14.12: Chapter Exercises
- Page ID
- 135956
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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: Symbolic Basics
1. Create symbolic variables a, b, and c.
2. Create the expression a^2 + 2*b - c.
3. Substitute a = 3, b = 4, and c = 5 into the expression.
4. Convert the result to a numeric value.
Exercise 2: Integration and Differentiation
Use symbolic math to complete the following tasks for f(x) = 4x^2 + 3:
5. Find the indefinite integral.
6. Find the definite integral from x = -1 to x = 3.
7. Find the derivative.
8. Evaluate the derivative at x = 3.
Exercise 3: Polynomial Derivative
Use diff to find the derivative of:
g(x) = 2x^5 + 6x^3 - x + 9
Then use subs to evaluate the derivative at x = 3.
Exercise 4: Solving an Equation
Use solve to solve the equation:
2x^2 + x = 6
Convert the symbolic answers to numeric answers using double.
Exercise 5: Solving a System
Solve the system below using solve. Then display the answers as numeric values.
· 4*x1 - 2*x2 + x3 = 7
· x1 + x2 + 5*x3 = 10
· -2*x1 + 3*x2 - x3 = 2
Exercise 6: Expression Manipulation
9. Create the symbolic expression (x + 4)*(x - 2).
10. Use expand to multiply it out.
11. Use factor to return it to factored form.
12. Create the expression sin(x)^2 + cos(x)^2 and simplify it.
Exercise 7: Plotting a Symbolic Expression
Use fplot to plot h(x) = x^3 + 3x^2 - 2 from x = -5 to x = 5. Add a title, x-axis label, y-axis label, and grid.
Exercise 8: Differential Equation
Use dsolve to solve the differential equation:
dx/dt = 3*sin(2*t), where x(0) = -1
Exercise 9: Biomedical Engineering Application
The concentration of insulin is modeled by:
C = C0 * exp(-30*t/m)
Write a script named insulin.m that determines how long it takes for a person with mass 65 kg to have insulin concentration decrease from 90 to 10. Convert the time from minutes to seconds, and use fprintf to display the result in a complete sentence.
Exercise 10: Challenge Problem
A spring-mass system has natural frequency described by:
omega = sqrt(k/m)
Use symbolic math to solve this equation for k. Then substitute omega = 12 rad/s and m = 3 kg to find k.