Avon High School
VU Calculus
nd
2 Semester FINAL EXAM REVIEW
Unit 5 (3.5-3.7, 3.9)
For 1-4, find the limit.
3

1.) lim  5  5 
x 
x 

 x3 
2.) lim 

x  7 x  5


 x7 
3.) lim  2

x  3 x  4


7 
 5
4.) lim   x  5 
x 
x 
 3
For 5 & 6, determine the slant asymptote of the graph of f  x  .
5.) f  x  
x2  6 x  5
x2
6.) f  x  
2 x2  x  2
x 1
7.) Find two positive numbers whose product is 185 and whose sum is a minimum.
8.) Find the length and width of a rectangle that has perimeter 16 meters and a maximum area.
9.) Find the length and width of a rectangle that has an area of 392 ft 2 and whose perimeter is a minimum.
10.) Find the equation of the tangent line to the graph of f  x  
22
 11 
at the point  2,  .
2
x
 2
11.) Find the differential dy of the function y  4 x 2  3x  4 .
Unit 6 (4.2 & 4.3)
9
1.) Find the sum:
  4i  7 
2.) Use sigma notation to write the sum:
i 3
 i
30
3.) Use the properties of summation to evaluate the sum:
i 1
2
4
4
4
4


 ... 
11 1 2 1 3
1  21
 7i 
4.) Use left endpoints and 6 rectangles to find the approximation of the area of the region between the graph of
 
the function y  cos  2 x  and the x-axis over the interval  0,  .
 2
4
5.) Find the limit of s  n  as n   for s  n   7
n
 n3  n  14 

.
5


6.) The graph of the function f  x   16  x2 is given. Write the definite integral that
yields the area of the shaded region.
2
7.) Evaluate the integral  12x dx given
3
3
5
 x dx  2 .
2
6
6
671
8.) Evaluate the integral   24 x  2  dx given  x dx 
,
4
5
5
2
3
6
91
5 x dx  3 ,
2
6
6
11
5 x dx  2 , 5 dx  1 .
9.) The graph of f consists of line segments, as shown in the figure. Evaluate
10
the definite integral
 f  x  dx using geometric formulas.
2
Unit 7 (4.1, 4.4, 4.5)
For 1-4, find the indefinite integral.
1.)
  20 x
3
 6 x  3 dx
2.)

5.) Solve the differential equation
7
x 4 dx
3.)
6t 2  8t  15
dt
 t4
4.)
  9sin x  3cos x  dx
dG
 16t 7 given G  2  3 .
dt
6.) A ball is thrown vertically upwards from a height of 7 ft with an initial velocity of 64 ft/sec. How high will the
ball go?
For 7-10, evaluate the definite integral.
8
5
3
7.)   5 z  2  dz
8.)  5 dx
x
1
2
5
5
 9

9.)   x 5  x 9  dx

2
6
10.)
  2 x  2 cos x  dx
0
11.) Find the area of the region bounded by the graphs of the equations y  x6  x, x  4, y  0 .
For 12 & 13, find the average value of the function on the given interval.
z2  4
, 3 z 7
12.) f  x   30  6 x2 ,  2  x  2
13.) f  z  
z2
14.) Find F   x  given F  x  
3 x2
  6t  1 dt .
2
For 15-20, find the indefinite integral.
15.)
 1  3x 
18.)
 7  x
4
 x 3  x  dx
2
3
dx
16.)
x dx
19.)   sin  2 x  dx
5
21.) Evaluate the definite integral:

3
5x
17.)

20.)
 sin
x
2
 4
dx
5
cos x
dx
8
x
1
dx
4x  3
Unit 8 (5.1-5.3)
For 1 & 2, state the domain of the function.
1.) f  x   16ln  4 x 
2.) f  x   6  ln  x  15
3.) Write 13ln x  15ln  x 2  7  as a single quantity.
4.) Find an equation of the tangent line to the graph of y  ln  x14  at the point 1,0  .
For 5-8, find the derivative of the function.
5.) f  x   ln  5x  3
6.) y  ln x 2  1
9.) Use implicit differentiation to find
 5x 
7.) f  x   ln  2

 x 7
8.) y  ln  ln x12 
dy
for x 4  7 ln y  6 .
dx
For 10-17, find the indefinite integral.
1
dx
10.) 
x 8
14.)
1
 x ln  x  dx
13
e
18.) Evaluate

 6  ln x 
1
 ln x 
6
x
dx
11.)  2
4x  7
x 2  6 x  11
dx
12.) 
x  16
13.)

15.)  tan  3  d
16.)  csc  36x  dx
17.)
 sin 12  d
20.) f  x   x3  9
21.) f  x   2 x2 ,  x  0
22.) f  x   3  3 2 x  10
x
dx
cos 12 
2
x
For 19-22, find f 1  x  .
19.) f  x   9 x  7
23.) Find
dy
at the point  6,1 for the equation x  y3  9 y 2  2
dx
Unit 9 (5.4, 5.5, 5.8)
For 1-4, solve the equation for x.
1.) eln9 x  4
2.) 4  3e3 x  10
4.) ln  x  6   6
3.) ln x 9  6
5
For 5-10, find the derivative.
5.) f  x   7e 4 x
2
8.) f  x   6e8 x  2e6 x
 e5 x  1 
7.) f  x   ln  x

 e 1 
6.) f  x   x5e x
9.) f  x  
ex
1  4 x2
10.) f  x   3e x cos x
For 11-13, solve the equation for x.
11.) log 2 x  log 2  x  1  1
12.) 5  43 x5   225
13.) log3  x  6   5
For 14-16, find the derivative.
14.) f  t   t 9 47t
 x3  5 
15.) f  x   log 7  3

 x 7
 x2  3 
16.) f  x   log 6 

 x 8 
17.) Find an equation of the tangent line to the graph of y  log 2 x at the point  32,5 .
For 18 & 19, find the indefinite integral.
18.)  75 x dx
For 20-22, find the derivative.
20.) y  coth 8x 
19.)
21.) y  ln  cosh 4  7 x  
x
8
5  dx
 x9
22.) g  x   2sec h2  9 x 
For 23 & 24, find the indefinite integral.
23.)  sinh  9  8x  dx
Unit 10 (6.2)
For 1-3, solve the differential equation.
dy
 x 8
1.)
dx
24.)
2.) y 
6x
y
9
8
2 x 
x
csc
h
  dx

9
3.) y 
4.) Find the function y  f  t  passing through the point  0,15 with the first derivative
x
8y
dy 1
 t.
dt 6
5.) The half-life of the carbon isotope C-14 is approximately 5715 years. If the initial quantity of the isotope is 30
grams, what is the amount left after 10,000 years?
6.) The half-life of the carbon isotope C-14 is approximately 5715 years. If the amount left after 3000 years is 1.6
grams, what is the amount after 6000 years?