1.5: Exercises
- Page ID
- 106802
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)To see if you are on track, solve the following exercises using only the formula sheet (no calculators, computers, books, etc!).
- Draw the straight line that has a \(y\)-intercept of 3/2 and a slope of 1/2.
- Express \(\frac{3}{4} - \frac{2}{3} + 1\) as a single fraction.
- Simplify \((a -4a^3)/a^{-2}\).
- Express \(\ln 8 − 5 \ln 2\) as the logarithm of a single number.
- Given \(\ln P = − \frac{a}{RT} + b \ln T + c\), where \(a\), \(b\), \(c\) and \(R\) are constants, obtain \(\frac{d(\ln P)}{dT}\)
- Obtain \(\frac{dy}{dx}\)
- \(y = \sin xe^{mx}\) (m is a constant).
- \(y = \frac{1}{\sqrt{1−x^2}}\)
- Obtain the first, second and third derivatives of
- \(y = e^{−2x}\)
- \(y = \cos(2x)\)
- \(y = 3 + 2x − 4x^2\)
- Evaluate \(\int_0^{\pi} \cos 3 \theta d \theta\).
- Use the properties of integrals and your previous result to evaluate \(\int_{\pi}^0 \cos 3 \theta d \theta\). What about \(\int_0^{\pi/4} \cos 3 \theta d \theta + \int_{\pi/4}^{\pi} \cos 3 \theta d \theta\)?
- Given \(f(x) = \left\{\begin{matrix} 0 & \text{if } x<0 \\ 3+2x & \text{if } 0 <x<1 \\ 0 & \text{if } x>1 \end{matrix}\right.\) Sketch \(f(x)\) and calculate \(\int_{- \infty}^{\infty} f(x) dx\)
- What is the value of this integral? \(\int_{- \infty}^{\infty} xe^{-x^2} dx\)
- Sketch \(\sin(x/2)\). What is the period of the function?
- The plots below (Figure \(\PageIndex{1}\)) represent the following functions: \(y = 3e^{−x/2}, ~ y = 3e^{−x}, ~ y = 3e^{−2x}\) and \(y = 2e^{−2x}\). Which one is which?
