# Unit 3 Exam

```Unit 5 Review (No Calculator)
1. Evaluate the integral:
  2x
3
 x  1 dx.
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2. Evaluate the integral:
x
2
 1 dx.
3
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3 / 2
3. Find the particular solution of f '  x   2 x
that satisfies the condition f  4   6
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4. Use properties of sigma notation and the summation
2
6
i
   i  8.
4
formulas to evauate the given sum:
i 1
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5. Given:
3
6
0
3
 f  x  dx  4 and  f  x  dx  1, find
6
a)
 f  x  dx
0
3
b)
 f  x  dx
6
4
c)
 f  x  dx
4
6
d)  10 f  x  dx
3
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1 
 1
6.   2  3  dx
x 
1 x
2
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 /2
7.

- /2
cos xdx
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x  2x  8
8. 
dx
x2
1
4
2
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3
9.

0
1
1 x
dx
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1
10.  x
0
2
x
3
 1 dx
3
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x
11.  cos dx
6
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12. Estimate the area bounded by f  x   x 2 +2 and the x-axis
between x  0 and x  4 using a lower sum (n  4).
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2

 2 
i
13. Let s  n    1    2    . Find lim s  n  .
n 
n
n




i 1 



n
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14. The rate of growth of a particular population is given by
dP
2
5/ 3
 30t  20t , where P is the population size and
dt
t is the time in years. The initial population is 300.
a) Determine the population function
b) Calculate the population after 32 years.
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14. The rate of growth of a particular population is given by
dP
 30t 2  20t 5/ 3 , where P is the population size and
dt
t is the time in years. The initial population is 300.
c) Graph the function. Then use the graph to estimate
the number of years until the population reaches 1200.
(Round to one decimal place.)
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15. If
10
10
4
8
 f  x  dx   and  f  x  dx  6 , find:
8
a)

 f  x  dx
10
10
b)
 f  x  dx
10
8
c)
 f  x  dx
4
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1
16. Find the average value of f  x  
on the interval 5,10.
2x 1
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17. Find the average value of f  x   x3 on the interval 0,2
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18. Evaluate the integral:  x
2
x  3dx.
3
7. Use the average of R4 &amp; L4 to approximate
2
1
1 1  x3 dx.
8. Use the average of R4 &amp; L4 to approximate
the area of the region bounded by the graphs of
y  cos x and y  0 on the interval 0, .
```