 dx e-x dx 2 Σarrow_forward
- Find a formula for the power series of f(x) = 2 ln (1 + x), −1 < x < 1 in the form Hint: First, find the power series for g(x) = 2 1 + x Then integrate. (Express numbers in exact form. Use symbolic notation and fractions where needed.) an = 8 0 n=1 an.arrow_forwardTypewritten for upvote. Thank youarrow_forward(a) Find a power series for the function f : (0, 0) → R given by f(x) = $in² about the point x = A. Hint: The Taylor series for xH sin x may be helpful. 2. = B. (b) Find the Taylor series for the function f : (0, 00) → R given by f(x) = log x about the point x =arrow_forward
- Use series to approximate the definite integral I to within the indicated accuracy. I = - 100.5 x³e-x² dx ([error] < 0.001) I =arrow_forward(a) Evaluate the integral: k Your answer should be in the form kä, where k is an integer. What is the value of k? 1 Hint: -arctan(2) d dx x² + 1 = a1 (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function 32 f(x) Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. x² + 4 a2 What are the first few terms of S? ao = = || az = 2 32 x² + 4 = = a4 = dx (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of 7 in terms of an infinite series. Approximate the value of π by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 7 terms? (Use the alternating series estimation.)arrow_forwarda. Series for sinh lx Find the first four nonzero terms of the Taylor series for sinh x = dt V1 + f° b. Use the first three terms of the series in part (a) to estimate sinh0.25. Give an upper bound for the magnitude of the estimation error.arrow_forward
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