Math 102-500  Exam 3 Review

1.a)  is one to one for . Find the inverse function.

b) Find the inverse function to the same function in a) for .

c) for  Find the inverse function.

d) Find the inverse for the function in c) on

2. a) Graph g(x)=ln(2x+4). Show the vertical asymptote and x-intercept.

b) Find the inverse function to g(x)=ln(2x+4) and graph it.

c) Graph h(x)=ln(3x-9). Show the vertical asymptote and x-intercept.

d) Find the inverse function to the function in c) and graph it.

e) Graph . Show the horizontal asymptote.

f) Find the inverse function to the function in e) and graph it.

3. $1000 is invested in an account earning 6% interest. How much is in the account after 10 years if interest is compounded

a) quarterly?   b) monthly ?   c) continuously?

4.a) If interest is 7% compounded continuously, how long will it take any investment to double?

b) If interest is 5% compounded continuously, how long will it take any investment to triple?

5. a) How much should be invested at 6% compounded continuously to amount to $30000 in 10 years?

b) How much should be invested at 8% compounded continuously to amount to $50000 in 15 years?

6. The population of a certain city grows continuously at annual rate 1.5%. If the population was 150000 in the year 2000, what will the population be in 2008?

7. A drug is eliminated from the body at a continuous rate so that after t hours the amount is . After 10 hours, half of the original amount is left.

a) Find c.    b) If initially there were 100mg, how much is present after 15 hours?

8. Solve each equation for x.

a)       b)     c) ln(x+1) + ln(x-1) = 1

d)     e)

9.  Evaluate using the change of base formula and a calculator.

a)    b)     c)

10. Solve without using a calculator.

   b) log0.001    c)     d)      e)     f)    g)

11. Use matrices to find the solution or state there is none.

2x + 3y = 11          b) x + 5y = 9                      c)    5x + 4y = 0

3x – 4y = -9              2x -  y  = -4                        2.5x  + 2y=1

12. Use matrices to solve the system of equations.

a)  5x + y + 2z = 4                   b) 2x + 2y + z = 1        c) 4x – 3y = 1

             y -   z = 3                          x -  y  + 2z =3            2x +  y = -7

                                                    2x + 3y + z = -1           -x + y =  -1

13. Perform the indicated matrix operations for

a)  AB     b)  BA

14. Find the inverse matrix to .   Use the inverse matrix to solve

a)  3x + 4y = 5                         b) 3x + 4y = 7

     4x + 5y = 4                             4x + 5y = -2

Rational Functions

Find all holes, vertical asymptotes and horizontal asymptote or oblique asymptote.

15.

16.

17.

18.

Key:

1a)       1b)

c) d)

2. a) vertical asymptote is x= -2, x-intercept is -3/2

b)

c) vertical asymptote is x = 3, x intercept is 10/3 

d)

e) horizontal asymptote is y=5 as x goes to minus infinity only.

f)

3. a) 1814.02   b) 1819.40   c) 1822.12

4a) 9.9 years   b) 21.97 years

5a) 16,464.35    b) 15059.71

6.  169,125

7. a)  c=.0693   b) 35.363 mg

8a) x=2 only, negatives cannot be used in logs.  b) x=1/24   c)

d) x= ln3 or x=0   e) x=5/3

9. a) .43   b) 1.58   c) -1.16

10 a) 0.5   b) -3   c) -2   d) 3   e) 1.5   f) 3   g) 1.5

11. a) x=1, y=3      b) x= -1, y=2    c) no solution

12. a)   t is any real number

b) x = 3, y = -2, z = -1

x = -2 , y = -3

13. a)       b) 

14.     a) x= -9, y = 8   b) x= - 43, y = 34

15. hole at (2,0) V.A. x= -2 H.A. y=3/2

16. no holes V.A. x= -3 x=2 H.A y=1

17. hole at (-3,1/9) V.A. x=3 H.A. y=0

18. no holes or V.A. oblique asymptote y=x `