Calculus 2

Test File

Spring 2005

 

Test #1

 

1.)        Without your calculator, find the area between the curves f(x) = px - x2 and g(x) = sin x.

 

2.)        Without your calculator, find the volume of revolution if the region bounded by f(x) = 2x - x2 + 3 and the x-axis is revolved around the x-axis.

 

3.)        Without your calculator, use cylindrical shells to find the volume of the region bounded by f(x) = 5x - x2 and the x-axis when it is revolved around the y-axis.

 

4.)        Without your calculator, use disks (or washers) to find the volume if the region bounded by y = x, y = 5x and y = 5 is revolved around the y-axis

 

5.)        Set up the integral (AND USE YOUR CALCULATOR TO EVALUATE IT) to find the arc length of the curve y = sin 3x, x = 0 to x = 2p.

 

6.)        Suppose we have the following pyramid.

            Suppose the base is a 4 foot by 4 foot square and the height is 5 feet.  Now suppose the top is cut off, parallel to the base, so the remaining solid has a height of 3 feet. 

            Use integration and cross-sections to find the volume of the resulting solid.

 

7.)        Set up the integral (AND USE YOUR CALCULATOR TO EVALUATE IT) to find the surface area of the surface generated by revolving the curve y = sin x, x = 0 to x = p, around the line y = 4.

 

8.)        Find the area of the region bounded by the curves y = sin x, y = 4 sin2x + 1, x = 0 and y = 4- 2x.  You may use your calculator

 

9.)        Suppose the region bounded by the curves y = 2 - x2 and y = x2 is revolved around the line x = 6.  Use cylindrical shells to find the volume of the resulting solid.  You may use your calculator.

 

10.)     Use disks or washers to find the volume if the region bounded by y = cos x, y = - cos x, x = -p/2 and x = p/2 is revolved around the line y = 3. You may use your calculator.

 

Test #2

 

1.)        Consider the function f(x) = 2cx5, where c is a constant. 

a.)        Find the value of c so that f(x) is a probability density function on the interval [0, 1].

b.)        Find the mean.

c.)        Find the median.

 

2.)        A rectangular plate that is 2 feet high and 3 feet wide is submerged in water (density = 62.4 pounds/cu. ft.).  Find the force exerted on the plate if the top edge of the plate is 3 feet below the surface.

 

3.)        Consider the full water tank shown below.

            What is the work done in pumping water of the tank to a height 5 feet above the top of the tank?

 

Do the following integrals.  You may not use your calculator.

 

4.)                     5.)                     6.)       

7.)                         8.)                  9.)       

 

Test #3

 1.)       A 6-inch rod has variable mass.  If the mass is given by (x) = x2, where x is the distance from the left hand end of the rod, find the center of mass of the rod.  Tell how far it is from the left hand end of the rod.

 

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

                                               

 

Evaluate the following integrals.  All work must be shown.  No calculators!

3.)                                               4.)       

 

5.)        EXACT ANSWER!!               6.)       

 

7.)       


 

 

Use the integral formulas in your book to work these three problems.  Write the number of any formula you use. 

8.)       

 

9.)       

10.)    

 

Test #4

 


1.)        Evaluate the following limits.

            a.)                                           b.)       

2.)        Evaluate the following integrals.

            a.)                                                  b.)       

3.)        Determine whether or not the following sequences converge.  If it converges, identify the limit.  If it does not converge, explain why.

            a.)                                        b.)        an =

4.)        Determine convergence or divergence for the following series.  If the series converges, to what does it converge.  If it does not converge, explain why.

            a.)                                                  b.)       

5.)        Use the integral test to determine convergence or divergence for the following series.  Be sure to show whether or not the integral test applies.

6.)        Determine convergence or divergence for the following series.  Be sure to make your reasons clear.

            a.)                                 b.)                        c.)       

 

 


Test #5

1.)        Determine convergence or divergence for the given series.  If it converges, determine if the convergence is absolute or conditional.

 

2.)        Determine convergence or divergence for the given series.

 

3.)        Determine convergence (absolute or conditional) or divergence for the given series. 

 

4.)        Consider the following series.

 

a.)        Find the sum of the first 50 terms of the series.  Feel free to use your calculator.

 

b.)        Determine convergence or divergence for the series.  If it converges, determine if the convergence is absolute or conditional.

 

5.)        For the following power series, use the root test to find the interval of convergence.  Do not forget to check the endpoints.

 

6.)        Find the Taylor Series for f(x) = ex-1 around c = 1.

 

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

 

8.)        Use the first 4 terms of the Maclaurin series for ex (hint: ) to approximate e0.1.  Then, find an upper bound for the error of this estimate.  You may use the following formula for the remainder.

 

9.)        Find the Maclaurin series for f(x) = tan-1x.

 

Test #6

1.)        Use the Maclaurin series  to evaluate the following limit.  Do NOT use L'Hopital.

 

2.)        Convert the following parametric equations to a single equation in x and y.

x = sin t, y = cos2t

 

3.)        Graph the following set of parametric equations.

x = sin (t/2), y = cos (t/3)

 

4.)        Graph the following set of parametric equations.  Show the starting point, ending point and direction of motion.

x = t2 + 2, y = 3t - 3, -2 < t < 2

 

5.)        Find the point(s) of intersection of the two sets of parametric equations.

 

6.)        Find the equation of the tangent line to the curve given by x = 2 cos t + sin (2t),   y = 2 sin t + cos (2t) at the point (2, 1).

 

7.)        Find dy/dx for x = t2 - t, y = t4 - 4t2.  Then find all points at which the curve has a horizontal tangent or a vertical tangent.  Be sure to label your answers so I know what represents what.

 

8.)        SET UP BUT DO NOT EVALUATE the integral for finding the arc length for x = 2 cos t + sin (2t),   y = 2 sin t + cos (2t) for 0 < t < 2p.

 

9.)        SET UP BUT DO NOT EVALUATE the integral for finding the surface area for the half-ellipse  when it is revolved around the x-axis.  (Hint:  Keep in mind that this half-ellipse can be graphed using x = 3 cos t, y = 2 sin t, 0 < t < .)

 

10.)     Now evaluate ONE of the integrals from #8 or #9.  Be sure to tell me which one you did.  You may use your calculator to evaluate it.

 

Test #7

 

1.)        a.)        Convert (3, -3) to polar coordinates.  Give the EXACT ANSWER.

            b.)        Convert (-3, 5p/6) to rectangular coordinates.  Give the EXACT ANSWER.

 

2.)        Convert r = 2 cos q - sin q to rectangular coordinates and graph it.

 

3.)        Graph r = sin(q/2) cos(q/3).

 

4.)        Find the slope of the tangent line to the three-leaf rose r = sin 3qat q = 0 and q = p/4.  EXACT ANSWERS ONLY.

 

5.)        Find the area that is inside BOTH of the curves r = 2 and r = 3 + 2 cos q.  Give the EXACT ANSWER.

 

6.)        Graph the following ellipse.  Fill in all blanks.  If a blank does not apply, write "DNA."

 

Focus_______________                            Focus__________________

Endpoint Minor Axis_______________    Endpoint Minor Axis________________

Vertex__________________                    Vertex__________________

Asymptote________________                  Asymptote________________

Center__________________

 

7.)        Graph the following hyperbola.  Fill in all blanks.  If a blank does not apply, write "DNA."

 

Focus_______________                            Focus__________________

Endpoint Minor Axis_______________    Endpoint Minor Axis________________

Vertex__________________                    Vertex__________________

Asymptote________________                  Asymptote________________

Center__________________

8.)        Identify and graph the following conic.

 

9.)        Identify the kind of curve for each of the following equations.

 

a.)        2x2 + 2y2 + Cx + Dy + E = 0

b.)        2x2 + 5y2 + Cx + Dy + E = 0

c.)        5x2 - 4y2 + Cx + Dy + E = 0

d.)        y2 - x + Dy + E = 0

e.)        Cx + 2y + E = 0

 

10.)     Graph the following rotated conic.  Fill in all the blanks.  If a blank does not apply, write "DNA."  SHOW ALL WORK!

 

Graph

 

 

 

Rotate Coordinate System

Regular Coordinate System

Focus

 

 

Focus

 

 

Vertex

 

 

Vertex

 

 

Center

 

 

Asymptote

 

 

Asymptote

 

 

 

Final Exam

1.)        Find the area between the curves y = x2 + x + 2 and y = 2x + 2.

 

2.)        Consider the region bounded by y = x2 and y = 4.  Find the volume of the resulting solid if this region is revolved around the line x = 3.

 

3.)        Consider the region bounded by y = x2 and y = 4.  Find the volume of the resulting solid if this region is revolved around the line y = 0.

 

4.)        A cylindrical tank (radius 6 ft., height 10 ft.) is filled with water.  Find the work done in pumping all of the water out through the top of the tank.  You may use the fact that water has a weight density of 62.4 lbs/ft3.

  

In #5-10, evaluate the given integral.  Do not use your calculator.

 

5.)                       6.)                        7.)       

8.)                                9.)                          10.)    

 

11.)     Determine convergence or divergence for the sequence {an} with an = tan-1n.  If it converges, state the limit.

 

12.)     Determine convergence or divergence for the sequence {an} with an = .  If it converges, state the limit.

 

13.)     Determine convergence or divergence for the following series.  Show all necessary work to justify your answer. 

 

14.)     Determine convergence or divergence for the following series.  Show all necessary work to justify your answer. 

 

15.)     Determine convergence or divergence for the following series.  If the series converges, determine if the convergence is conditional or absolute.  Show all necessary work to justify your answer.

 

16.)     Evaluate the limit.

 

17.)     Evaluate the limit.

 

18.)     Evaluate the integral. 

 

19.)     Use integration or differentiation of a known power series to find a power series for ln(1+x).

 

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

 

Center       _____                                         Vertex or vertices____________              

Focus or Foci____________                     Asymptote(s)______________                 

End pts. Minor Axis__________________                      _