Math 105 Practice Exam 31.(a) Evaluatelim(x,y)→(1,-1)sin(x2+y)x2+yor show that it doesn’t exist.(b) Consider the area functionA(x) =Rx1f(t)dt, withA(2) = 6 andA(3) = 5.ComputeZ23f(t)dt.(c) A self-employed software engineer estimates that her annual income over the next10 years will steadily increase according to the formula 70,000e0.1t, wheretis thetime in years. She decides to save 10% of her income in an account paying 6%annual interest, compounded continuously. Treating the savings as a continuousincome stream over a 10-year period, find the present value.(d) Draw the level curves of the graph off(x, y) = 2x2+y2at the heights 0,1,2.(e) EvaluateZ10cos(√x)√xdx.(f) Letf(x, y) =x+yx-y. Use linear approximation to estimatef(2.95,2.05).2. EvaluateZx+ 2x(x2-1)dx.3. Find the area of the region in the first quadrant bounded byy=1x,y= 4x, andy=12x.4. Findksuch thatf(x) =k(x+1)3is a probability density function on the interval [0,∞),for some random variableX. Then compute the probability that 1
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Probability theory, lim, probability density function, self-employed software engineer