14.12: Chapter Exercises

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.