Name: __________________________ Date: _____________



1.
Find the general solution of the differential equation below and check the result by differentiation.

A.
B.
C.
D.
E.


2.
Find the indefinite integral of the following function.

A.
B.
C.
D.
E.


3.
Find the indefinite integral and check the result by differentiation.

A.
B.
C.
D.
E.
None of the above


4.
Evaluate the following definite integral.



Use a graphing utility to check your answer.
A.
B.
C.
D.
E.


5.
Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to four decimal places and compare the result with the exact value of the definite integral.

A.
The Trapezoidal rule gives –0.803708 and Simpson's rule gives –0.85332.
B.
The Trapezoidal rule gives 0.846008 and Simpson's rule gives 0.844876.
C.
The Trapezoidal rule gives 0.849404 and Simpson's rule gives 0.888309.
D.
The Trapezoidal rule gives 0.846008 and Simpson's rule gives 0.888309.
E.
The Trapezoidal rule gives 0.849404 and Simpson's rule gives 0.844876.


6.
Find the indefinite integral of the following function.


A.
B.
C.
D.
E.


7.
Write the following limit as a definite integral on the interval [2 , 5], where ci is any point in the ith subinterval.

A.
B.
C.
D.
E.
Both B and D


8.
Use the properties of summation and Theorem 4.2 to evaluate the sum.

A.
2077
B.
1984
C.
1891
D.
1950
E.
2070


9.
Find the indefinite integral and check the result by differentiation.


A.
B.
C.
D.
E.
None of the above


10.
Solve the differential equation.

A.
B.
C.
D.
E.


11.
Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.

A.
3
B.
2
C.
2.5
D.
1
E.
12.5


12.
Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 16 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.

A.
The Trapezoidal rule gives 3.833678 and Simpson's rule gives 4.025912.
B.
The Trapezoidal rule gives 3.834202 and Simpson's rule gives 4.025912.
C.
The Trapezoidal rule gives 3.834202 and Simpson's rule gives 3.834376.
D.
The Trapezoidal rule gives –3.834202 and Simpson's rule gives –3.834376.
E.
The Trapezoidal rule gives 3.833678 and Simpson's rule gives 3.834376.


13.
Solve the differential equation.

,    
A.
B.
C.
D.
E.


14.
Find the sum given below.

A.
B.
C.
D.
E.


15.
Use the properties of summation and Theorem 4.2 to evaluate the sum.

A.
1510
B.
159
C.
3789
D.
1799
E.
1785


16.
Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.



A.
B.
C.
D.
E.
None of the above


17.
Determine the area of the given region.



A.
B.
C.
D.
E.
None of the above


18.
Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule when approximating the following integral.

A.
The error for the Trapezoidal rule is 0.010204 and for Simpson's rule it is 0.0300.
B.
The error for the Trapezoidal rule is 0.0000 and for Simpson's rule it is 0.0300.
C.
The error for the Trapezoidal rule is 0.010204 and for Simpson's rule it is 0.0000.
D.
The error for the Trapezoidal rule is 0.0000 and for Simpson's rule it is 0.0000.
E.
The error for the Trapezoidal rule is 0.0026 and for Simpson's rule it is 0.0019.


19.
Solve the differential equation.

A.
B.
C.
D.
E.


20.
Find F'(x) given

.
A.
B.
C.
D.
E.


21.
Evaluate the definite integral of the algebraic function.



Use a graphing utility to verify your results.
A.
194
B.
–126
C.
126
D.
354
E.
–1534


22.
Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule when approximating the following integral.




The approximate errors for each method are as follows.
Trapezoidal rule
Simpson's rule

                  
A.
0.00003263
0.00000029
               
B.
0.00065255
0.00002640
               
C.
0.00005438
0.00000015
D.
0.00005438
0.00000029
 
E.
0.00003263
0.00000015
 


23.
Evaluate the definite integral of the algebraic function.



Use a graphing utility to verify your results.
A.
B.
C.
D.
E.


24.
Find the indefinite integral and check the result by differentiation.


A.
B.
C.
D.
E.
None of the above


25.
Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 4 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.

A.
The Trapezoidal rule gives 25.501465 and Simpson's rule gives 24.816382.
B.
The Trapezoidal rule gives 25.501465 and Simpson's rule gives 26.776538.
C.
The Trapezoidal rule gives 27.556714 and Simpson's rule gives 26.776538.
D.
The Trapezoidal rule gives –25.501465 and Simpson's rule gives –24.816382.
E.
The Trapezoidal rule gives 27.556714 and Simpson's rule gives 24.816382.


26.
Evaluate the following definite integral by the limit definition.

A.
B.
C.
D.
E.


27.
Find the indefinite integral of the following function and check the result by differentiation.

A.
B.
C.
D.
E.
None of the above


28.
The diagram below shows upper and lower sums for the function using 4 subintervals.


                    Lower sum (n=4)                                            Upper sum (n=4)

Use upper and lower sums to approximate the area of the region using 4 subintervals.
A.
Lower sum = 0.768283
Upper sum = 1.036566
      
B.
Lower sum = 1.036566
Upper sum = 1.088394
        
C.
Lower sum = 0.518283  
Upper sum = 1.088394
        
D.
Lower sum = 0.518283
Upper sum = 0.768283
       
E.
Lower sum = 0.518283
Upper sum = 0.570111


29.
Find the indefinite integral of the following function and check the result by differentiation.


A.
B.
C.
D.
E.
   


30.
Find the indefinite integral of the following function and check the result by differentiation.


A.
B.
C.
D.
E.
None of the above


31.
Find the indefinite integral of the following function and check the result by differentiation.



A.
B.
C.
D.
E.


32.
Solve the differential equation.

,    
A.
B.
C.
D.
E.


33.
Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 16 subintervals. Round your answer to four decimal places and compare the result with the exact value of the definite integral.

A.
The Trapezoidal rule gives –0.036059 and Simpson's rule gives –0.039835.
B.
The Trapezoidal rule gives 0.037957 and Simpson's rule gives 0.039855.
C.
The Trapezoidal rule gives 0.038014 and Simpson's rule gives 0.039855.
D.
The Trapezoidal rule gives 0.037957 and Simpson's rule gives 0.037938.
E.
The Trapezoidal rule gives 0.038014 and Simpson's rule gives 0.037938.


34.
Evaluate the integral



given

A.
–107
B.
–2702
C.
–2238
D.
–2232
E.
–108


35.
Evaluate the definite integral of the trig function.



Use a graphing utility to verify your results.
A.
–3
B.
–5
C.
–6
D.
–4
E.
–2


36.
Use the properties of summation and Theorem 4.2 to evaluate the sum.

A.
12760
B.
318648
C.
12522
D.
282480
E.
318640


37.
Find the sum given below.

A.
996
B.
969
C.
39
D.
1983
E.
957


38.
Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.


A.
B.
C.
D.
E.
None of the above


39.
Evaluate the definite integral of the function.



Use a graphing utility to verify your results.
A.
B.
C.
D.
E.


40.
Find the sum given below.

A.
280
B.
245
C.
210
D.
225
E.
255


41.
Find the limit of s(n) as n.



                                             
A.
0
B.
12/7
C.
1/7
D.
Unbounded
E.
6/7


42.
Find F'(x) given

.
A.
B.
C.
D.
E.


43.
Find the average value of the function over the given interval and all values z in the interval for which the function equals its average value.



Use a graphing utility to verify your results.
A.
The average is and the point at which the function is equal to its mean value is .
B.
The average is and the point at which the function is equal to its mean value is .
C.
The average is and the point at which the function is equal to its mean value is and .
D.
The average is and the point at which the function is equal to its mean value is and .
E.
The average is and the point at which the function is equal to its mean value is .


44.
Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.

A.
–365
B.
730
C.
480
D.
240
E.
10


45.
Write the following limit as a definite integral on the interval [5 , 11], where ci is any point in the ith subinterval.

A.
B.
C.
D.
E.
Either B or D


46.
Evaluate the definite integral of the algebraic function.



Use a graphing utility to verify your results.
A.
B.
C.
D.
E.


47.
Evaluate the following definite integral by the limit definition.

A.
B.
C.
D.
E.


48.
Find the indefinite integral of the following function.

A.
B.
C.
D.
E.


49.
Find the indefinite integral and check the result by differentiation.

A.
B.
C.
D.
E.


50.
valuate the definite integral of the algebraic function.



Use a graphing utility to verify your results.
A.
B.
C.
D.
E.


51.
Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 10 subintervals. Round your answer to four decimal places and compare the result with the exact value of the definite integral.

A.
The Trapezoidal rule gives 261.1800 and Simpson's rule gives 261.0000.
B.
The Trapezoidal rule gives 261.1800 and Simpson's rule gives 274.2390.
C.
The Trapezoidal rule gives 261.7200 and Simpson's rule gives 274.2390.
D.
The Trapezoidal rule gives –261.18 and Simpson's rule gives –261.
E.
The Trapezoidal rule gives 261.7200 and Simpson's rule gives 261.0000.



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