CALCULUS 2 Name: _____________________________

WORKSHEET 7.1-1

1. Graph x5y and xlogy 5 . 2. Graph xey and xlny .

Convert to log form. Convert to exponential form.

3. x5y 7. xlogy 9

4. 53243 8. 1000log3

5. 7ex 9. 10lnx

6. 2

14 2

1

10. 2

3

27

1log9

Evaluate.

11. 64log4 12. 7

1log 7

13. 100

1log 14. 3 2

5 5log

15. 27log9 16. 16

1log 64

17. 21log1111 18.

42log2 2log

2

CALCULUS 2 Name: _____________________________

WORKSHEET 7.1-2

Expand each logarithm.

1. log6 3x 2. log2 x

7

3. log4 xy2 4. log3

3

2y3x

5. log3 zyx 4 6. log2 4

2

3z

yx3

7. log5

4

2

(xy)

z 8. log8

2 3

34

x y

z

Write each logarithmic expression as a single logarithm.

9. log3 7 – log3 x 10. 2 log5 x + log5 3

11. 23

log2 x – 3 log2 y 12. 12

(log3 4 + log3 y) – 3 log3 z

13. 12

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) - 34

log2 (x-1) 16. 12

[log6 (x-2) + 2 log6 (x+1) – log6 (x+2) – 5 log6 x]

Evaluate.

17. 16log 2 18. 2

a alog 19. 7log3

20. 7log66 21. 62.14log3 22. a

1log a

Solve.

23. 32x 24. 253 1x 25. 52 4x3

26. 162 2x 27. x1x 43 28. x32x1 53

29. 314e x6 30. x07.e31225 31. x2.3eln1.2ln

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-1

Find the derivative of each of the following:

1. lny x x 2. lny t t t 3. 2

lny x

4. 2lny x 5. 3ln 3 1y x x 6. ln2xy

7. ln sin3 1y t 8. 2 lny x x 9. ln x

yx

10. ln lny x 11. 3

ln lny x 12. 2ln( 3 )y x x

13 3ln1

xy

x

14. 3ln( 4)y x 15. ln 3 2y x

16. 2 2 3ln( ) (ln )y x x x 17.

3

2 2

ln 1ln

ln

xy

x x x

18. 2ln( 1)y x x

19. 2logy x 20. 3 210logy x x 21. 3log siny t

Find the equation of the tangent line to the graph of f at the given point.

22. ( ) ln ; 5f x t t 23. ( ) ln(8 4 ); 1f x t t

24. 2

x ;sinln)(

xxf 25. 2( ) 3 ln ; 1f x x x x

Find dy

dx using implicit differentiation.

26. 2 23ln 10x y y 27. ln 5 30xy x

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-2

Find dy

dxusing logarithmic differentiation.

1. ( 2)( 4)y x x 2. ( 1)( 2)( 4)y x x x

3. 3

2

( 1)

(3 1)

x xy

x

4.

2( 1)

1

x xy

x

5. 22 1 4 9y x x x 6. y =( 2)

(2 1)(2 2)

x x

x x

7. 2 2 2 2( 1)( 2)( 3)y x x x 8. cos

( 1)sin

x xy

x x

9. 2xy x 10. cos xy x

11. 2xy x 12.

xxy e

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-3

Find the derivative of each of the following:

1. 5xy 2. 2

3x xy 3. cos4 xy

4. 2xy e 5. 2xy e 6. xy e

7. 2 xy x e 8. 3

t ty e e 9. 23/xy e

10. 2ln 1 xy e 11. 1

ln1

x

x

ey

e

12.

2

x xe ey

13. (sin cos )xy e x x 14. ln xy e 15. 2

ln xy e

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

16. 1( ) ; 1xf x e x 17. 22( ) ; 2x xf x e x

18. 2

( ) ln ; 2xf x e x 19. ( ) ln ; 02

x xe ef x x

20. 2( ) 2 2 ; 1x x xf x x e xe e x 21. ( ) ln ; 1xf x e x x

Find dy

dxusing implicit differentiation.

22. 10 3 0yxe x y 23. 2 2 10xye x y

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

24. 1; (0,1)y xxe ye 25. 1 ln ; (1,1)x yxy e

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-4

Expand the logarithmic expression.

1. 4

4

z

y xln 2. 3 2ln 1x 3.

2ln ( 1)z z

Write each expression as a single logarithm.

4. 3ln 2ln 4lnx y z 5. 212ln3 ln 1

2x 6. 2 ln ln( 1) ln( 1)x x x

Solve.

7. ln 1 2x 8. 0x4lnxln 9. 57 100te

10. 2 22 8x x 11.

4 2ln( ) ln( ) 2x x 12. 23 3log 3log 14y y

Find dy

dxfor each of the following:

13. lny x 14. lny x x 15. 2 xy x e

16. ln1

x

x

ey

e

17. 2 2x xy e e 18.

2

2

x

y e

19. 2

x

xy

e 20. sin21

2xy e 21.

36 xy

Find dy

dxfor each of the following using logarithmic differentiation:

22.

2

2

3

1

xy

x

23.

2 1

4

x xy

x

24. sin

xy x

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

25. 2 1

( ) 4 ln 1 ; 02

f x x x x

26. 21( ) 2 ; 1xf x e x

Find dy

dxfor each of the following using implicit differentiation:

27. 2cos yx xe 28. x yye xe xy

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

29. 2ln 0; ( , 1)y x y e 30. ln( ) ; (0,1)x y x

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. xlnxxf 2. xlnxxf

3. x

xlnxf 4. 3x2xlnxf 2

5. xx e2

1e

2

1xf 6. xxexf

7. x2exxf 8. xex21xf

Find the intervals where f(x) is concave up and concave down and the x-coordinates of any

points of inflection.

9. 2xexf 2x 10. xxexf

11. 2xxexf 12.

x

xlnxf

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-6

Integrate:

1. dx

1x

1 2.

dx 23x

1 3. dx

2x-3

1

4.

dx

xx

1x33

2

5.

dx 1x

x2

6.

dx

x9x3x

3x2x23

2

7.

dx 1x

1 8.

dx

x

lnx4

9. lnxx

dx

10. dx xlnx

1 11.

dx

2x

ln2x2

12. 2

1-3xln1-3x

dx

13. dx tan x

xsec2

14. dx cot x

xcsc2

15. dx

sin x1

x cos

16. dx 1- x sec

xtanx sec 17. dxcot x 18. dx2x tan

19. dx 1-3xcot

20. dx sinxlncot x

21.

dx x

lnxtan

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-7

Integrate:

1. dx 5e5x 2. dx e4x-4-x3 3. dx

x

e x

4. dx x

e3

x

12

5.

dx

e1

ex

x

6.

dx e1

ex2

x2

7. dx e-1e xx 8.

dx

ee

e-exx

xx

9.

dx

ee

eexx

xx

10.

dx

ee

2e-2e2xx

xx

11. dx e

e-52x

x

12. dx etane -x-x

13. dx eln 1-2x 14.

dx e

1e2ex

x2x

15.

dx e1

e4x

x

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-8

Evaluate the definite integrals:

1. dx2x

11

1 2. dx

4x

29

7

3. dx

1x3

54

0

4.

dxx

xln1e

1

2

5. dx

xlnx

12e

e 6. dx

xsinx

xcos12

1

7. dxx

125

1 8. dx

4x

x84

0 2 9. dx

x

1x

4

1

10. dxxtan4

0

11. dxe

1

0

x2

12. dxe

4

3

x3

13. dxxe1

0

x 2

14. dxex

0

2

2x2 3

15. dx

x

e3

1 2

x3

16. dxxe2

0

2x 2

17. dxexcos

2

0

xsin

18. dxxe2tanx2sec2

3

x2sec

CALCULUS 2 Name: _____________________________

WORKSHEET 7.2-9

1. Find the area of the region between the curve 21

2

x

xy

and the x-axis over the interval 2,2 .

2. Find the area between the curves x

y1

and x

y1

over the interval e,1 .

3. Find the area of the region above the curve 2

2

xy and below the x-axis over the interval 1,4 .

4. Find the volume of the solid obtained by rotating the region under the curve 1

1

xy ,

over the interval 1,0 , about the x-axis.

5. Find the area of the region enclosed by 3, yey xand .0x

6. Find the volume of the region enclosed by 3ln,0, xyey xand 0x , 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: Write the expression as a single logarithm:

2.

4z

2yxln

24

2. yln51xln

3

12ln4

Solve:

3. 2xln24ln 4. 11e5 t3 5. 82 4x3

Find dy

dxfor each of the following:

6. x2xlny 3 7. 3

2

x23

1x2lny

8. 4x

3ln2y 9.

x

x2

e1

elny

10. 1x2

ey 11. x

5

e

xy

12. x4cosxe3y 13. 3xx2 eey

Find dy

dxfor each of the following using logarithmic differentiation:

14.

32

5

x21

xy

15. xsinxy

Find the equation of the tangent line to the graph of the function at the given point:

16. 4 x; xlny 17. 0 x; xey x2

Find dy

dx using implicit differentiation:

18. x22 eyyx

Find the x coordinates of any maxima, minima and points of inflection:

19. xxey 20. xlnxy 2

Integrate:

21. dx 1x2

4

22. dx xtan

xsec3 2

23.

dx x

xln3

24. dx e

3eex2

xx3

25. dx ee

eex5x5

x5x5

26. dx ex

3x342

Evaluate the definite integrals:

27. dx xlnx

14

2

e

e 28. dx

x

e1

21 3

x1

2

29. Find the area enclosed by

xey , x = 0, y = 0 and x = ln4.

30. Find the volume formed by rotating the area enclosed by 1x

1y

, 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?