Mathematics

1(20 pts)—(Integration)—Carefully evaluate the following integrals (4 pts each): (a) r 1x ln(x) dx (b) J(x2+2x+3)ln(x)dx (c) f (x2 + x —12) dx (d) Je )71 (e) dx 0 2(10 pts) )—(Improper Integration)—Evaluate the following integrals (S pts each)
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w 1 (a) directly: f—ex x2 X – 1 dx (b) by comparison test show that dx actually converges
3(8 pts)–(Mouse Population)—The mouse population follows a certain mathematical behavior with respect of time and is related to things such as food availability, shelter as well as the proximity of a house cat. In such case, the behavior of the mouse population is described by the differential equation: dP = 0.04P — 0.01P2 —0.03 di (a) Find P(t) assuming that we are starting with five mice (b) Describe the general behavior of the mouse population (graph) (c) What happen to the mouse population if left for ever?
4(8 pts)-(Crime Investigation)-At noon the body of a murdered man is found in a room that has been maintained at a constant temperature of 70 F. The coroner arrives at 1:00 PM and notes that the body temperature is 90 F. An hour later the temperature of the body is noted to be 87 F. If the body temperature is 98.6 F at the time of death, find the time of death. Hint: (a) what math phenomena is this? (b) write the differential equation of such phenomena (c) solve (d) answer the last question
5(8 pts)–(Volume)-Forf(x,y) = x, within the fourth quadrant and over the region D bounded by x = 0, x = 2, y = —x2 and y = —3x —1 find the double integral if f (x, y)dA
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6(11 pts)—(Volume)—A cylinder is standing on the x-y plane aroundNth::-Eacxils.m cylinder has height 3 and radius 2. a(lpt)—Using basic geometry what is the volume of such cylinder? TiAlLet b(2 pts)–What is the f(x,y) function? c(1 pt)—Draw a picture of the domain d(3 pts)—What is the Algebraic expression of the domain? e(4 pt)—Now, find the volume using double integration (same answer as (a)). 2 I Hint: At some point, assume the result: (V22 — y2)dy = -2
7(12 pts)–(Approximation of solution of differential equation by Euler’s Method)— dy y Consider the differential equation — = —±1 dx x a(1 pt)—Is this a separable differential equation? Explain why or why not. b(3 pts)–Verify that the solution to such an equation is y = xln(x) c(5 pts)—Using Euler’s approximation in the interval [1,2] with n = 10 (that is Ax=0.1) and with the initial value condition being yo = y(1) = 0 , find y10 . (Use a table) d(1 pt)—Compute the corresponding values of the real solution (b) at above increments e(1 pt)—Together, sketch roughly the graphs of both (c) and (d) f(1 pt)—Comment on Euler’s approximation usefulness and how to improve it
8(5 pts)–(Rational?)—Prove that 1.5234234234234234….. actually converges to 1691
1110 Make sure you give the proper argument(s).
9(8 pts)—(Econ/Optimization)—You make x units of a product that sells for $15 per unit and y units of a product that sells for $10 per unit. The cost is: C(x, y) — (3×2 + xY + 3y2) +7x +4y+ 400 Find x & y that maximizes your profit P(x,y). 100
10(8 pts)—(Carpentry/Optimization)—Find the dimensions of the absolute cheapest rectangular box with an open top and a volume 96 cubic feet for which the cost per square feet of the base is three times that of the cost of the sides. 11(6 pts) )–(Approximation of functions)—By computing the first (say six) Taylor polynomial coefficients for the function f(x). — x) at x = 0, figure out a formula for the n-th Taylor coefficient c.„ and write the expanded polynomial that approximate f(x) 12(6 pts) )–(Approximation of integrals)—Consider the integral J exit& . Since we cannot evaluate it as written (why?), then let’s, instead, first approximate f(x) to by a Taylor polynomial of order 4 around x = 0 and then integrate the resulting approximation.

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