Name: __________________________ Date: _____________



1.
Complete the table and use the result to estimate the limit.

x
f(x)
 
 
 
 
 
 
 
A.
0.316228
B.
–0.191228
C.
–0.316228
D.
0.149561
E.
0.066228


2.
Determine the following limit. (Hint: Use the graph of the function.)



A.
8
B.
6
C.
2
D.
4
E.
Does not exist


3.
Determine the following limit. (Hint: Use the graph of the function.)



A.
Does not exist
B.
2
C.
3
D.
7
E.
0


4.
Let and . Find the limits:

(a)     (b)     (c)
A.
–3 , 2 , 41
B.
5 , –4 , 50
C.
21 , –8 , 42
D.
5 , –4 , 16
E.
1 , –4 , 1


5.
Find the limit:

A.
B.
C.
D.
Does not exist
E.


6.
Suppose that and . Find the following limit:
A.
14
B.
17
C.
11
D.
42
E.
–3


7.
Use the graph to determine the following limits, and discuss the continuity of the function at .

(i)     (ii)     (iii)

A.
1 , 0 , Does not exist , Not continuous
B.
0 , 1 , Does not exist , Not continuous
C.
0 , 1 , 0 , Continuous
D.
–3 , 0 , Does not exist , Not continuous
E.
1 , –1 , Does not exist , Not continuous


8.
Find the x-values (if any) at which the function is not continuous. Which of the discontinuitites are removable?
A.
No points of discontinuity.
B.
(Not removable), (Removable)
C.
(Removable), (Not removable)
D.
No points of continuity.
E.
(Not removable), (Not removable)


9.
Determine whether approaches or as x approaches –3 from the left and from the right by completing the tables below.
x
f(x)
 
 
 
 
 
x
f(x)
 
 
 
 
 
A.
B.
C.
D.
E.
Both A and C
F.
Both B and D


10.
Find the vertical asymptotes (if any) of the function .
A.
x = 6
B.
x = 2
C.
x = –6
D.
x = –2
E.
A and B
F.
C and D


11.
Find the limit:

A.
0
B.
C.
1
D.
–1
E.


12.
Find the slope of the tangent line to the graph of the function below at the given point.

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


13.
Find the derivative of the following function using the limiting process.

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


14.
Find the derivative of the function.


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


15.
Find the slope of the graph of the function at the given value.

when
A.
B.
C.
D.
E.


16.
Determine the point(s), (if any), at which the graph of the function has a horizontal tangent.


A.
B.
and
C.
and
D.
E.
There are no points at which the graph has a horizontal tangent.


17.
Use the product rule to differentiate.



A.
B.
C.
D.
E.


18.
Use the quotient rule to differentiate.



A.
B.
C.
D.
E.


19.
Find the derivative of the trigonometric function.



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


20.
Find the derivative of the function.


A.
B.
C.
D.
E.


21.
Find the derivative of the function.


A.
B.
C.
D.
E.


22.
Find the derivative of the function.


A.
B.
C.
D.
E.


23.
Find the second derivative of the function.




A.
B.
C.
D.
E.


24.
Find dy/dx by implicit differentiation.



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


25.
Find an equation of the tangent line to the graph of the function given below at the given point.

,    

(The coefficients below are given to two decimal places.)
A.
B.
C.
D.
E.


26.
A point is moving along the graph of the function



such that dx/dt = 5 centimeters per second.

Find dy/dt when.
A.

B.
 
C.

D.
 
E.


27.
Area The radius, r, of a circle is increasing at a rate of 4 centimeters per minute.

Find the rate of change of area, A, when the radius is
A.
B.
C.
D.
E.


28.
Find the value of the derivative (if it exists) of the function at the extremum point (0,0).
A.
0
B.
1
C.
-1
D.
E.


29.
Find any critical numbers of the function , t < 10.
A.
0
B.
C.
D.
Both A and B
E.
Both A and C


30.
Find any critical numbers of of the function , .
A.
B.
C.
D.
E.
Both A and B
F.
Both A and C


31.
Locate the absolute extrema of the function over the following intervals:
(a) [–5,2]   (b) [–5,2)   (c) (–5,2]   (d) (–5,2).
A.
 
Absolute Maximum
Absolute Minimum
(a)
7
–7
(b)
None
–7
(c)
7
None
(d)
None
None
 
B.
 
Absolute Maximum
Absolute Minimum
(a)
7
None
(b)
7
–7
(c)
None
None
(d)
None
–7
 
C.
 
Absolute Maximum
Absolute Minimum
(a)
7
–7
(b)
7
–7
(c)
7
None
(d)
7
None
 
D.
 
Absolute Maximum
Absolute Minimum
(a)
None
–7
(b)
None
None
(c)
7
None
(d)
None
None
 
E.
None of the above.


32.
Determine whether Rolle's Theorem can be applied to the function on the closed interval [1,9]. If Rolle´s Theorem can be applied, find all values of c in the open interval (1,9) such that .
A.
Rolle's Theorem applies; c = –5
B.
Rolle's Theorem applies; c = 3
C.
Rolle's Theorem does not apply
D.
Rolle's Theorem applies; c = 5
E.
Both A and D


33.
Determine whether the Mean Value Theorem can be applied to the function on the closed interval [–2,4]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (–2,4) such that .
A.
MVT applies; 0
B.
MVT applies; 2
C.
MVT applies; 1
D.
MVT applies; 3
E.
MVT applies; –1


34.
The graph of f is shown in the figure. Sketch a graph of the derivative of f.

A.


B.


C.


D.


E.




35.
Determine the open intervals on which the graph of is concave downward or concave upward.
A.
Concave upward on ; concave downward on
B.
Concave downward on
C.
Concave upward on
D.
Concave downward on ; concave upward on
E.
Concave upward on ; concave downward on


36.
Find the points of inflection and discuss the concavity of the function.
A.
Inflection point at ; concave upward on ; concave downward on
B.
Inflection point at ; concave downward on ; concave upward on
C.
Inflection point at ; concave upward on ; concave downward on
D.
Inflection point at ; concave downward on ; concave upward on
E.
None of the above


37.
Find all relative extrema of the function Use the Second Derivative Test where applicable.
A.
Relative max: ; no relative min
B.
Relative max: ; no relative min
C.
No relative max or min
D.
Relative min: ; no relative max
E.
Relative min: ; no relative max


38.
The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.

A.


B.


C.


D.


E.
None of the above


39.
Sketch the graph of the function using any extrema, intercepts, symmetry, and asymptotes.
A.


B.


C.


D.


E.




40.
Analyze and sketch a graph of the function .
A.


B.


C.


D.


E.




41.
Determine the slant asymptote of the graph of .
A.
B.
C.
D.
E.
No slant asymptote


42.
A rectangular page is to contain square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.
A.
B.
C.
D.
E.


43.
Find the differential dy of the function .
A.
B.
C.
D.
E.


44.
Find the indefinite integral and check the result by differentiation.


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


45.
Find the sum given below.

A.
175
B.
126
C.
77
D.
141
E.
159


46.
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 3 subintervals.
A.
Lower sum = 0.797949
Upper sum = 0.804738
      
B.
Lower sum = 0.804738
Upper sum = 0.844975
        
C.
Lower sum = 0.464616  
Upper sum = 0.844975
        
D.
Lower sum = 0.464616
Upper sum = 0.797949
       
E.
Lower sum = 0.464616
Upper sum = 0.511077


47.
The diagram below shows upper and lower sums for the function using 5 subintervals.


   

                         Lower sum (n=5)                                                                      Upper sum (n=5)

Use upper and lower sums to approximate the area of the region using 10 subintervals.
A.
Lower sum = 0.682833
Upper sum = 0.668771
       
B.
Lower sum = 0.668771
Upper sum = 0.718771
     
C.
Lower sum = 0.292897
Upper sum = 0.668771
   
D.
Lower sum = 0.292897
Upper sum = 0.307542
      
E.
Lower sum = 0.718771
Upper sum = 0.790649
      


48.
Find the limit of s(n) as n.



                                             
A.
2/3
B.
4/3
C.
1/3
D.
2
E.
Unbounded


49.
Evaluate the following definite integral by the limit definition.

A.
B.
C.
D.
E.


50.
Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.

A.
6
B.
12.5
C.
13
D.
9.5
E.
41.5


51.
Evaluate the integral



given

A.
8
B.
1346
C.
293
D.
1074
E.
84


52.
Evaluate the definite integral of the algebraic function.



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


53.
Evaluate the definite integral of the algebraic function.



Use a graphing utility to verify your results.
A.
659
B.
–637
C.
637
D.
1307
E.
403


54.
Determine the area of the given region.



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


55.
Find the average value of the function over the interval and find all values in the interval for which the function equals its average value.
A.
The mean is and the points at which the function is equal to its mean value are and .
B.
The mean is and the points at which the function is equal to its mean value are and .
C.
The mean is and the points at which the function is equal to its mean value are and .
D.
The mean is and the points at which the function is equal to its mean value are and .
E.
The mean is and the points at which the function is equal to its mean value are and .


56.
Find the indefinite integral of the following function.

A.
B.
C.
D.
E.


57.
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 390.4219 and Simpson's rule gives 390.0000.
B.
The Trapezoidal rule gives 390.4219 and Simpson's rule gives 409.9430.
C.
The Trapezoidal rule gives 391.6875 and Simpson's rule gives 409.9430.
D.
The Trapezoidal rule gives –390.421875 and Simpson's rule gives –390.
E.
The Trapezoidal rule gives 391.6875 and Simpson's rule gives 390.0000.


58.
Find an equation of the tangent line to the graph of at the point (1,0).
A.
B.
C.
D.
E.


59.
Find the derivative of the function .
A.
B.
C.
D.
E.


60.
Locate any relative extrema and inflection points of the function . Use a graphing utility to confirm your results.
A.
Relative maximum value at ; inflection point at x = 0.
B.
Relative minimum value at ; inflection point at x = 0.
C.
Relative minimum value at ; no inflection points.
D.
Relative minimum value at ; no inflection points.
E.
Relative maximum value at ; no inflection points.


61.
Use logarithmic differentiation to find the derivative of .
A.
B.
C.
D.
E.
None of the above


62.
Find the indefinite integral.

A.
B.
C.
D.
Integral does not exist
E.
None of the above


63.
Find the indefinite integral.

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


64.
Evaluate the definite integral.

A.
Integral does not exist
B.
C.
D.
E.


65.
Use the Horizontal Line Test to answer the following.

True or False: The function is one-to-one on its entire domain and therefore has an inverse function.
A.
 
B.
 


66.
True or False: The function is one-to-one on its entire domain.
A.
 
B.
 


67.
Find if () .
A.
B.
C.
D.
E.


68.
Find the indefinite integral.

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


69.
Find the following indefinite integral.

A.
B.
C.
D.
E.


70.
Find an equation of the tangent line to the graph of at the point .
A.
B.
C.
D.
E.
None of the above


71.
Find the indefinite integral.

A.
B.
C.
D.
E.


72.
Find the indefinite integral.

A.
B.
C.
D.
E.


73.
Find the indefinite integral.

A.
B.
C.
D.
E.


74.
Find the indefinite integral.

A.
B.
C.
D.
E.



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