Calculus 2

Calculus 2

Test File

Test #1

1.) Find the exact area between the curves f(x) = x² and g(x) = 2x + 3.

2.) Find the center of mass of the region bounded by f(x) = sin x and the x-axis between x = 0 and x = p. Feel free to use your calculator to help with integrating.

For problems #3 - 6, consider the region bounded by f(x) = cos x, x = 0, y = 0.

3.) Suppose the region is revolved around the x-axis. SET UP BUT DO NOT EVALUATE the integral for finding the volume of the resulting solid using cylindrical shells.

4.) Suppose the region is revolved around the x-axis. SET UP BUT DO NOT EVALUATE the integral for finding the volume of the resulting solid using disks.

5.) Suppose the region is revolved around the line y = 2. SET UP BUT DO NOT EVALUATE the integral for finding the volume of the resulting solid and use your calculator to evaluate it.

6.) Consider the top edge of the region. (the part formed by the cosine curve) Set up the integral needed for finding the length of that edge. USE YOU CALCULATOR TO EVALUATE IT.

7.) A force of 500 pounds stretches a spring 5 inches from its natural length of 18 inches. SET UP BUT DO NOT EVALUATE the integral needed to find the work done in stretching the spring 3 inches beyond its natural length.

8.) A vertical wall of a swimming pool is a rectangle. The wall is 30 feet and 10 feet high. SET UP BUT DO NOT EVALUATE the integral needed to find the fluid force on the wall if the pool is filled to a depth of 9 feet.

Test #2

1.) ò ln x dx 2.) ò sin³x cos²x dx

3.) ò e^2x sin 3x dx 4.) ò tan³x sec³x dx

5.) 6.) ò sec³x dx

7.) 8.) EXACT ANSWER!!

Test #3

1.) Set up BUT DO NOT SOLVE, the partial fraction decomposition for .

2.) 3.)

4.) 5.)

6.) 7.)

8.) Find the volume if the region from #7 is revolved around the y-axis.

9.) 10.)

Test #4

A.) Determine convergence or divergence forfor a_n as given. Give reasons for your answers.

1.) a_n = 2.) a_n =

3.) a_n = tan^-1n 4.) a_n =

B.) True - False. Answer "True" if the statement is ALWAYS true. If the statement is true, tell why it is true. If it is false, give an example to show that it is false.

5.) A bounded sequence converges.

6.) If a_n £ b_n£ 0 for all n and converges then

converges.

7.) Ifconverges thenconverges to 0.

8.) If converges to 0 then converges.

9.) If diverges then also diverges.

C.) Determine whether or not the sequenceconverges when a_n is as given. Show work to justify your answers.

10.) a_n = 11.) a_n = n e^-n

12.) a_n =where is a convergent series.

Test #5

For each of the series in #1 ‑ 4, determine whether it converges or diverges. If it converges, determine whether the convergence is conditional or absolute.

1.) see representative problems in the text

2.) see representative problems in the text

3.) see representative problems in the text

4.) see representative problems in the text

In #5 ‑ 6, find the interval and radius of convergence. Don't forget to check the endpoints of the interval of convergence.

5.) see representative problems in the text

6.) see representative problems in the text

7.) Find the 3rd degree Taylor polynomial for f(x) = with c = 1. Use it to approximate .

Find an upper bound for the error.

8.) Determine the degree of the Maclaurin polynomial required so that the error in the

approximating sin(0.5) will be less than 0.00001.

Test #6

1.) Use the fact that sin 2x = 2 sin x cos x to find a power series representation for the function f(x) = sin 3x cos 3x.

2.) Graph the conic xy + x + 1 = 0. Find the (x,y) coordinates for any vertices.

3.) Find a power series representation for the function

f(x) =. Find the interval of convergence. Don't

forget to check the endpoints.

4.) Graph the conic 3x² - 6x + y² + 6y = 0. Fill in all the following blanks. If one does not apply to this particular conic, write "DNA."

Vertices Foci

Endpoints of Minor Axis

Center Asymptotes

Directrix

Graph

5.) The ellipse x² + 4y² = 0 is revolved around the y-axis. Find the volume of the resulting solid.

6.) Write the first five terms of the Taylor series for the function f(x) = e^x centered on c = 1.

Test #7

1.) Let x = e^t and y = e^-t.

a.) Find dy/dx. b.) Find d²y/dx²

c.) Find all points of horizontal tangency.

d.) Graph the curve.

2.) Find the length of the curve represented by x = 6 cos q and y = 6 sin q with 0 £ q £ 2p. EXACT ANSWER

3.) Set up BUT DO NOT EVALUATE the integral(s) necessary for finding the common interior of r = 1 and r² = 2 sin 2q.

4.) The curve given by x = a cos t and y = a sin t with 0 £ t £ p is half of a circle of radius a. Use the curve revolved around the x-axis to find the surface area of a sphere. EXACT ANSWER

5.) Convert the point (4, -p/3) into rectangular coordinates. EXACT ANSWER

6.) Convert the point (-3, -3) into polar coordinates. EXACT ANSWER

7.) Graph r = 2 + 4 sin(q/3). Use a window with

xMin = -10.1982, xMax = 10.1982, yMin = -6, yMax = 6.

8.) Set up BUT DO NOT EVALUATE the integral(s) necessary for finding the arc length of the curve r = q, 0 £ q £ p.

Final Exam

1.) Consider the region bounded by the graphs of y = x² + 1, y = 0, x = 0 and x = 1.

a.) Find the volume of the solid when that region is revolved around the x-axis.

b.) Find the volume of the solid when that region is revolved around the y-axis.

2.) Find the area between the curves y = x^3/2 and y = x.

3.) The graph below shows one leaf of the curve r = 3 cos 2q. Find the surface area if that leaf is revolved around the x-axis.

4.) Find the arc length of the curve given by x = t² and y = sin t, with 0 £ t £ p.

5.) Determine convergence or divergence for the sequence. .

6.) Find the centroid of the region bounded by y = x, y = cos x and x = 0.

7.) A vertical wall of a swimming pool is a rectangle. The wall is 30 feet and 10 feet high. Find the fluid force on the wall if the pool is filled to a depth of 9 feet.

8.) Graph the following conic and fill in the blanks. For any blank that is not applicable to the particular conic, write "DNA."