1.6.E: Exercises
- Page ID
- 157147
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.6 Exercises
Find the degree and leading coefficient of each polynomial
| 1. \(4x^7\) | 2. \(5x^6\) |
| 3. \(5-x^2\) | 4. \(6+3x-4x^3\) |
| 5. \(-2x^4-3x^2+x-1\) | 6. \(6x^5-2x^4+x^2+3\) |
Find the vertical and horizontal intercepts of each function.
| 7. \(f(t) = 2(t-1)(t+2)(t-3)\) | 8. \(f(x)=3(x+1)(x-4)(x+5)\) |
| 9. \(g(n) = -2(3n-1)(2n+1)\) | 10. \(k(u)=-3(4-n)(4n+3)\) |
| 11. \(C(t) = 2t^4-8t^3+6t^2\) | 12. \(C(t)=4t^4+12t^3-40t^2\) |
Use your calculator or other graphing technology to solve graphically for the zeros of the function.
| 13. \(f(x) = x^3 - 7x^2+4x+30\) | 14. \(g(x)=x^3-6x^2+x+28\) |
Solve each inequality.
| 15. \((x-3)(x-2)^2>0\) | 16. \((x-5)(x+1)^2>0\) |
| 17. \((x-1)(x+2)(x-3)<0\) | 18. \((x-4)(x+3)(x+6)<0\) |
For each function, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal asymptote.
| 19. \(p(x) = \frac{2x-3}{x+4}\) | 20. \(q(x) = \frac{x-5}{3x-1}\) |
| 21. \(s(x) = \frac{4}{(x-2)^2}\) | 22. \(r(x) = \frac{5}{(x+1)^2}\) |
| 23. \(f(x) = \frac{3x^2-14x-5}{3x^2+8x-16}\) | 24. \(g(x) = \frac{2x^2+7x-15}{3x^2-14+15}\) |
| 25. \(h(x) = \frac{2x^2+x-1}{x-4}\) | 26. \(k(x) = \frac{2x^2-3x-20}{x-5}\) |
A scientist has a beaker containing 20 mL of a solution containing 20% acid. To dilute this, she adds pure water.
- Write an equation for the concentration in the beaker after adding \(n\) mL of water.
- Find the concentration if 10 mL of water has been added.
- How many mL of water must be added to obtain a 4% solution?
- What is the behavior as \(n \to \infty\), and what is the physical significance of this?
A scientist has a beaker containing 30 mL of a solution containing 3 grams of potassium hydroxide. To this, she mixes a solution containing 8 milligrams per mL of potassium hydroxide.
- Write an equation for the concentration in the tank after adding \(n\) mL of the second solution.
- Find the concentration if 10 mL of the second solution has been added.
- How many mL of water must be added to obtain a 50 mg/mL solution?
- What is the behavior as \(n \to \infty\), and what is the physical significance of this?