2a) Let a, b e R with a < b and let g [a, bR be continuous. Show that g(x) cos(nx) dx→ 0 n →oo. as Hint: Let ε > 0, By uniform continuity of g, there exists δ > 0 such that 2(b - a Choose points a = xo < x1 < . . . < Xm such that Irh-1-2k| < δ. Then we may write rb g (z) cos(nx) dx = An + Bn where 7m (g(x)...
3i)16 in polar form: z r(cos 0isin 0) where (1 Write the complex number z and e= The angle should satisfy 0 0 < 2«.
1 (b) g(x)- х? cos 1 (c) If g(0) 3, and g'(0) -2, what is h'(0) where h(x) ? Show your work. = (x)8) 1 (b) g(x)- х? cos 1 (c) If g(0) 3, and g'(0) -2, what is h'(0) where h(x) ? Show your work. = (x)8)
(Leave your answer in the exact form - no decimals) 4. [16 points) Given that cos p = following: <a<n. Find the exact values of each (a) 4 points) sinſ and cos a (b) (4 points) cos() (c) (4 points) sin(2a) (d) (4 points) In what quadrant does 2a lie? Justify your answer.
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
Problem 6. Describe the surface r(u, u)-R cos u x + R sin u ý + uz where 0 < u < 2π and 0 < u-H. and R and H are positive constants. What is the surface element and what is the total surface area? Show that Or/au, or/àv are continuous across the "cut at 2T coS W T
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
if csc x -13 and 3 <x< 2л. Use a half-angle identity to find cos Enter the exact answer. Enclose numerators and denominators in parentheses. For example, (a - b)/ (1 n). Equation Editor Common sin(a) tan(a) cos(a) Y 00 b sec(a) csc(a) cot(a) к Va Va a1 sina) cos (a tan о Ф X COS o
Bartle The Elements of Integration and Lebesgue Measure: 4.R. If fe M*(X, X) and is due < +00, then the set N = {xe X: f(x) > 0} is o-finite (that is, there exists a sequence (Fa) in X such that N CU Fn and u(F.) < too).
upson's Rule with n=4 #5 (9) Use Simpson's Rule wi intervals to estimate ex-l at 16 Find the exact value - A the error. drant of the integral linpartas) and the =0: #5, Use integration by. Seť Int de #7 (a) Evaluate St sin x cos x dic (6) If g(x) = 5 Je tidt, find g'(x) and g'(o). |#8 use partial fractions to find substitution to evaluate $3x(x-3) dx #0 (a). Find 52 sin o do (6) Find the...