Final Exam Name -------------MAT 286 Spring 2013

Circle your instructor's name:

Shaw (9:30) Shaw (11:00) Bruce Ozal Rhodes Snyder

Show your work and justify your answers. You may use a TI 83/84 Plus or other approved calculator, but you must write the steps you take with the calculator. A correct answer with no work will not receive credit. Point values are indicated.

Prob Possible Earned

1 10

2 15

3 30

4 10

5 10

6 10

7 10

8 10

9 10

Total 115

1. Suppose that the slope of the tangent line to a function f(x) at any xis given by

J' ( x) = ex - x3 + 1.

If the graph y = J(x) goes through the point (0, 1), find the function f(x).

I.-----------(10 points)

2. Consider the following function J(x} on the interval [O, 1].

J(x) = x3 + x.

a) Use calculus to find the exact area below the curve and above the x-axis on the given interval. ( Show all steps including the appropriate antiderivative and the evaluations. No credit will be given for providing only the final numerical value.)

2a. __________ _

(10 points)

b) Set up, but do not evaluate the integral that gives the volume of the solid formed by revolving the region below the curve f(x) = x3 +x and above the x-axis on the interval (0, 1] about the x- axis. Circle your integral.

(5 points)

3. Evaluate the following indefinite integrals. Circle your answers.

a)

J cos 2x sin 2x dx

(10 points)

b)

/ 1 d

x (lnx)2 x

(10 points)

3c)

j xcos(-x) dx

(10 points)

4. Determine whether the following improper integrals converge or diverge. If the integral converges, give the value of the convergent integral. Make sure your answer is easy to find.

a) f°o x2 dx.

lo Jx3 + 1

(5 points)

b}

(5 points)

5. Find the general solution of the differential equation

Be sure to solve for y.

(10 points)

dy y dx - x3 '

y > 0.

5. -------------

6. Find fx(x, y) and f1r11(x, y) for

f(x, y) = cos(3xy2) - 4x2y + 6y.

(10 points)

fx(x, y) = ________ _

f1111(x,y) = ________ _

7. Evaluate the double integral of the function J(x, y) = x2 + xy over the region R, where R = {(x, y) : 0 < x $ I, x2 < y :;; x }. Give the exact answer (no decimals).

7. ------------

(10 points)

8. Find the particular solution of the following differential equation subject to the initial condition that y = 2 when x = l.

dy x dx - 3y + I .:: 0.

8. ------------{10 points)

9. A tank initially contains 40 gallons of sugar water having a sugar concen­tration of 5 pounds per gallon. A sugar solution with a concentration of 3 pounds per gallon is poured into the tank at a rate of 4 gal per minute and is thoroughly mixed at the same time. A drain is opened at the bottom of the tank so that the thoroughly mixed sugar water is leaving the tank at the same rate of 4 gal per minute. How many pounds of sugar are in solution in the tank 5 minutes after the flow begins? Round your answer to two decimal places, if necessary.

9. ---------

(10 points)

Final Exam Name -------------MAT 286 Spring 2013

Circle your instructor's name:

Shaw (9:30) Shaw (11:00) Bruce Ozal Rhodes Snyder

Show your work and justify your answers. You may use a TI 83/84 Plus or other approved calculator, but you must write the steps you take with the calculator. A correct answer with no work will not receive credit. Point values are indicated.

Prob Possible Earned

1 10

2 15

3 30

4 10

5 10

6 10

7 10

8 10

9 10

Total 115

1. Suppose that the slope of the tangent line to a function J(x) at any x is given by

J'(x) = x4 - ex + 3.

If the graph y = f(x) goes through the point (0, 2), find the function J(x).

I. -----------(10 points)

2. Consider the following function J(x) on the interval [O, 1].

f(x) = x2 + 2x.

a) Use calculus to find the exact area below the curve and above the x-axis on the given interval. ( Show all steps including the appropriate antiderivative and the evaluations. No credit will be given for providing only the final numerical value.)

2a. __________ _

(10 points)

b) Set up, but do not evaluate the integral that gives the volume of the solid formed by revolving the region below the curve f(x) = x2 + 2x and above the x-axis on the interval [O, lj about the x-axis. Circle your integral.

(5 points)

3. Evaluate the following indefinite integrals. Circle your answers.

a)

j sin 1rx cos rrx dx

(10 points)

b)

J 1 d x (lnx)3 x

(10 points)

3c) J xe- 2x dx

(IO points)

4. Determine whether the following improper integrals converge or diverge. If the integral converges, give the value of the convergent integral. Make sure your answer is easy to find.

a) {'"' X

Jo (x2 + 1)2 dx.

(5 points)

b)

(5 points)

5. Find the general solution of the differential equation

dy y -=-, dx x4

y > 0.

Be sure to solve for y.

5. -------------

(10 points)

6. Find fy(x, y) and fxx(x, y) for

f(x,y) = sin(5x2y)-4xy2 + 6x.

(10 points)

fy(x,y)= _______ _

fxx(x,y) =---------

7. Evaluate the double integral of the function f(x, y) = x + xy2 over the region R, where R = {(x, y): 0 ~ x ~ 1, x2 ~ y ~ x}. Give the exact answer ( no decimals).

7. ------------

(10 points)

8. Solve the following differential equation subject to the initial condition that y = 2 when x = 1.

dy x dx - 2y + 3 = 0.

8. -----------

(10 points)

9. A tank initially contains 40 gallons of sugar water having a sugar concen­tration of 5 pounds per gallon. A sugar solution with a concentration of 3 pounds per gallon is poured into the tank at a rate of 4 gal per minute and is thoroughly mixed at the same time. A drain is opened at the bottom of the tank so that the thoroughly mixed sugar water is leaving the tank at the same rate of 4 gal per minute. How many pounds of sugar are in solution in the tank 5 minutes after the flow begins? Round your answer to two decimal places, if necessary.

9. ---------

(10 points)