Disc5 - Math 17B Kouba Discussion Sheet 5 1 $2+1 1 1.)...

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Unformatted text preview: Math 17B Kouba Discussion Sheet 5 1 $2+1 1 1.) Compute the Midpoint Estimate, A16, for / dac . Compare your answer with 0 the exact value of the integral. 1 2.) Compute the Trapezoidal Estimate, M5, for / \/1 — xda: . Compare your answer —1 with the exact value of the integral. 3.) Determine the value of n so that the Trapezoidal Estimate, Tn, estimates the exact 1/2 . value of / 6‘sz da: with absolute error at most 0.00001. 0 4.) Determine the value of n so that the Midpoint Estimate, Tn, estimates the exact value 3 x + 1 of / d3: with absolute error at Inost 0.00001. 0 in + 5 5.) Compute the following improper integrals. 00 1 0 3 00 1 . ——-—— d b. m d . d 8L)/1 1:(:r+4) m e a: C)/_1 :I:+1 x d. d . d )/_Oo$2+9 a: e)1 93—1 3: . 1 6.) Consider the region R (in the first quadrant) bounded by the graphs of y = E, m = 1, and y 2: 0. a.) Determine if R has finite or infinite area. b.) Form a solid by revolving R about the x-axis. Determine if the resulting volume is finite or infinite. 7.) Find the following Taylor polynomials of degree n about a = 0, Pn(a:), for the indicated functions. A a.) f(a:)::z:4+333—$2+3a:—5,n:2 b.) fx)=m4+a:3~:c2+39:—5,n=4 c.) f(:r)=xex,n=3 d.) f(;r)= m+4,n=2 93—2 :3 + 1 : P0(:c), 131(23), P2(:r), P3(:L') . Compare the values of the function and its Taylor polynomials at a: = .1 and :c = 2. What conclusion do you draw ? 8.) Find the following Taylor polynomials about a = 0 for the function f = 1 9.) It is well known that the integral / 6332 dm has no closed-form anti-derivative. Replace 0 g 1 f = 6””2 with 134(17), its fourth-degree Taylor Polynomial centered at as = 0, to get an estimate for this definite integral. Compare this value with one obtained by a calculator which computes definite integrals and determine the absolute percentage error in your estimate. ****>|<******************************>ls*********************************** THE FOLLOWING} PROBLEM IS FOR RECREATIONAL PURPOSES ONLY. 10.) A nonnegative integer I is a perfect square, triangular (PST) number ifI is equal to the square of a nonnegative integer AND is also equal to one—half the product of consecutive nonnegative integers. Find the first four PST numbers. ...
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Disc5 - Math 17B Kouba Discussion Sheet 5 1 $2+1 1 1.)...

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