Post on 15-May-2019
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?