12.11: Chapter Exercises
- Page ID
- 135914
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise 1: Polynomial Representation
Represent each polynomial as a MATLAB coefficient vector: (a) 5*x^3 - 2*x + 8, (b) -3*x^4 + x^2 - 9, (c) 7*x - 4. Then use polyval to evaluate each polynomial at x = 2.
Exercise 2: Temperature Interpolation
Create arrays for time = [8 10 12 14 16 18] and temperature = [55 70 78 76 70 63]. Use interp1 to estimate the temperature at 13:00 and 15:30.
Exercise 3: Ideal Gas Interpolation
Using the volume and pressure data from the ideal gas example, write a script named idealGas.m that uses cubic spline interpolation to estimate the pressure when V = 3.8 m^3 and the volume when P = 1000 kPa. Plot the original data and the two estimated points.
Exercise 4: Predict Voltage
The voltage in a circuit is measured at times 1 through 10 seconds with voltage values [1.1 1.9 3.3 3.4 3.1 3.3 5.1 4.3 5.0 3.9]. Write predictVoltage.m to fit a straight line, plot the data and line, and estimate the voltage at t = 3.5 seconds.
Exercise 5: Channel Flow Curve Fits
Use the height and flow data from the chapter slides. Fit linear, quadratic, and cubic models to the data and plot all three fits on the same graph. Which model appears to represent the data best? Explain your choice in a comment.
Exercise 6: Compare Integrals
In a script called compareIntegrals.m, use both integral and trapz to calculate the definite integral of f(x) = 4*x^2 + 3 from x = -1 to x = 3. Compare the results.
Exercise 7: Polynomial Derivative
In a script called findDiff.m, calculate the derivative of f(x) = 2*x^5 + 6*x^3 - 9*x + 5. Then find the derivative at x = 2.
Exercise 8: Polynomial Roots
Use roots to find the roots of f(x) = (x - 1)*(x - 2)*(x - 3). Then plot the function from x = -2 to x = 4 and verify where the function crosses the x-axis.
Exercise 9: Nonlinear Root with fzero
Define f(x) = cos(x) - x as an anonymous function. Use fzero to estimate one root. Plot the function over a reasonable interval to check your answer.
Exercise 10: Mini Project - Sensor Calibration
A sensor produces voltage readings for known distances. Create a realistic data set with at least six distance and voltage values. Fit a line or polynomial model, plot the data and fitted curve, and use the model to estimate the voltage at a distance not included in the original data.