1.(10)  Find the area beneath f(x) = xe^x, between x = 0 and x = 1, and above the x-axis.
 

2.(10)  a.  State the Mean Value Theorem

b. Sketch a graph that shows what it is telling us.

c. Explain in two or three sentences, in your own words, what it says.
 

3.(10)  Integrate:  ò[((x-1)³)/( x²)]dx
 

4.(10)  Approximate ò₀^2p[(sinx)/( x)]dx using the Midpoint Rule with three intervals.
 

5.(10)  Use the definition of Taylor polynomial to compute P₃(x) for the function Öx expanded about a = 4.
 

6.(10)  Find the general solution to   y¢¢-3y¢+2y = 0
 

7.(10)  Find the sum of the series:  ²/₃+²/₉+[2/ 27]+[2/ 81]+¼

1.(10) a. Write the Taylor series for f(x) = e^x.

b.  Find a series expansion for g(x) = e^-x2.

c.  On what interval does the series in (b) converge to e^-x2?

d.  Find a series expansion for h(x) = -2xe^-x2.

e.  What is the interval of convergence for the series in (d)?  Why?

f.  Find a series representation for òe^-x2dx
 

2.(10)  Show all of your work to compute   òx²sin(px)dx
 

3.(10) a. Why is  ò_1/3¹[1/( [Ö(3x-1)])]dx an improper integral?

b.  Determine if the integral in (a) converges or diverges.  Show your work.
 

4.(10)  SET UP a definite integral to compute the volume of a solid of revolution if ln(x+1), 0 £ x £ 4 is rotated about the horizontal line y = -2.

a. How much work does it take to pump a liquid with density w lb/ft³ from a full upright, right circular cylindrical tank of radius 5 ft and height 10 feet to a level 4 feet above the top of the tank?

b. A tank contains 10000 gallons of fresh water.  Water that has a salt concentration of 0.5 lb/gal is flowing into the tank at a rate of 10 gal/min.  The salt water is mixed and the tank is draining water also at 10 gal/min.  How long will it take for the water in the pool have a concentration of 0.25 lb/gal?

6.(10)  Consider the system [(dy)/( dt)] = y(3-y-x), [(dx)/( dt)] = x(2-x-2y).

a. Locate all nullclines and equilibrium points and sketch them in the phase plane of this system.

b. Sketch the three trajectories that start from (1, 3), (1,1) and (.5, .5).
 

7.(10) a.  Find the quadratic Taylor polynomial P₂(x) for the function f(x) = e^-2x expanded about a = 0 .

b.  Give an upper bound for the error that comes from using the Taylor polynomial P₂(x) to approximate e^-2x at x = 0.3 .

c.  Use your calculator to find the actual error.
 

8.(10)  Consider the differential equation  dy/dx = xy+y and initial condition  y(0) = 1.

a.  What is the Euler's approximation to y(1) found using a step size of 0.1?  Show the first two steps explicitly.  After that, you may simply use your calculator.

b.  Find the exact value of y(1) by solving the differential equation.
 

9.(10)   (a) Using two subdivisions, find the left, right and trapezoid approximations to ò₀¹(1-e^-x) dx.

left:

right:

trapezoidal:

(b)  Draw 3 sketches showing what each approximation represents.

Given only the information in (a), what is your best estimate for the value of this integral?  Explain your answer.
 

10.(2@)  True/False Questions. If your answer is ``False,'' explain why the statement is false.

_____ a.  If  ò₀^¥ f(x) dx converges and if g(x) > f(x), then ò₀^¥ g(x) dx diverges.

Reason:

_____ b.  Integration by parts cannot be used if the integrand is a quotient.

Reason:

_____ c.  The Riemann sum å_{k = 1}ⁿp z_ksin(z_k)Dz on the interval 1 £ z £ 5 may be approximated by pò₁⁵zsinz dz.

Reason:

_____ d.  The arc length of [1/( sinx)] from [(p)/ 4] to [(3p)/ 4] equals

ò_{[(p)/ 4]}^{[(3p)/ 4]}Ö{1+([1/( cosx)])²} dx.

Reason:

_____ e.  The series  1+x+x²+x³+¼ converges for all x in the interval (-1,1).

Reason: