Sinclair Community College
MAT 1470-College Algebra: Review sheet for Test 3
 13 
x
P age |1
1.
Graph both functions on one set of axes: y   3 and y 
2.
Graph the function g  x   2
3.
form a x . State the domain, range, and asymptote of the function.
Graph the function y  2 x , not by plotting points but by applying your knowledge of the general shape of graphs of the
4.
form a x . State the domain, range, and asymptote of the function.
Graph the function, not by plotting points, but by starting from the graph of y  e x . State the domain, range, and asymptote.
5.
6.
x
x 1
.
, not by plotting points but by applying your knowledge of the general shape of graphs of the
h( x)  e x 1  5
A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function
m(t )  13e0.015t where m(t ) is measured in kilograms. Round your answers to three decimal places.
Find the mass at time t  0 .
How much of the mass remains after 50 days?
If $10,000 is invested in an account at 5% per year, find the amount of the investment at the end of 12 years for the following
compounding methods.
7.
(a) Continuously
(b) Daily
Express the equation log3  19   2 in exponential form.
8.
Express the equation log7  2 x   2 in exponential form.
9. Express the equation 63  216 in logarithmic form.
10. Evaluate the expression.
ln  e 1 
11. Evaluate the expression.
ln  e2 
e
12. Evaluate the expression.
log 7 49
13. Use the definition of the logarithmic function to find x if log x 512  3 .
14. Use the definition of the logarithmic function to find x if log x 7  1/ 3 .
15. Find the domain of the function h  x   log  x 2  x  .
16. Rewrite the expression log  x2  4   log  x  2   log  x 4  8x 2  16  as a single logarithm.
 x3 
17. Expend the expression ln  3 3 
y z
 x 
18. Expend the expression log a  3 2 
y z 
19.
20.
21.
22.
Find the solution of the equation 31 x  7 correct to four decimal places.
Find the solution of the equation 3x  2  52 x correct to four decimal places.
Find the solution of the equation e13 x  7 correct to four decimal places.
Solve the logarithmic equation log7  x  4  log7  x  2   1 for x .
23. How long will it take for an investment of $2550 to double in value if the interest rate is 5.65% per year compounded
continuously?
24. A sum of $1250 was invested for 8 years, and the interest compounded quarterly. If this sum amounted to $6583.32 in the
given time, what was the interest rate?
25. A 12-g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by m(t )  12e0.087t ,
where m(t ) is measured in grams. After how many days is there only 4 g remaining?
26. Rudy wants to invest $1000 in savings certificates that bear an interest rate of 7.5%
compounded semiannually. How long will it take for the amount to be $1500?
27. How long will it take an investment of $300 to triple, if the interest rate is 3.5% per year compounded continuously?
Sinclair Community College
MAT 1470-College Algebra: Review sheet for Test 3
P age |2
28. The frog population in a pond grows exponentially. The current population is 235 frogs, and the relative growth rate is 12%
per year. (a) Find a formula for the population n  t  after t years. (b) Find the projected population after 5 years. (c) Find
29.
30.
31.
32.
the number of years required for the frog population to reach 1000 .
The half-life of strontium- 90 is 25 years. How long will it take for a 12 mg sample to decay to a mass of 8 mg?
If after one day a sample of radioactive element decays to 96% of its original amount, find its half-life.
How much more intense is an earthquake with a magnitude of 5.4 on the Richter scale than one with a magnitude of 3.2 ?
Solve the system.
 4 x  12 y  24

11x  4 y  153
33. Solve the system.
25 x  75 y  100

20 x  60 y  80
34. Use back-substitution to solve the triangular system.
x  2 y  z  0

y  3z  1


4 z  4

35. Find the complete solution of the linear system or show that it is inconsistent.
x y z  2


2 x  3 y  2 z  4
 4 x  y  3z  1

36. Find the complete solution of the linear system or show that it is inconsistent.
 x  2 y  z  2

 2 x  3 y  5 z  4
4 x  8 y  4 z  8

37. Solve the system of linear equations using Gaussian elimination.
x  3y  0


2 x  7 y  2 z  1
 x  4 y  3z  4

38. Solve the system of linear equations using Gaussian elimination.
2 x  8 y  z  2

 x  3y  z  1
 x  4y  z  0

39. Solve the system of linear equations using Gauss-Jordan elimination.
 xz  2

 y  2z  7
 x  z  4

40. Solve the system of linear equations.
2 x  y  3z  3


 x  3 y  2 z  4
7 x  7 y  12 z  5

Sinclair Community College
MAT 1470-College Algebra: Review sheet for Test 3
2
41. Given A   4
 3
2
42. Given A   3
 4
3 1
3

2 0  and B   5
 7
5 2 
5 3
 1
2 0  and B  
0
2 2 
5 3
2 2  , find 3 A  2B , or explain why the operation cannot be performed.
4 2 
5
, find A  B , or explain why the operation cannot be performed.
25
5
 4 1 1
and B  
 , find AB , or explain why the operation cannot be performed.
3
 3 5 2 
5
 4 1 1
and B  

 , find BA , or explain why the operation cannot be performed.
3
 3 5 2 
45. Find A2 , or explain why the operation cannot be performed.
1
43. Given A  
 2
1
44. Given A  
 2
2 0
A  0 2
0 0
46. Solve for x
1
0 
2 
and y .
3 
 2 x   y 3  1
6 0   2 2   6 18


 

47.
P age |3
Sinclair Community College
MAT 1470-College Algebra: Review sheet for Test 3
Answer Key
1.
The graph is
2.
Domain
 ,  
, Range
 0,  
3.
Domain
 ,  
, Range
 , 0
, Asymptote y  0
, Asymptote y  0
2
–4
–2
0
2
4
–2
–4
–6
–8
4.
Domain: (, ) ; Range: (5, ) ; Asymptote: y  5
5.
6.
7.
a) 13 kg b) 6.141 kg
(a) $18221.19 (b) $18220.44
log3  19   2  32  19
8.
72  2x
9.
log6 216  3
10.
ln  e 1     1
P age |4
Sinclair Community College
MAT 1470-College Algebra: Review sheet for Test 3
11.
e
 
ln e2
P age |5
 e2
4
4
 1
12. log 7 49  log 7 7  log 7 7  log 7  7 2   log 7 7  4
 
3
3
13. log x 512  3  x  512  x  83  x  8
4
2
2
14. log x 7  13

7  x1/ 3

 
x  343 .
15. x  x  0  x  x  1  0 . The domain is  , 1   0,   .
2
log  x 2  4   log  x  2   log  x 4  8 x 2  16   log
16.
 log
x
2
 4  x  2
x  8 x 2  16
4
 x  2  x  2  x  2 
1
 log
  log  x  2 
2
2
x2
 x  2  x  2
 x3 
1
17. ln  3 3   ln x3  ln y 3  ln z 3  3ln x  3ln y  13 ln z
y z

18.



log a x /  y3 z 2   loga x   3loga y  2loga z   loga x  3loga y  2loga z
19. log 31 x  log 7  1  x  log 3  log 7  1  x 
20.
21.
3x  2  52 x
1 3 x
e
7
 x  2 log 3  2x log 5


13 x
ln e

 ln 7
22. log7  x  4  log7  x  2   1


log 7
log 7
 0.7712
 x  1
log 3
log 3
2 x log5  x log3  2log3
1  3x  ln 7

x
log7   x  4 x  2   1

x  2log 3 /  2log 5  log 3  1.0363
1  ln 7  0.3153
  x  4 x  2  7
1
3

x2  2 x  15  0

x5
or x  3 , but x  5 is the only solution.

23. 5100  2550e0.0565t
 r
24. Use A  P 1  
 n
nt

2  e0.0565t

ln 2  ln e0.0565t
 r
6583.32  1250 1  
 4
32
32
 r
5.27  1  
 ln 5.27  ln 1  4r 
 4
 r  0.213 or 21.3%
25. 13 days

4  8

t
ln 2
 12.27 , about 12 years, and 3 months.
0.0565
 r
. Now solve for r : 6583.32  1250 1  
 4
 r
ln 5.27  32ln  1  
 4

4  8
 r
0.0519  ln  1  
 4


 r
e0.0519  1  
 4
nt
26.
 r
P  1000 , A  t   1500 , r  0.075 , and n  2 . Since A  t   P 1   ,
 n
2t
27.
28.
29.
30.
log 32
log 32
3
 0.075 
2t
therefore 1500  1000 1 
 5.51 . So in approximately
 2t 
 t
  1.0375 
2 
2log1.0375
log1.0375
2

5.51 years the amount will be $1500 .
ln 3
 31.39 years.
A  t   Pert  900  300e0.035t  3  e0.035t  ln 3  0.035  t  t 
0.035
1
1000
ln
 12.07 years.
(a) n  t   235e0.12t (b) n  5  235e0.125  428 (c) n  t   1000  1000  235e0.12t  t 
0.12 235
25ln  2 / 3
 14.62 years.
m  t   m0 et ln 2 / 25 , so m  t   8  8  12et ln 2 / 25  t ln 2 / 25  ln 32  t  
ln 2
ln 2
 16.98 days.
Let t be measured in days. Then m 1  m0 e ln 2 / h  0.96m0   ln 2 / h  ln 0.96  h  
ln 0.96
log I / I
31. log  I1 / I 2   5.4  3.2  2.2 . So it is 10  1 2   102.2  158 times as intense.
32. (15,3)
Sinclair Community College
MAT 1470-College Algebra: Review sheet for Test 3
P age |6
 1 4
33.  t , t  
 3 3
34. x  5 , y  2 , z  1
x yz  2


35. 2 x  3 y  2 z  4
 4 x  y  3z  1

 x yz  2

 
5 y  0  x  1 , y  0 , z  1
3 y  7 z  7

 x  2 y  z  2
 x  2 y  z  2



36.  2 x  3 y  5 z  4
 7 y  7 z  0  dependent, so let z  t .
4 x  8 y  4 z  8

00


Then y  t , x  t  2 .
1 3 0 0 
37.  2 7 2 1
1 4 3 4 
R2  2 R1  R2

R3  R1  R3
1 3 0 0 
0 1 2 1


0 1 3 4

R3  R2  R3
1 3 0 0 
0 1 2 1 , so the solution is z  3 ,


0 0 1 3

R1  R3
1 4 1 0 
0 1 0 1  , so the solution is z  2 , y  1 ,


0 0 1 2
y  1  2 z  5 , x  3 y  15 .
 2 8 1 2  R1  2 R3  R1
38.  1 3 1 1 

 1 4 1 0  R2  R3  R2
x  z  4 y  2  4  2 .
0 0 1 2
0 1 0 1 


1 4 1 0 
 1 0 1 2
1 0 1 2  R1  12 R3  R1

0 1 2 7 

39.  0 1 2 7 

R3  R1  R3 
 1 0 1 4 
0 0 2 6  R2  R3  R2
x  1 , y  1 , z  3 .
40.
 2 1 3 3  R1  2 R2  R1 0 7 1 11 
1 3 2 4
1 3 2 4 





7 7 12 5 R3  7 R2  R3 0 14 2 21
so the system is inconsistent, and there is no solution.
1 0 0 1
0 1 0 1  

 1R
0 0 2 6  2 3
1 0 0 1
0 1 0 1  , so the solution is


0 0 1 3 
0 7 1 11 
R3  2R1  R3 1 3 2 4  ,
0 0 0 1
9 3  6 10 6  0 1 3 
 2 3 1
 3 5 3  6
41. 3 A  2 B  3  4 2 0   2  5 2 2   12 6 0    10 4 4   2 10 4 
 3 5 2 
 7 4 2   9 15 6   14 8 4  5 23 2 
42. The operation cannot be performed because the matrices are different sizes.
 1 5  4 1 1 19 24 9
43. AB  



 2 3  3 5 2   1 17 8
44. The operation BA cannot be performed, because B has 3 columns and A only has 2 rows.
 2 0 1   2 0 1   4 0 4
2
45. A  0 2 0  0 2 0    0 4 0 
0 0 2  0 0 2  0 0 4 
3 
 2 y  2 x 6  2 x  1
2 x   y 3  2 y  2 x 6  2 x 
2 y  2 x  1  x   3 , y  1
46. 
 


2

  




6
y
18

6
18
6
0

2
2
6
y
18

 



 

6  2 x  3

OS2006(inutake)fnl

Practice 4-4 Level B

Notes - Arlington Local Schools

esercizi sulle derivate 2

Foglio derivate 2