1.7.E: Exercises
- Page ID
- 157148
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.7 Exercises
A population numbers 11,000 organisms initially and grows by 8.5% each year. Write an exponential model for the population.
A population is currently 6,000 and has been increasing by 1.2% each day. Write an exponential model for the population.
A vehicle purchased for $32,500 depreciates at a constant rate of 5% each year. Determine the approximate value of the vehicle 12 years after purchase.
A business purchases $125,000 of office furniture which depreciates at a constant rate of 12% each year. Find the residual value of the furniture 6 years after purchase.
If $4,000 is invested in a bank account at an interest rate of 7 per cent per year, find the amount in the bank after 9 years if interest is compounded annually, quarterly, monthly, and continuously.
If $6,000 is invested in a bank account at an interest rate of 9 per cent per year, find the amount in the bank after 5 years if interest is compounded annually, quarterly, monthly, and continuously.
Match each function with one of the graphs below.
| 7. \(f(x) = 2(0.69)^x\) |
| 8. \(f(x) = 2(1.28)^x\) |
| 9. \(f(x) = 2(0.81)^x\) |
| 10 \(f(x) = 4(1.28)^x\) |
| 11. \(f(x) = 2(1.59)^x\) |
| 12. \(f(x) = 4(0.69)^x\) |
If all the graphs to the right have equations with form \(f(t) = ae^{kt}\),
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13. Which graph has the largest value for \(k\)? |
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14. Which graph has the smallest value for \(k\)? |
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15. Which graph has the largest value for \(a\)? |
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16. Which graph has the smallest value for \(a\)? |
1.8 Exercises
Rewrite each equation in exponential form
| 1. \(\log (v) = t\) | 2. \(\log (r) = s\) | 3. \(\ln (w) = n\) | 4. \(\ln (x) = y\) |
Rewrite each equation in logarithmic form.
| 5. \(10^a = b\) | 6. \(10^p = v\) | 7. \(e^k = h\) | 8. \(e^y = x\) |
Solve each equation for the variable.
| 9. \(5^{x} = 14\) | 10. \(3^x = 23\) | 11. \(7^x = \frac{1}{15}\) | 12. \(3^x = \frac{1}{4}\) |
| 13. \(e^{5x} = 17\) | 14. \(e^{3x} = 12\) | 15. \(3^{4x-5} = 38\) | 16. \(4^{2x-3} = 44\) |
| 17. \(1000(1.03)^t = 5000\) | 18. \(200(1.06)^t = 550\) |
| 19. \(3(1.04)^{3t} = 8\) | 20. \(2(1.08)^{4t} = 7\) |
| 21. \(50e^{-0.12t} = 10\) | 22. \(10e^{-0.03t} = 4\) |
| 23. \(10 - 8 \left(\frac{1}{2}\right)^x = 5\) | 24. \(100-100\left(\frac{1}{4}\right)^x = 70\) |
The population of Kenya was 39.8 million in 2009 and has been growing by about 2.6% each year. If this trend continues, when will the population exceed 45 million?
The population of Algeria was 34.9 million in 2009 and has been growing by about 1.5% each year. If this trend continues, when will the population exceed 45 million?
If $1000 is invested in an account earning 3% compounded monthly, how long will it take the account to grow in value to $1500?
If $1000 is invested in an account earning 2% compounded quarterly, how long will it take the account to grow in value to $1300?
Sketch a graph of: \(f(x) = \log (x)\), \(g(x) = \ln (x)\)
Find the domain of each function.
| 30. \(f(x) = \log (x-5)\) | 31. \(f(x) = \ln (3-x)\) |
| 32. \(f(x) = \ln (1-3x)\) | 33. \(f(x) = \log (2x+5)\) |