Download Chapter 7 Worksheet Packet

CALCULUS 2
Name: _____________________________
WORKSHEET 7.1-1
1. Graph y  5 x and y  log 5 x .
2. Graph y  e x and y  ln x .
Convert to log form.
Convert to exponential form.
3. y  5 x
7. y  log 9 x
4. 243  35
8. 3  log 1000
5. x  e 7
9. x  ln 10

1
2
1
2
10. log 9
1
3

27
2
11. log 4 64
12. log 7
1
7
6. 4

Evaluate.
13. log
1
100
14. log 5 3 5 2
1
16
15. log 9 27
16. log 64
17. 11log11 21
18. 2log2 2
log2 4
CALCULUS 2
Name: _____________________________
WORKSHEET 7.1-2
Expand each logarithm.
1. log6 3x
2. log2
3. log4 xy2
4. log3
5. log3
xy4z
6. log2
(xy) 4
7. log5 2
z
8. log8
x
7
x  3y  2
3
3x 2 y
z  34
3
x 2 y3
z4
Write each logarithmic expression as a single logarithm.
9. log3 7 – log3 x
10. 2 log5 x + log5 3
11.
2
3
log2 x – 3 log2 y
12.
13.
1
2
log7 x + 13 log7 y – 2 log7 z
14. log5 x – 4(log5 y + 2 log5 z)
15. log2 (x – 4) + 5 log2 (x+1) -
3
4
log2 (x-1)
16.
1
2
1
2
(log3 4 + log3 y) – 3 log3 z
[log6 (x-2) + 2 log6 (x+1) – log6 (x+2) – 5 log6 x]
Evaluate.
17. log 2 16
18. log a a 2
19. log 3 7
20. 6 log6 7
21. log 3 14.62
22. log a
23. 2 x  3
24. 3x 1  25
25. 23 x  4  5
26. 2 x  2  16
27. 3x 1  4 x
28. 31 x  52  3 x
29. e6 x  314
30. 225  31e .07 x
31. ln 2.1  ln e3.2 x
1
a
Solve.
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-1
Find the derivative of each of the following:
1.
y  x ln x
2.
y  t ln t  t
4.
y  ln x 2
5.
y  ln x 3  3x  1
7.
y  ln  sin3t   1
8.
y  x 2 ln x

10. y  ln  ln x 
11. y  ln  ln x 
 x 
13 y  3ln 

 x 1
14. y  ln( x  4)3
16. y  x 2 ln( x 2 )  (ln x )3
17. y 
19. y  log2 x
20. y  log10 x 3  x 2

3
3.
y   ln x 
6.
y  ln2x
9.
y
2
ln x
x
12. y  ln( x 2  3x   )
15. y  ln 3x  2
3
ln x
 1
  ln 
2
2
x ln x  x 


18. y  ln( x  x 2  1)
21. y  log3  sin t 
Find the equation of the tangent line to the graph of f at the given point.
22. f ( x )  ln t ; t  5
24. f ( x )  lnsin x ;
Find
23. f ( x )  ln(8  4t ); t  1
x

2
25. f ( x )  3x 2  ln x ; x  1
dy
using implicit differentiation.
dx
26. x 2  3ln y  y2  10
27. ln  xy   5x  30
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-2
Find
dy
using logarithmic differentiation.
dx
1.
y  ( x  2)( x  4)
3.
x( x  1)3
y
(3x  1)2
5.
y   2x  1  4 x 2
7.
9.
2.
y  ( x  1)( x  2)( x  4)
x( x 2  1)
4.
y
6.
y=
y  ( x 2  1)( x 2  2)( x 2  3)2
8.
y
y  x 2x
10. y  x cos x

11. y  x x
2

x 9
x 1
x( x  2)
(2x  1)(2x  2)
x cos x
( x  1)sin x
12. y  e x
x
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-3
Find the derivative of each of the following:
1.
y  5x
4.
y  e x
7.
y  x 2e  x
2

10. y  ln 1  e
2x

13. y  e x (sin x  cos x )
2
x
2.
y  3x
5.
y  e2x
8.
y  e t  et


3
3.
y  4cos x
6.
y e
9.
y  e 3/x
 1  ex 
11. y  ln 
x 
1e 
12. y 
14. y  ln e x
15. y  e 
x
2
e x  e x
2
ln x 
2
Find the equation of the tangent line to the graph of the function at the given point.
16. f ( x )  e1 x ; x  1
2
17. f ( x )  e2x  x ; x  2
18. f ( x )  ln e x ; x  2
 e x  e x
19. f ( x )  ln 
 2
20. f ( x )  x 2e x  2xe x  2e x ; x  1
21. f ( x )  e  x ln x ; x  1
 
2
Find

; x  0

dy
using implicit differentiation.
dx
22. xe y  10x  3 y  0
23. e xy  x 2  y2  10
Find the equation of the tangent line to the graph of the function at the given point.
24. xe y  ye x  1; (0,1)
25. 1  ln xy  e x  y ; (1,1)
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-4
Expand the logarithmic expression.
x4 y
1. ln 4
2. ln 3 x 2  1
z
3. ln z( z  1)2
Write each expression as a single logarithm.
1
5. 2ln3  ln  x 2  1 
2
6. 2ln x  ln( x  1)  ln( x  1)
8. ln x  ln4x   0
9. 7e5t  100
11. ln( x 4 )  ln( x 2 )  2
12. log3 y  3log3 y 2  14
13. y  ln x
14. y  x ln x
15. y  x 2e x
ex
16. y  ln
1  ex
17. y  e  e
x2
19. y  x
e
1
20. y  e sin2 x
2
4. 3ln x  2ln y  4ln z
Solve.
7. ln x  1  2
10. 2x
Find
Find
2
2 x
8
 
dy
for each of the following:
dx
2x
2 x
18. y  e
 x2
2
21. y  63x
dy
for each of the following using logarithmic differentiation:
dx
 x 3 
22. y   2

 x 1
2
23. y 
x x2  1
x 4
24. y   sin x 
x
Find the equation of the tangent line to the graph of the function at the given point.
2
1

25. f ( x )  4  x 2  ln  x  1  ; x  0
26. f ( x )  2e1 x ; x  1
2

Find
dy
for each of the following using implicit differentiation:
dx
27. cos x 2  xe y
28. ye x  xe y  xy
Find the equation of the tangent line to the graph of the function at the given point.
29. y ln x  y2  0; (e , 1)
30. ln( x  y )  x ; (0,1)
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-5
Find the intervals where f(x) is increasing and decreasing and the x-coordinates of any extrema.
1. f x   x  ln x
2. f x   x ln x
3. f x  
ln x
x
4. f x   ln x 2  2x  3
5. f x  
1 x 1 x
e  e
2
2
6. f x   xe  x
7. f x   x 2 e  x
8. f x   1  2  x e  x
Find the intervals where f(x) is concave up and concave down and the x-coordinates of any
points of inflection.
9. f x   e x  x 2  2
10. f x   xe  x
11. f x   xe x
12. f x  
2
ln x
x
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-6
Integrate:
1
1
1
1.
 x  1 dx
2.
 3x  2 dx
3.
 3 - 2x dx
4.
3x 2  1
 x 3  x dx
5.
x
x
dx
2
1
6.
x 2  2x  3
 x 3  3x 2  9x dx
9.
x
12.
 3x - 1ln 3x - 1
csc 2 x
dx
14. 
cot x
15.
 1  sin x
17.  cot x dx
18.  tan 2x dx
20.  cot x ln sinx  dx
21.
7.
10.

1
x 1
dx
1
 xlnx dx
sec 2 x
dx
13. 
tan x
16.

sec x tan x
dx
sec x - 1
19.  cot 3x - 1 dx
8.
11.


lnx 4
x
dx
ln2x 2
2x
dx
dx
lnx
dx
cos x

dx
tan lnx 
dx
x
2
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-7
Integrate:
1.
5x
 5e dx
2.
3 -x
 - 4x e dx
3.
4

e
x
dx
x
1
4.
e
 x 3 dx
5.
e x
 1  e  x dx
6.
e 2x
 1  e 2 x dx
7.
x
x
 e 1 - e dx
8.
e x - e x
 e x  e  x dx
9.
e x  e x
 e x  e  x dx
x2
10.
13.
2e x - 2e  x
 e
x

e

x 2

2x -1
 ln e dx
dx
5-ex
11.  2x dx
e
14.
e 2x  2e x  1
dx

ex
 
12.  e -x tan e -x dx
15.
ex
 1  e 
x 4
dx
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-8
Evaluate the definite integrals:
1.
1
1 x  2dx
1
e2
6.

2
8x
dx
4
9.

4

12
dx
1 x
8.
x

4
0
11.

7.

e
1
x
5

tan xdx
1
13.
 xe
16.

0
0
2
x 2
dx
14.
xe  x 2dx
17.
2
e
4
0
1

0
e 2 x dx

0

2
2 x3 2
xe
2
0
2
12.
dx
cosx esinx dx
4
1
1
0
1  cos x
dx
x  sin x
x
4
e
3
3 x
1
dx
x
dx
e3 x
dx
x2
15.

3
18.

2
1
5
 3x  1dx
3.
1
dx
x ln x
5.
4.
10.
1  ln x 2 dx
2
7 x  4dx
9
2.
3
sec 2x tan 2xe sec 2 x dx
CALCULUS 2
Name: _____________________________
WORKSHEET 7.2-9
1. Find the area of the region between the curve y 
2. Find the area between the curves y 
2x
and the x-axis over the interval  2,2.
1 x2
1
1
and y 
over the interval 1, e .
x
x
3. Find the area of the region above the curve y 
2
and below the x-axis over the interval  4,1 .
x2
4. Find the volume of the solid obtained by rotating the region under the curve y 
1
x 1
,
over the interval 0,1 , about the x-axis.
5. Find the area of the region enclosed by y  e x , y  3 and x  0.
6. Find the volume of the region enclosed by y  e x , y  0, x  ln 3 and x  0 , rotated about the x-axis.
CALCULUS 2
Name: _____________________________
WORKSHEET 7.7
Assume exponential growth or decay for each of the following:
1. A bacterial culture starts with 500 bacteria. After 3 hours, there are 8000 bacteria.
a. Find the number of bacteria after 4 hours.
b. When will the population reach 30,000 bacteria?
2. A cell of a particular bacterium divides into 2 cells every 1/3 of an hour. The initial population of
bacteria is 100 cells.
a. Find the number of cells after 10 hours.
b. When will the population reach 10,000 cells?
3. The population of a particular city doubled from 1890 to 1950. The population was 60,000 in 1950.
What was the population in 2000?
4. The half life of carbon-14 is 5730 years. How old is a specimen when it contains 40% of its original
quantity of carbon-14?
5. 30% of a radioactive substance disappears in 15 years. Find the half-life of the substance.
6. A particular substance triples in size every hour. At the end of 4 hours, the substance has a size of 10
units. What was the substance’s initial size?
CALCULUS 2
Name: _____________________________
CHAPTER 7 PRACTICE TEST
Expand the logarithmic expression:
x 4 y  2 
2
2. ln
z4
Solve:
3. ln 4  2 ln x  2
Find
Write the expression as a single logarithm:
1
2. 4 ln 2  ln x  1  5 ln y
3
4. 5e 3 t  11
5. 2 3x  4  8
dy
for each of the following:
dx

6. y  ln x 3  2x

7. y  ln
2x  12
3
3  2x
3
8. y  2 ln 4
x
e 2x
9. y  ln
1  e x
10. y  e  x
11. y 
2
1

12. y  3xe cos 4 x
Find
x5
ex
13. y  e 2 x  e  x

3
dy
for each of the following using logarithmic differentiation:
dx
14. y  3
x5
1  2x 
2
15. y  x sin x
Find the equation of the tangent line to the graph of the function at the given point:
16. y  ln x ; x  4
Find
17. y  xe 2 x ; x  0
dy
using implicit differentiation:
dx
18. x 2 y  y 2  e x
Find the x coordinates of any maxima, minima and points of inflection:
19. y  xe  x
20. y  x 2 ln x
Integrate:
4
dx
21. 
2x  1
ln x 3
23.

25.
e 5 x  e 5 x
 e 5x  e 5x dx
x
dx
 3 sec 2 x
22. 
dx
tan x
24.
e 3x  e x  3
dx
 e 2x
26.
x
2
e 43x dx
3
Evaluate the definite integrals:
1
27.

e4
e2
1
dx
x ln x
2
e x
28. 1
dx
3
2 x
1
29. Find the area enclosed by y  e x , x = 0, y = 0 and x = ln4.
30. Find the volume formed by rotating the area enclosed by y 
1
x 1
, x = 2, x = 5 and y = 0
about the x-axis.
31. The half life of a particular radioactive substance is 876 years. If the initial size was 13 grams,
what will be the size in 500 years?
32. The population of a certain organism tripled between 1920 and 1980. In what year will the
population have quadrupled?