Calculus 2

Test File

Fall 2000

 

Test #1

 

1.)        Consider the region bounded by f(x) = x2, y = 0, x = 2.  Suppose the region is revolved around the x-axis.  Find the volume of the resulting solid.  (Exact answer only.)

2.)        Suppose the region bounded by the curves y = 0, y = x2, 0 £ x £ 2, is revolved around the line y = 6.  Find the volume of the resulting solid.  (Exact answer only.)

3.)        Consider the region bounded by y =x2 + 1, y = sin x, x = 0, and x = p.

            a.)        Set up and use your calculator to evaluate the integral needed to find the area of the region.

            b.)        Set up and use your calculator to evaluate the integral needed to find the volume of the solid obtained by revolving the region around the x-axis.

4.)        Consider a square based pyramid, with base area, A, and height, h.  Use cross-sections and integration to find the volume.

5.)        Find the following antiderivative. 

6.)          EXACT ANSWER!!

7.)        A certain radioactive substance has a half-life of 4 years.  If you have 20 grams of the substance, how much will you have after 5 years?  You may use the equation dy/dt = ky if you think it applies.

8.)        Use Simpson's rule, with n = 4, to approximate the area under the curve y = sin x from x = 0 to x = 1.  Give at least 4 places to the right of the decimal point.  Also, find an upper bound for the error.  You may use the following formulae if you think you might find them helpful.

                       

                       

 

Test #2

 

For problems #1 - 5, consider the region bounded by f(x) = sin x, x = 0, x = p/2. 

1.)        Suppose the region is revolved around the x-axis.  SET UP AND USE YOUR CALCULATOR TO EVALUATE the integral for finding the volume of the resulting solid using cylindrical shells.

2.)        Suppose the region is revolved around the x-axis.  SET UP AND USE YOUR CALCULATOR TO EVALUATE the integral for finding the volume of the resulting solid using disks.

 

3.)        Suppose the region is revolved around the line y = 2.  SET UP AND USE YOUR CALCULATOR TO EVALUATE the integral for finding the volume of the resulting solid.

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

5.)        Consider the top edge of the region. (the part formed by the sine curve)  Set up the integral needed for finding the area of the surface formed by revolving that curve around the line y = 3.  USE YOUR CALCULATOR TO EVALUATE IT.

6.)        Consider the region bounded by the graphs of y = x2 + 1, y = 5, x = 0 and x = 1.

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

For problems number 7 and 8, consider the region bounded by the graphs of y = 2x2 and x = 2y2. 

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

8.)        Find the volume of the solid when that region is revolved around the line y = 3.. Exact Answer.

 

Test #3

 

1.)        A tank has the shape of a cone with the vertex down.  The height of the tank is 10 feet and the radius is 5 feet.  If the water level is 5 feet high, how much work is done in emptying the tank through a pipe that pumps the water to a point 5 feet above the top of the tank?  The density of water is 62.4 pounds per cubic foot.

2.)        (exact answer only)

3.)        ò  sec3 x  tan3 x dx                              4.)        ò  sin4 x  cos3 x dx

5.)        ò  sin2 (2x)  cos2 (2x) dx                     6.)        ò  x4 sin(2x) dx

7.)        ò   ln x dx                                              8.)       

 

Test #4

 

1.)                                                    2.)       

3.)                                               4.)       

5.)        Set up BUT DO NOT EVALUATE the partial fraction decomposition for the following.

                                               

6.)        Solve the following differential equation.

 

           

7.)       

8.)        A tank contains 10 lb. of salt dissolved in 30 gallons of water.  Suppose 2 gallons of brine containing 1 lb. of dissolved salt per gallon runs into the tank every minute and that the mixture (kept uniform by stirring) runs out at the same rate.

            a.)        Find the amount of salt in the tank at time t.

b.)        How long does it take (to the nearest second) for the tank to contain 15 lb. of salt?

 

Test #5

1.)        EXACT ANSWER!!                   2.)        EXACT ANSWER!!

3.)       

In #4 - 6, determine whether the sequenceconverges when an is as given.  Show all necessary work.

4.)        an =                     5.)        an =         6.)       

7.)        Determine convergence or divergence for the following series.  Show all necessary work to justify your answer.  If the series converges, find the sum.

8.)        Find the fractional representation for 1.242424… .

Do the following integrals.  All work must be shown to receive credit.

9.)                                  10.)    

Test #6 - Sorry, I lost the file.  When we get there, remind me to see if I can find a paper copy to run off.

 

Test #7

 

For each of the following power series, find the interval of convergence.  Do not forget to check the endpoints.

 

1.)                               2.)                   3.)       

 

 

4.)        Use a familiar power series to find a power series for the following function.  Then find the interval of convergence.  Include endpoints.

                                   

5.)        Use the first 6 terms of the Maclaurin series for sin x to approximate sin(0.1).  Then, find an upper bound for the error of this estimate.  You may use the following formula for the remainder.

6.)        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.

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

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

9.)        Graph r = 2 + 4 sin(q/3).  Use a window with xMin = -10.1982, xMax = 10.1982, yMin = -6, yMax = 6.

 

Final Exam

 

In #1 - 6, evaluate the given integral.  In the case of the definite integral, find the EXACT ANSWER.

1.)                    2.)                         3.)        ò x2 sin x dx

4.)                      5.)                                      6.)       

7.)        Determine convergence or divergence for the following series.

                         

8.)        Determine absolute convergence, conditional convergence or divergence for the following series.             

9.)        Determine convergence or divergence for the following series.

10.)     Use differentiation to find a power series for the following function.

                       

 

 

11.)     Determine convergence or divergence of the following sequence.

                         

12.)     Solve the following differential equation.

                       

13.)     Find an explicit relationship between x and y by eliminating the parameter.  Sketch the path described by the parametric equations.

                        x = - cos t

                        y = sin2 t

14.)     Use disks or washers to find the volume of the solid generated by revolving the region bounded by y = x + 1, x = 0, y = 0 and x = 2 around the line y = 4.

15.)     Use cylindrical shells to find the volume of the solid generated by revolving the region bounded by y = x-2, the x-axis, x = 1 and x = 2 around the y-axis.

16.)     An object located x ft. from a fixed starting position is moved along a straight road by a force of F(x) = 3x2 + 5 lb.  What work is done by the force to move the object through the first 4 feet?