Master Cheat.pdf - Math 104A Homework 1 Section 1.1 2 4a 4c 8 12a 12b 24 2 Find intervals containing solutions to the following equations(a(b(c(d x \u2212

Master Cheat.pdf - Math 104A Homework 1 Section 1.1 2 4a 4c...

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Math 104A - Homework 1Section 1.1 - 2, 4a, 4c, 8, 12a, 12b, 242 Find intervals containing solutions to the following equations.(a)x-3-x= 0(b) 4x2-ex= 0(c)x3-2x2-4x+ 2 = 0(d)x3+ 4.001x2+ 4.002x+ 1.101 = 04a Find maxaxb|f(x)|forf(x) = (2-ex+ 2x)/3, x[0,1].4c Find maxaxb|f(x)|forf(x) = 2xcos(2x)-(x-2)2, x[2,4].8 Find the third Taylor polynomialP3(x) for the functionf(x) =x+ 1aboutx0= 0. Approximate0.5,0.75,1.25,and1.5 usingP3(x), andfind the actual (absolute and relative) errors.12a Find the third Taylor polynomialP3(x) forf(x) = 2xcos(2x)-(x-2)2andx0= 0, and use it to approximatef(0.4).12b Use the error formula in Taylor’s theorem to find an upper bound for theabsolute error|f(0.4)-P3(0.4)|. Compute the actual absolute error.24 Theerror functiondefined byerf(x) =2πZx0e-t2dtgives the probability that any one of a series of trials will lie withinxunitsof the mean, assuming that the trials have a normal distribution with mean0 and standard deviation22. This integral cannot be evaluated in terms ofelementary functions, so an approximating technique must be used.(a) Integrate the Maclaurin series fore-x2to show thaterf(x) =2πXk=0(-1)kx2k+1(2k+ 1)k!.(b) The error function can also be expressed in the formerf(x) =2πe-x2Xk=02kx2k+11·3·5· · ·(2k+ 1).Verify that the two series agree fork= 1,2,3,and 4. (Hint: Use theMaclaurin series fore-x2).(c) Use the series in part (a) to approximate erf(1) to within 10-7.(d) Use the same number of terms as in part (c) to approximate erf(1) withthe series in part (b).(e) Explain why difficulties occur using the series in part (b) to approximateerf(1).1
Math 104A - Homework 1Section 1.1 - 2, 4a, 4c, 8, 12a, 12b, 242Find intervals containing solutions to the following equations.(a)f(x) =x-3-x= 0.(b)f(x) = 4x2-ex= 0.

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