3.5.1 Classwork

Use a calculator to fill in the tables and sketch the two functions on the same graph.

1. [latex]f(x)=e^x[/latex]
   

   [latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol x}[/latex]	[latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol y}{\color[rgb]{1.0, 1.0, 1.0}\mathbf=}{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol e}^{\color[rgb]{1.0, 1.0, 1.0}\mathbf x}[/latex]
   [latex]-3[/latex]
   [latex]-2[/latex]
   [latex]-1[/latex]
   [latex]0[/latex]
   [latex]1[/latex]
   [latex]2[/latex]

   

2. [latex]g(x)=\ln(x)[/latex]
   

   [latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol x}[/latex]	[latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol y}{\color[rgb]{1.0, 1.0, 1.0}\mathbf=}{\color[rgb]{1.0, 1.0, 1.0}\mathbf ln}\mathbf{\color[rgb]{1.0, 1.0, 1.0}\left(x\right)}[/latex]
   [latex]0.05[/latex]
   [latex]0.14[/latex]
   [latex]0.37[/latex]
   [latex]1[/latex]
   [latex]2.72[/latex]
   [latex]7.39[/latex]

3. How are the x- and y-coordinates of these two functions related? Give the domain and range for each function.

Use a calculator to evaluate. Give exact answers when possible, and round approximate answers to two decimal places.

4. [latex]ln(1)[/latex]
5. [latex]ln(8)[/latex]
6. [latex]ln(800)[/latex]
7. [latex]ln(8,000)[/latex]
   1. How are the inputs changing in problems 4 through 7 above?
   2. How are the outputs changing in problems 4 through 7 above?
   3. What is the difference between how the inputs are changing versus how the outputs are changing in problems 4 through 7 above?

Evaluate.

9. [latex]\sqrt{3^2}[/latex]
10. [latex]\sqrt{13^2}[/latex]
11. [latex]e^{In\left(7\right)}[/latex]
12. [latex]\ln(e^5)[/latex]
13. [latex]\ln(e^8)[/latex]
14. [latex]e^{\ln(15)}[/latex]

Identify the variable, then solve for the exact answer.

15. [latex]400=50e^{k10}[/latex]
16. [latex]\ln(2x)=0[/latex]
17. [latex]\ln(x-2)=1[/latex]
18. [latex]\ln(x-2)=0[/latex]
19. [latex]50=Pe^{0.06⋅10}[/latex]
20. [latex]25=50e^{k7}[/latex]

Solve. Round approximate answers to two decimal places.

21. If [latex]\$1,000[/latex] is invested, how long will it take to double to [latex]\$2,000[/latex] at a [latex]3\%[/latex] rate of interest compounded continuously?
22. Stan takes out an [latex]\$800[/latex] student loan. In three years and after making no payments but accruing interest continuously, he owes [latex]\$1,100[/latex]. What rate of interest is Stan being charged?
23. Jill wants to invest some money for her new nephew. She finds a CD that will pay [latex]4.25\%[/latex] interest compounded continuously for the next twenty years. If she wants the future value of the CD to be [latex]\$5,000[/latex], how much does she need to initially deposit?

3.5.2 Homework

Solve.

1. [latex]e^{3x}=12[/latex]
2. [latex]e^x=72[/latex]
3. [latex]250=50e^{0.06t}[/latex]
4. [latex]25=50e^{-0.0000124t}[/latex]
5. [latex]25=50e^{15r}[/latex]
6. [latex]\ln(x)=3.548[/latex]
7. [latex]2\ln(3x+1)=10[/latex]
8. [latex]ln(4x-7)=1[/latex]
9. [latex]2300=1000(1+r⋅15)[/latex]
10. [latex]I=1000\cdot r\cdot12,\;\text{where}\;r\;\text{is}\;0.2\%[/latex]

Spot the error and correct it.

11. [latex]\ln\left(x\right)+20=35[/latex]

[latex]x+20=e^{35}[/latex]

[latex]e\wedge\ln\left(x\right)+20=e\wedge35[/latex]

[latex]x=e^{35}-20[/latex]

12. [latex]\ln\left(50x\right)=0[/latex]

[latex]e\ast\ln\left(50x\right)=e\ast0[/latex]

[latex]50x=0[/latex]

[latex]x=0[/latex]

Solve. Round approximate answers to two decimal places.

13. [latex]\$7,000[/latex] is invested at [latex]8.5\%[/latex]. How long will it take the investment to grow to [latex]\$10,000[/latex] if it is compounded continuously?
14. [latex]\$7,000[/latex] is invested at [latex]8.5\%[/latex]. How long will it take the investment to double if it is compounded continuously?
15. Earning a [latex]6\%[/latex] interest rate, compounded continuously, how long will it take an investment to:
    1. double?
    2. triple?
16. Jeff owes [latex]\$838,13[/latex] to a debt collector after borrowing money at [latex]7.25\%[/latex] interest compounded continuously and failing to make payments for two years. How much did he originally borrow?
17. If your money grew to [latex]\$12,000[/latex] invested under simple interest on an investment of [latex]\$10,000[/latex] in five years, what is the rate of interest?
18. Jack would like to invest [latex]\$3,000[/latex] for 15 years and end up with [latex]\$6,500[/latex]. Assuming he will earn interest compounded continuously, what rate of return will he need to earn?