Chapter 10, Symbolic Processing with MATLAB Video Solutions, Engineering Student Survival Guide | Numerade (2024)

Use MATLAB to prove the following identities:
a. $\sin ^2 x+\cos ^2 x=1$
b. $\sin (x+y)=\sin x \cos y+\cos x \sin y$
c. $\sin 2 x=2 \sin x \cos x$
d. $\cosh ^2 x-\sinh ^2 x=1$

Use MATLAB to express $\cos 5 \theta$ as a polynomial in $x$, where $x=\cos \theta$.

Two polynomials in the variable $x$ are represented by the coefficient vectors $p 1=\lfloor 6,2,7,-3\rfloor$ and $p 2=\lfloor 10,-5,8]$.
a. Use MATLAB to find the product of these two polynomials; express the product in its simplest form.
b. Use MATLAB to find the numeric value of the product if $x=2$.

The equation of a circle of radius $r$ centered at $x=0, y=0$ is $$ x^2+y^2=r^2 $$ Use the subs and other MATLAB functions to find the equation of a circle of radius $r$ centered at the point $x=a, y=b$. Rearrange the equation into the form $A x^2+B x+C x y+D y+E y^2=F$ and find the expressions for the coefficients in terms of $a, b$, and $r$.

The equation for a curve called the "lemniscate" in polar coordinates $(r, \theta)$ is $$ r^2=a^2 \cos (2 \theta) $$ Use MATLAB to find the equation for the curve in terms of Cartesian $\operatorname{coordinates}(x, y)$, where $x=r \cos \theta$ and $y=r \sin \theta$.

The law of cosines for a triangle states that $a^2=b^2+c^2-2 b c \cos A$, where $a$ is the length of the side opposite the angle $A$, and $b$ and $c$ are the lengths of the other sides.
a. Use MATLAB to solve for $b$.
b. Suppose that $A=60^{\circ}, a=5 \mathrm{~m}$, and $c=2 \mathrm{~m}$. Determine $b$.

Use MATLAB to solve the polynomial equation $x^3+8 x^2+a x+10=0$ for $x$ in terms of the parameter $a$, and evaluate your solution for the case $a=17$. Use MATLAB to cheek the answer.

The equation for an ellipse centered at the origin of the Cartesian coordinates $(x, y)$ is $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$ where $a$ and $b$ are constants that determine the shape of the ellipse.

The equation $$ r=\frac{p}{1-\epsilon \cos \theta} $$ describes the polar coordinates of an orbit with the coordinate origin at the sun. If $\epsilon=0$, the orbit is circular; if $0<\epsilon<1$, the orbit is elliptical. The planets have orbits that are nearly circular; comets have orbits that are highly elongated with $\epsilon$ nearer to 1. It is of obvious interest to determine whether or not a comet's or an asteroid's orbit will intersect that of a planet. For each of the following two cases, use MATLAB to determine whether or not orbits A and B intersect. If they do, determine the polar coordinates of the intersection point. The units of distance are AU, where 1 AU is the mean distance of the Earth from the sun.
a. Orbit $A: p=1, \epsilon=0.01$. Orbit $B: p=0.1, \epsilon=0.9$.
b. Orbit $\mathrm{A}: p=1, \epsilon=0.01$. Orbit $\mathrm{B}: p=1.1, \epsilon=0.5$.

Figure $10.2-2$ on page 601 shows a robot arm having two joints and two links. The angles of rotation of the motors at the joints are $\theta_1$ and $\theta_2$. From trigonometry we can derive the following expressions for the $(x, y)$ coordinates of the hand.
$$ \begin{aligned} & x=L_1 \cos \theta_1+L_2 \cos \left(\theta_1+\theta_2\right) \\ & y=L_1 \sin \theta_1+L_2 \sin \left(\theta_1+\theta_2\right) \end{aligned} $$
Suppose that the link lengths are $L_1=3 \mathrm{ft}$ and $L_2=2 \mathrm{ft}$.
a. Compute the motor angles required to position the hand at $x=3 \mathrm{ft}$, $y=1 \mathrm{ft}$. Identify the elbow-up and elbow-down solutions.
b. Suppose you want to move the hand along a straight, horizontal line at $y=1$ for $2 \leq x \leq 4$. Plot the required motor angles versus $x$. Label the elbow-up and elbow-down solutions.

Use MATLAB to find all the values of $x$ where the graph of $y=3^x-2 x$ has a horizontal tangent line.

Use MATLAB to determine all the local minima and local maxima and all the inflection points where $d y / d x=0$ of the following function: $$ y=x^4-\frac{16}{3} x^3+8 x^2-4 $$

The surface area of a sphere of radius $r$ is $S=4 \pi r^2$. It volume is $V=4 \pi r^3 / 3$.
a. Use MATLAB to find the expression for $d S / d V$.
b. A spherical balloon expands as air is pumped into it. What is the rate of increase in the balloon's surface area with volume when its volume is $30 \mathrm{in}^3{ }^3$ ?

Use MATLAB to find the point on the line $y=2-x / 3$ that is closest to the point $x=-3, y=1$.

A particular circle is centered at the origin and has a radius of 5 . Use MATLAB to find the equation of the line that is tangent to the circle at the point $x=3, y=4$.

Ship $A$ is traveling north at $6 \mathrm{mi} / \mathrm{hr}$, and ship $B$ is traveling west at $12 \mathrm{mi} / \mathrm{hr}$. When ship A was dead ahead of ship B, it was 6 mi away. Use MATLAB to determine how close the ships come to each other.

Suppose you have a wire of length $L$. You cut a length $x$ to make a square, and use the remaining length $L-x$ to make a circle. Use MATLAB to find the length $x$ that maximizes the sum of the areas enclosed by the square and the circle.

A certain spherical street lamp emits light in all directions. It is mounted on a pole of height $h$ (see Figure P18). The brightness $B$ at point $P$ on the sidewalk is directly proportional to $\sin \theta$ and inversely proportional to the square of the distance $d$ from the light to the point. Thus $$ B=\frac{c}{d^2} \sin \theta $$ where $c$ is a constant. Use MATLAB to determine how high $h$ should be to maximize the brightness at point $P$, which is 30 ft from the base of the pole.

A certain object has a mass $m=100 \mathrm{~kg}$ and is acted on by a force $f(t)=$ $500\left[2-e^{-t} \sin (5 \pi t)\right] \mathrm{N}$. The mass is at rest at $t=0$. Use MATLAB to compute the object's velocity $v$ at $l=5 \mathrm{~s}$. The equation of motion is $m \dot{v}=f(t)$.

A rocket's mass decreases as it burns fuel. The equation of motion for a rocket in vertical flight can be obtained from Newton's law and is $$ m(t) \frac{d v}{d t}=T-m(t) g $$ where $T$ is the rocket's thrust and its mass as a function of time is given by $m(t)=m_0(1-r t / b)$. The rocket's initial mass is $m_0$, the burn time is $b$. and $r$ is the fraction of the total mass accounted for by the fuel. Use the values $T=48,000 \mathrm{~N} ; m_0=2200 \mathrm{~kg} ; r=0.8 ; g=9.81 \mathrm{~m} / \mathrm{s}^2$; and $b=40 \mathrm{~s}$.
a. Use MATLAB to compute the rocket's velocity as a function of time for $t \leq b$.
b. Use MATLAB to compute the rocket's velocity at burnout.

The equation for the voltage $v(t)$ across a capacitor as a function of time is $$ v(t)=\frac{1}{C}\left(\int_0^t i(t) d t+Q_0^{\prime}\right) $$ where $i(t)$ is the applied current and $Q_0$ is the initial charge. Suppose that $C=10^{-6} \mathrm{~F}$ and that $Q_0=0$. If the applied current is $i(t)=$ $\left[0.01+0.3 e^{-5 t} \sin (25 \pi t)\right] 10^{-3} \mathrm{~A}$, use MATLAB to compute and plot the voltage $v(t)$ for $0 \leq t \leq 0.3 \mathrm{~s}$.

The power $P$ dissipated as heat in a resistor $R$ as a function of the current $i(t)$ passing through it is $P=i^2 R$. The energy $E(t)$ lost as a function of time is the time integral of the power. Thus $$ E(t)=\int_0^t P(t) d t=R \int_0^t i^2(t) d t $$ If the current is measured in amperes, the power is in watts and the energy is in joules ( $1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}$ ). Suppose that a current $i(t)=0.2[1+\sin (0.2 t)] \mathrm{A}$ is applied to the resistor.
a. Determine the energy $E(t)$ dissipated as a function of time.
b. Determine the energy dissipated in 1 min if $R=1000 \Omega$.

The RLC circuit shown in Figure P23 can be used as a narrow-band filter. If the input voltage $v_i(t)$ consists of a sum of sinusoidally varying voltages with different frequencies, the narrow-band filter will allow to pass only those voltages whose frequencies lie within a narrow range. These filters are used in tuning circuits, such as those used in AM radios, to allow reception only of the carrier signal of the desired radio station. The magnification ratio $M$ of a circuit is the ratio of the amplitude of the output voltage $v_o(t)$ to the amplitude of the input voltage $v_i(t)$. It is a function of the radian frequency $\omega$ of the input voltage. Formulas for $M$ are derived in elementary electrical circuits courses. For this particular circuit, $M$ is given by $$ M=\frac{R C \omega}{\sqrt{\left(1-L C \omega^2\right)^2+(R C \omega)^2}} $$ The frequency at which $M$ is a maximum is the frequency of the desired carrier signal.
a. Determine this frequency as a function of $R, C$, and $L$.
b. Plot $M$ versus $\omega$ for two cases where $C=10^{-5} \mathrm{~F}$ and $L=5 \times 10^{-3} \mathrm{H}$. For the first case, $R=1000 \Omega$. For the second case, $R=10 \Omega$. Comment on the filtering capability of each case.

The shape of a cable hanging with no load other than its own weight is a catenary curve. A particular bridge cable is described by the catenary $y(x)=10 \cosh ((x-20) / 10)$ for $0 \leq x \leq 50$, where $x$ and $y$ are the horizontal and vertical coordinates measured in feet. (See Figure P24.) It is desired to hang plastic sheeting from the cable to protect passersby while the bridge is being repainted. Use MATLAB to determine how many square feet of sheeting are required. Assume that the bottom edge of the sheeting is located along the $x$-axis at $y=0$.

The shape of a cable hanging with no load other than its own weight is a catenary curve. A particular bridge cable is described by the catenary $y(x)=10 \cosh ((x-20) / 10)$ for $0 \leq x \leq 50$, where $x$ and $y$ are the horizontal and vertical coordinates measured in feet.
The length $L$ of a curve described by $y(x)$ for $a \leq x \leq b$ can be found from the following integral:
$$ L=\int_d^b \sqrt{1+\left(\frac{d y}{d x}\right)^2} d x $$ Determine the length of the cable.

Use the first five nonzero terms in the Taylor series for $e^{i x}, \sin x$, and $\cos x$ about $x=0$ to demonstrate the validity of Euler's formula $e^{i x}=$ $\cos x+i \sin x$.

Find the Taylor series for $e^x \sin x$ about $x=0$ in two ways: $a$. by multiplying the Taylor series for $e^x$ and that for $\sin x$, and $b$. by using the taylor function directly on $e^x \sin x$.

Integrals that cannot be evaluated in closed form sometimes can be evaluated approximately by using a series representation for the integrand. For example, the following integral is used for some probability calculations (see Chapter 7, Section 7.2):
$$ I=\int_0^1 e^{-x^2} d x $$
a. Obtain the Taylor series for $e^{-x^2}$ about $x=0$ and integrate the first six nonzero terms in the series to find $I$. Use the seventh term to estimate the error.
b. Compare your answer with that obtained with the MATLAB erf ( $t$ ) function, defined as
$$ \operatorname{erf}(t)=\frac{2}{\sqrt{\pi}} \int_0^t e^{-t^2} d t $$

Use MATLAB to compute the following limits:
a. $\lim _{x \rightarrow 1} \frac{x^2-1}{x^2-x}$
b. $\lim _{x \rightarrow-2} \frac{x^2-4}{x^2+4}$
c. $\lim _{x \rightarrow 0} \frac{x^4+2 x^2}{x^3+x}$

Use MATLAB to compute the following limits:
a. $\lim _{x \rightarrow 0+} x^x$
b. $\lim _{x \rightarrow 0+}(\cos x)^{1 / \tan x}$
c. $\lim _{x \rightarrow 0+}\left(\frac{1}{1-x}\right)^{-1 / x^2}$
d. $\lim _{x \rightarrow 0-} \frac{\sin x^2}{x^3}$
e. $\lim _{x \rightarrow 5-} \frac{x^2-25}{x^2-10 x+25}$
f. $\lim _{x \rightarrow 1+} \frac{x^2-1}{\sin (x-1)^2}$

Use MATLAB to compute the following limits:
a. $\lim _{x \rightarrow \infty} \frac{x+1}{x}$
b. $\lim _{x \rightarrow-\infty} \frac{3 x^3-2 x}{2 x^3+3}$

Find the expression for the sum of the geometric series $$ \sum_{k=0}^{n-1} r^k $$ for $r \neq 1$.

A particular rubber ball rebounds to one-half its original height when dropped on a floor.
a. If the ball is initially dropped from a height $h$ and is allowed to continue to bounce, find the expression for the total distance traveled by the ball after the ball hits the floor for the $n$th time.
b. If it is initially dropped from a height of 10 ft , how far will the ball have traveled after it hits the floor for the eighth time?

The equation for the voltage $y$ across the capacitor of an RC circuit is $$ R C \frac{d y}{d t}+y=v(t) $$
where $v(t)$ is the applied voltage. Suppose that $R C=0.2 \mathrm{~s}$ and that the capacitor voltage is initially 2 V . If the applied voltage goes from 0 to 10 V at $t=0$, use MATLAB to determine and plot the voltage $y(t)$ for $0 \leq t \leq 1 \mathrm{~s}$.

The following equation describes the temperature $T(t)$ of a certain object immersed in a liquid bath of temperature $T_b(t)$ :
$$ 10 \frac{d T}{d t}+T=T_b $$
Suppose the object's temperature is initially $T(0)=70 . \mathrm{F}$ and the bath temperature is 170 F . Use MATLAB to answer the following questions:
a. Determine $T(t)$.
b. How long will it take for the object's temperature $T$ to reach 168 F ?
c. Plot the object's temperature $T(t)$ as a function of time.

This equation describes the motion of a mass connected to a spring with viscous friction on the surface $$ m \dot{y}+c \dot{y}+k y=f(t) $$ where $f(t)$ is an applied force. The position and velocity of the mass at $t=0$ are denoted by $x_0$ and $v_0$. Use MATLAB to answer the following questions:
a. What is the free response in terms of $x_0$ and $v_0$ if $m=3, c=18$, and $k=102$ ?
b. What is the free response in terms of $x_0$ and $v_0$ if $m=3, c=39$, and $k=120$ ?

The equation for the voltage $y$ across the capacitor of an RC circuit is $$ R C \frac{d y}{d t}+y=v(t) $$
where $v(t)$ is the applied voltage. Suppose that $R C=0.2 \mathrm{~s}$ and that the capacitor voltage is initially 2 V . If the applied voltage is $v(t)=$ $10\left[2-e^{-t} \sin (5 \pi t)\right]$, use MATLAB to compute and plot the voltage $y(t)$ for $0 \leq t \leq 5 \mathrm{~s}$.

The following equation describes a certain dilution process, where $y(t)$ is the concentration of salt in a tank of fresh water to which salt brine is being added: $$ \frac{d y}{d t}+\frac{2}{10+2 t} y=4 $$
Suppose that $y(0)=0$. Use MATLAB to compute and plot $y(t)$ for $0 \leq t \leq 10$.

This equation describes the motion of a certain mass connected to a spring with viscous friction on the surface
$$ 3 y+18 \dot{y}+102 y=f(t) $$
where $f(t)$ is an applied force. Suppose that $f(t)=0$ for $t<0$ and $f(t)=10$ for $t \geq 0$.
a. Use MATLAB to compute and plot $y(t)$ when $y(0)=\dot{y}(0)=0$.
b. Use MATLAB to compute and plot $y(t)$ when $y(0)=0$ and $\hat{y}(0)=10$.

This equation describes the motion of a certain mass connected to a spring with viscous friction on the surface
where $f(t)$ is an applied force. Suppose that $f(t)=0$ for $t<0$ and $f(t)=10$ for $t \geq 0$.
a. Use MATLAB to compute and plot $y(t)$ when $y(0)=\dot{y}(0)=0$.
b. Use MATLAB to compute and plot $y(t)$ when $y(0)=0$ and $\dot{y}(0)=10$.

The equations for an armature-controlled de motor follow. The motor's current is $i$ and its rotational velocity is $\omega$.
$$ \begin{aligned} & L \frac{d i}{d t}=-R i-K_e \omega+v(t) \\ & I \frac{d \omega}{d t}=K_T i-c \omega \end{aligned} $$
$L, R$, and $I$ are the motor's inductance, resistance, and inertia; $K_7$ and $K_e$ are the torque constant and back emf constant; $c$ is a viscous damping constant; and $v(t)$ is the applied voltage.
Use the values $R=0.8 \Omega, L=0.003 \mathrm{H}, K_T=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}$, $K_e=0.05 \mathrm{~V} \cdot$ s/rad, $c=0$, and $I=8 \times 10^{-5} \mathrm{~N} \cdot \mathrm{m} \cdot \mathrm{s}^2$.
Suppose the applied voltage is 20 V . Use MATLAB to compute and plot the motor's speed and current versus time for zero initial conditions. Choose a final time large enough to show the motor's speed becoming constant.

The RLC circuit described in Problem 23 and shown in Figure P23 on page 640 has the following differential equation model: $$ L C \dot{v}_o+R C \dot{v}_j+v_o=R C \dot{v}_i(t) $$ Use the Laplace transform method to solve for the unit-step response of $v_0(t)$ for zero initial conditions, where $C=10^{-5} \mathrm{~F}$ and $L=5 \times 10^{-3} \mathrm{H}$. For the first case (a broadband filter), $R=1000 \Omega$. For the second case (a narrow-band filter), $R=10 \Omega$. Compare the step responses of the two cases.

The differential equation model for a certain speed control system for a vehicle is $$ i+\left(1+K_p\right) \dot{v}+K_I v=K_p \dot{v}_d+K_I v_d $$ where the actual speed is $v$, the desired speed is $v_d(t)$, and $K_p$ and $K_t$ are constants called the "control gains." Use the Laplace transform method to find the unit-step response (that is, $v_3(t)$ is a unit-step function). Use zero initial conditions. Compare the response for three cases:
a. $K_p=9, K_I=50$
b. $K_p=9, K_I=25$
c. $K_p=54, K_l=250$

The differential equation model for a certain position control system for a metal cutting tool is
$$ \begin{aligned} & \frac{d^3 x}{d t^3}+\left(6+K_D\right) \frac{d^2 x}{d t^2}+\left(11+K_p\right) \frac{d x}{d t}+\left(6+K_l\right) x \\
& =K_D \frac{d^2 x_d}{d t^2}+K_p \frac{d x_d}{d t}+K_I x_d \end{aligned} $$
where the actual tool position is $x$; the desired position is $x_d(t)$; and $K_p$. $K_I$, and $K_D$ are constants called the control gains. Use the Laplace transform method to find the unit-step response (that is, $x_d(t)$ is a unit-step function). Use zero initial conditions. Compare the response for three cases:
a. $K_p=30, K_I=K_D=0$
b. $K_p=27, K_l=17.18, K_D=0$
c. $K_p=36, K_I=38.1, K_D=8.52$

The differential equation model for the motor torque $m(t)$ required for a certain speed control system is
$$ 4 \ddot{m}+4 K \dot{m}+K^2 m=K^2 \dot{v}_d $$
where the desired speed is $v_d(t)$, and $K$ is a constant called the control gain.
a. Use the Laplace transform method to find the unit-step response (that is, $v_d(t)$ is a unit-step function). Use zero initial conditions.
b. Use symbolic manipulation in MATLAB to find the value of the peak torque in terms of the gain $K$.

Show that $\mathbf{R}^{-1}(a) \mathbf{R}(a)=\mathbf{I}$, where $\mathbf{I}$ is the identity matrix and $\mathbf{R}(a)$ is the rotation matrix given by $(10.6-1)$. This equation shows that the inverse coordinate transformation returns you to the original coordinate system.

Show that $\mathbf{R}^{-1}(a)=\mathbf{R}(-a)$. This equation shows that a rotation through a negative angle is equivalent to an inverse transformation.

Find the characteristic polynomial and roots of the following matrix: $$ A=\left[\begin{array}{rr} -6 & 2 \\ 3 k & -7 \end{array}\right] $$

Use the matrix inverse and the left-division method to solve the following set for $x$ and $y$ in terms of $c$ :
$$ \begin{aligned} & 4 c x+5 y=43 \\ & 3 x-4 y=-22 \end{aligned} $$

The currents $i_1, i_2$, and $i_3$ in the circuit shown in Figure P50 are described by the following equation set if all the resistances are equal to $R$. $$ \left[\begin{array}{rrr} 2 R & -R & 0 \\ -R & 3 R & -R \\ 0 & R & -2 R \end{array}\right]\left[\begin{array}{l} i_1 \\ i_2 \\ i_3 \end{array}\right]=\left[\begin{array}{c} v_1 \\ 0 \\ v_2 \end{array}\right] $$
$v_1$ and $v_2$ are applied voltages; the other two currents can be found from $i_4=i_1-i_2$ and $i_5=i_2-i_3$.
a. Use both the matrix inverse method and the left-division method to solve for the currents in terms of the resistance $R$ and the voltages $v_1$ and $v_2$.
b. Find the numerical values for the currents if $R=1000 \Omega, v_1=100 \mathrm{~V}$, and $v_2=25 \mathrm{~V}$.

The equations for the armature-controlled do motor shown in Figure P51 follow. The motor's current is $i$, and its rotational velocity is $\omega$. $$ \begin{aligned} & L \frac{d i}{d t}=-R i-K_{,} \omega+v(t) \\ & I \frac{d \omega}{d t}=K_T i-c \omega \end{aligned} $$
$L, R$, and $I$ are the motor's inductance, resistance, and inertia; $K_T$ and $K_c$ are the torque constant and back emf constant; $c$ is a viscous damping constant; and $v(t)$ is the applied voltage.
a. Find the characteristic polynomial and the characteristic roots.
b. Use the values $R=0.8 \Omega, L=0.003 \mathrm{H}, K_T=0.05 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A}$, $K_e=0.05 \mathrm{~V}$-s/rad, and $I=8 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^2$. The damping constant $c$ is often difficult to determine with accuracy. For these values find the expressions for the two characteristic roots in terms of $c$.
$c$. Using the parameter values in part $b$, determine the roots for the following values of $c$ (in newton meter second): $c=0, c=0.01$, $c=0.1$, and $c=0.2$. For each case, use the roots to estimate how long the motor's speed will take to become constant; also discuss whether or not the speed will oscillate before it becomes constant.