Practice Final Please work out each of the problems below. Credit will be based on the steps towards the final answer. Show your work. Problem 1 Sketch the following. A. The point (3,4,1). Solution We draw the xyzaxes, the shadow at the point (3,4) in the xyplane and move it up 1 unit.
B. The surface z = x^{2} + y^{2} Solution This is a paraboloid with the zaxis as its central axis. The graph is shown below.
C. Five level curves to the surface z = x  y^{2} Solution Draw a table as follows
Notice that these are all sideways parabolas with distinct xintercepts that correspond to the value of k. The graphs below show these level curves.
Problem 2 Find f_{xz} for f(x,y,z) = x^{2}z + y^{2}  y cos(xz) Solution We first find the derivative with respect to x by letting y and z be constant. f_{x} = 2xz + yz sin(xz) Note the chain rule, the derivative of xz is z Now find the derivative of this with respect to z, f_{xz} = 2x + y sin(xz) + xyz cos(xz) We used the product rule for the second term
Problem 3 A swimwear store sells both men's and women's swimsuits and has determined that the profit, P, it can make in stocking x men's suits and y women's suits is P(x,y) = 2x^{2} + y^{2} + xy  900x  1100y + 400,000 Determine the number of men's and the number of women's suits that should be stocked in order to maximize profit. Then determine the maximum profit. Solution We find the partial derivatives and set them equal to 0. P_{x} = 4x + y  900 P_{y} = 2y + x  1100 This gives us the two equations 4x + y  900 = 0 x + 2y  1100 = 0 The first equation can be written as y = 900  4x Substituting into the second equation gives x + 2(900  4x)  1100 = 0 x + 1800  8x  1100 = 0 7x = 700 x = 100 Hence y = 900  4(100) = 500 The store should stock 100 men's suits and 500 women's suits. The profit will be P(100,500) = 2(100)^{2} + (500)^{2} + (100)(500)  900(100)  1100(500) + 400,000 = 80,000 Problem 4 A medical researcher is studying the effect that exercise has on a women's percent body fat. The table below shows the results of the corresponding study done. Determine the least squares regression line for this data and use it to estimate the mean percent body fat for a women that exercises 3 hours per week.
Solution We use a calculator to find the equation of the regression line. y = 3.155x +31.414 We plug in x = 3 to get y(3) = 3.155(3) + 31.414 = 21.95 We predict that the expected percent body fat for a women who exercises three hours per week is 21.95%.
Problem 5 Switch the order of integration to evaluate the following
Solution To change the order we first draw the region shown below.
Now we switch the order using dydx. Notice that y goes from 0 to x and x goes from 0 to 5. We have
This integral can be solved using substitution with u = 1 + x^{2} du = 2xdx This gives
Problem 6 Consider the following game. Pick a card. If you select a face card (Jack, Queen, or King) you win $3. Otherwise, you lose $1. Find the expected value and comment on how this number shows whether playing this game many times is a good idea.
Solution We write down the probability distribution table
The expected value is just
3
10
1 Since this is a negative value, we would expect to lose money on the average.
Problem 7 Consider the following probability density function f(x) = x + 0.5 0 < x < 1 A. Find P(0.5 < x < 1). Solution We just find the integral
B. Find the standard deviation. Solution First find the mean.
Now calculate the integral using a calculator.
Finally take a square root to get the standard deviation s = 0.276 C. Find the median. To find the median we set the following integral equal to 1/2.
Multiplying by 2 and bringing all the terms to the left hand side gives m^{2} + m  1 = 0 Using the quadratic formula gives m = 0.618
Problem 8 A pair of baby rabbits is introduced to an island. Assume that rabbits take two months to reproduce when they have 2 pairs of new baby rabbits. Each month thereafter, the adult rabbits produce two additional pairs of rabbits. Assume that the rabbits never die. A. Write down the number of pairs of rabbits on the island during the first 6 months. Solution We have 1, 1, 3, 5, 11, 21
B. Find constants u and v such that a_{n} = u a_{n1} + v a_{n2} Where a_{n} represents the number of rabbits after the n^{th} month. Solution We just notice that to find the current generation, add the newborns to the prior population. The newborns are coming from rabbits that were alive two months previously. Hence u = 1 and v = 2
Problem 9 A very long river that runs along a vast area of farming communities has 3000 trout in its first mile. Because of pollutants, each mile contains only 80% of the fish that the previous mile had. so that the second mile has 2400 trout and the third has 1820 trout. Considering the river as running an infinite distance, how many trout live in the river? Solution We have a geometric sequence 3000, 2400, 1820, ... with r = 0.8 and a = 3000 Hence the sum is
3000 There are 15,000 trout in the river.
Problem 10 Determine whether the following series converge or diverge. Be sure to cite the test that you are using. A. Solution We use the n^{th} term test.
Since the limit does not go to 0, by nth term test, the series diverges.
We can rewrite this as
This is a Pseries with p = 3 > 1. Hence by the PSeries Test the series converges.
Problem 11 Find a power series representation for
using the fact that
then find the radius of convergence for your solution
Solution We substitute x^{2} in for x to get
Now integrate to get
Problem 12 Find the third degree Taylor polynomial centered at x = 0 for f(x) = x e^{x}
Solution We find the derivative of f and evaluate them at x = 0. f '(x) = e^{x} + x e^{x} = e^{x}(1 + x) f '(0) = 1 f ''(x) = e^{x} + (1+ x) e^{x} = e^{x}(2 + x) f '(0) = 2 f '''(x) = e^{x} + (2+ x) e^{x} = e^{x}(3 + x) f '(0) = 3 We now plug this into the Taylor polynomial formula to get
0
1
2
3
1
Problem 13 Use Newton's method with initial guess of 1 to estimate
within three decimal places of accuracy by finding the solution to x^{2}  2 = 0 Solution With f(x) = x^{2}  2 we have f '(x) = 2x so that
x_{n}^{2}  2 With our initial guess of x_{0} = 1, the table below gives the rest of the values
Hence 1.414 approximates the square root of 2 accurate up to three decimal places.
Problem 14 Solve the following differential equation
1 Solution We first find the integrating factor
now multiply through by the integrating factor to get
1
1
1 so that
Now integrate both sides to get
1 and finally multiply both sides by x to get y = x^{3} + x lnx + Cx
Problem 15 The rate of growth of a bacteria is equal to the quotient of the number of bacteria present and one more than the number of hour that the bacteria has been growing. If initially there were 5 grams of bacteria, how many grams of bacteria will there be in 8 hours? Solution We have the differential equation
dx x Now separate the variables to get
dx dt Now integrate to get ln x = ln(1 + t) + C_{1} Exponentiate both sides to get x = e^{ln(1 + t) + C1} = e^{ln(1 + t) }e^{c1} = C(1 + t) We have that when t = 0, x = 5 so that 5 = C(1 + 0) C = 5 so that x(8) = 5(1 + 8) = 45 There will be 45 grams of bacteria after 8 hours.
