The equation of a circle is xz + yz = Rz,where R is the radius of the circle. Write

Define x as a symbolic variable and create the two symbolic expressions S1= x2(x- 6) + 4(3x- 2) and S2= (x + 2)2- 8x Use symbolic operations to determine the simplest form of each of following expressions: (a) s, S2 (b) s, (c) s s2 , + S2 (d) Use the subs command to evaluate the numerical value of the result from part (c) for x = 5.

Define y as a symbolic variable and create the two symbolic expressions S1 = x(x2 + 6x + 12) + 8 and S2 = (x- 3 )2 + lOx- 5 Use symbolic operations to determine the simplest form of each of following expressions: (a) s, S2 (b) s, (c) s, + S2 s2 (d) Use the subs command to evaluate the numerical value of the result from part (c) for x = 3.

Define x andy as symbolic variables and create the two symbolic expressions S = x + .{xy2 + y4 and T = JX- y2 Use symbolic operations to determine the simplest form of S T . Use the subs command to evaluate the numerical value of the result for x = 9 and y = 2 .

Defme x as a symbolic variable. (a) Derive the equation of the polynomial that has the roots x = -2, X = -0.5, X = 2, and X = 4.5 . (b) Determine the roots of the polynomial fix) = x6- 6.5x5- 58x4 + 167.5x3 + 728x2- 890x- 1400 by using the factor command.

Use the commands from Section 11.2 to show that: (a) sin(4x) = 4sinxcosx- 8sin3xcosx 1 (b) cosxcosy = 2[cos(x- y) + cos(x + y)]

Use the commands from Section 11.2 to show that: (a) tan(3x) = 3tanx- tan3x 1- 3tan2x (b) sin(x+y+z) = sinxcosycosz+ cosxsinycosz + cosxcosysinz- sinxsinysinz

The folium of Descartes is the graph shown in the figure. In parametric form its equation is given by: 3t 3t2 x = -- and y = -- for t :;t: -1 1+P 1+P (a) Use MATLAB to show that the equation of the folium of Descartes can also be written as: x3 + y3 = 3xy J<"+y"-3 X y = 0 X (b) Make a plot of the folium for the domain shown in the figure by using the ezplot command.

A water tower has the geometry shown in the figure (the lower part is a cylinder with radius Rand height h, and the upper part is a half sphere with radius R). Determine the radius R if h = 10m and the volume is 1,050 m3 (Write an equation for the volume in terms of the radius and the height. Solve the equation for the radius, and use the double command to obtain a numerical value.

The relation between the tension T and the steady shortening velocity v in a muscle is given by the Hill equation: (T+a)(v+b) = (T0+a)b where a and b are positive constants and T0 is the isometric tension, i.e., the tension in the muscle when v = 0. The maximum shortening velocity occurs when T = 0. (a) Using symbolic operations, create the Hill equation as a symbolic expression. Then use subs to substitute T = 0, and finally solve for v to show that vmax = (bT0)1 a. (b) Use vmax from part (a) to eliminate the constant b from the Hill equation, a(T0- T) and show that v = To(T+ a) vmax

Consider the two ellipses in the x y plane given by the equations (x-1)2 v2 _ (x+2)2 f"-5)2 _ ---+L-- 1 and + ":.!..........: - 1 62 32 22 42 (a) Use the ezplot command to plot the two ellipses in the same figure. (b) Determine the coordinates of the points where the ellipses intersect.

A 120 in.-long beam AB is attached to the wall with a pin at point A and to a 66 in.long cable CD. A load W = 200 lb is attached to the beam at point B. The tension in the cable T and the x andy components of the force at A (FAx and FAy) can be calculated from the equations: F -T-d = Ax 0 Lc F + T.JfT2d2.Lc - u--W = Ay 0 L c Tji0F d- WL = 0 Lc where Land Lc are the lengths of the beam and the cable, respectively, and d is the distance from point A to point D where the cable is attached. (a) Use MATLAB to solve the equations for the forces T, FAx, and FAy in terms of d, L, Lc, and W. Determine FA given by FA = jFx + FY. (b) Use the subs command to substitute W = 200 lb, L = 120 in., and Lc = 66 in. into the expressions derived in part (a). This will give the forces as a function of the distance d. (c) Use the ezplot command to plot the forces T and FA (both in the same figure as functions of d, for d starting at 20 and ending at 70 in. (d) Determine the distance d where the tension in the cable is the smallest. Determine the value of this force.

A box of mass m is being pulled by a rope as shown. The force Fin the rope as a function of x can be calculated from the equations: -F Jx2X+ h2 +JlN=O -mg+N+F h = Jx2+ h2 0 where N and Jl are the normal force and friction coefficient between the box and surface, respectively. Consider the case where m = 18 kg, h = 10 m, 11 = 0.55, and g = 9.81 mJsz. (a) Use MATLAB to derive an expression for F, in terms of x, h, m, g, and ll (b) Use the subs command to substitute m = 18kg, h = 1 0 m, ll = 0.55, and g = 9.81 m/s2 into the expressions that were derived in part (a). This will give the force as a function of the distance x. (c) Use the ezplot command to plot the force F as a function of x, for x starting at 5 and ending at 30 m. (d) Determine the distance x where the force that is required to pull the box is the smallest, and determine the magnitude of that force.

The mechanical power output Pin a contracting muscle is given by: kvT0( 1- v v ) p = Tv= max k+...2:.... vmax where Tis the muscle tension, v is the shortening velocity (max of v max), T0 is the isometric tension (i.e., tension at zero velocity), and k is a non-dimensional constant that ranges between 0.15 and 0.25 for most muscles. The equation can be written in non-dimensional form: _ ku(l-u) p - k+u where p = (Tv)I(T0vmax), and u = vlvmax Consider the case k = 0.25. (a) Plotp versus u for 0:5: u:5: 1. (b) Use differentiation to fmd the value of u where p is maximum. (c) Find the maximum value ofp.

The equation of a circle is xz + yz = Rz, where R is the radius of the circle. Write a program in a script file that ftrst derives the equation (symbolically) of the tangent line to the circle at the point (x0, y0) on the upper part of the circle (i.e., for -R < x0 < R and 0 < y0 ). Then for specific values of R, x0, and y0 the program makes a plot, like the one shown on the right, of the circle and the tangent line. Execute the program with R = 10 and x0 = 7.

A tracking radar antenna is locked on an airplane flying at a constant altitude of 5 km, and a constant speed of 540 km/h. The airplane travels along a path that passes exactly above the radar station. The radar starts the tracking when the airplane is 100 km away. Chapter 11: Symbolic Math (a) Derive an expression for the angle e of the radar antenna as a function of time. (b) Derive an expression for the angular velocity of the antenna, , as a function of time. (c) Make two plots on the same page, one of e versus time and the other of versus time, where the angle is in degrees and the time is in minutes for 0 t 20 min.

Evaluate the following indefmite integrals: (a) I= J dx (b) I= Jx2cosx dx

Defme x as a symbolic variable and create the symbolic expression S = cos2x 1 + sin2x J lt 2 Plot S in the domain 0 x 1t and calculate the integral I = cos . dx . ol+smx

The parametric equations of an ellipsoid are: x = a cosusinv, y = b sinusinv, z = c cosv where 0 u 2n and -7t v 0 Show that the differential volume element of the ellipsoid shown is given by: dV = -nabcsin3vdv Use MATLAB to evaluate the integral of dV from -7t to 0 symbolically and show that the volume of the ellipsoid 4 is V = 3nabc .

The one-dimensional diffusion equation is given by: du d2u -=mdt dx2 Show that the following are solutions to the diffusion equation. (a) u = A} exp(;2t ) + B , where A and B are constants. (b) u=Aexp(-ax)cos(a.x- 2ma2t+B)+C, where A, B, C, and a are constants.

A ceramic tile has the design shown in the figure. The shaded area is painted red and the rest of the tile is white. The border line between the red and the white areas follows the equation y = -kxZ + 12kx Determine k such that the areas of the white and the red colors will be the same.

Show that the location of the centroid y c of the half-circle area shown is given by Yc = .The coordinate Yc can be calculated by:

For the half-circle area shown in the previous problem, show that the moment of inertia about the x axis, Ix, is given by Ix = 1tR4 The moment of inertia Ix can be calculated by:

The rms value of an AC voltage is defined by T !J v2(f)dt' To where Tis the period of the waveform. (a) A voltage is given by v(t) = Vcos(rot). Show that vrms = ; and is inde-pendent of ro. (The relationship between the period T and the radian fre- . 27t quency ro ts T = - . ) (I) (b) A voltage is given by v(t) = 2.5cos(350t) + 3 V. Determine vrms.

The spread of an infection from a single individual to a population of Nuninfected persons can be described by the equation = -Rx(N + l - x) with initial condition x(O) = N where x is the number of uninfected individuals and R is a positive rate constant. Solve this differential equation symbolically for x(t). Also, determine symbolically the timet at which the infection rate dx!dt is maximum.

The Maxwell-Boltzmann probability density function f(v) is given by f(v) = v2exp(-2fJ where m (kg) is the mass of each molecule, v (m/s) is the speed, T (K) is the temperature, and k = 1.38 x 10-23 J/K is Boltzmann's constant. The most probable speed v P corresponds to the maximum value of f( v) and can be determined from d) = 0. Create a symbolic expression for f(v), differentiate it with respect to v and show that vP = . Calculate vP for oxygen molecules (m = 5.3 x 10-26 kg) at T = 300 K (k = 1.38 x 10-23 J/K). Make a plot of f( v) versus v for 0 v 2500 m/s for oxygen molecules

The velocity of a skydiver whose parachute is still closed can be modeled by assuming that the air resistance is proportional to the velocity. From Newton's second law of motion the relationship between the mass m of the skydiver and his velocity v is given by (down is positive) dv mg-cv = m dt where cis a drag constant and g is the gravitational constant (g = 9.81 m/s2). (a) Solve the equation for v in terms of m, g, c, and t, assuming that the initial velocity of the skydiver is zero. (b) It is observed that 4 s after a 90 kg skydiver jumps out of an airplane, his velocity is 28 m/s. Determine the constant c. (c) Make a plot of the skydiver velocity as a function of time for 0 t 30 s.

A resistor R (R = 0.4Q) and an inductor L (L = 0.08 H) are connected as shown. Initially, the switch is connected to point A and there is no current in the circuit. At t = 0 the L switch is moved from A to B, so that the resistor and the inductor are con- l nected to Vs ( Vs = 6 V), and current starts flowing in the circuit. The switch remains connected to B until the voltage on the resistor reaches 5 V. At that time (tBA) the switch is moved back to A. The current i in the circuit can be calculated from solving the differential equations: .R L di z + - dt = vs during the time from t = 0 and until the time when the switch is moved back to A. iR + L dt!:!:.t = 0 from the time when the switch is moved back to A and on. The voltage across the resistor, vR , at any time is given by vR = iR. (a) Derive an expression for the current i in terms of R, L, Vs, and t for 0 t tBA by solving the first differential equation. (b) Substitute the values of R, L, and Vs in the solution fori, and determine the time t BA when the voltage across the resistor reaches 5 V. (c) Derive an expression for the current i in terms of R, L, and t, for tBA t by solving the second differential equation. (d) Make two plots (on the same page), one for vR versus t for 0 t tBA and the other for vR versus t for tBA t 2tBA.

Determine the general solution of the differential equation tlJ!. = x4-2y dx 2x Show that the solution is correct. (Derive the first derivative of the solution, and then substitute back into the equation.)

Determine the solution of the following differential equation that satisfies the given initial conditions. Plot the solution for 0 t 7 . rf2!. 008tl1!.+06 - 0 (0) 2 dyl = 3

The current, i, in a series RLC circuit when the switch is closed at t = 0 can be determined from the solution of the 2nd-order ODE Ld2i dt2 +RdcJ.!:.t +!i = 0 c Chapter 11: Symbolic Math where R, L, and C are the resistance of the resistor, the inductance of the inductor, and the capacitance of the capacitor, respectively. (a) Solve the equation fori in terms of L, R, C, and t, assuming that at t = 0 i = 0 and di I dt = 8. (b) Use the subs command to substitute L = 3 H, R = 10 Q, and C = 80 jlF into the expression that were derived in part (a). Make a plot of i versus t for 0 t 1 s. (Underdamped response.) (c) Use the subs command to substitute L = 3 H, R = 200 Q, and C = 1200 jlF into the expression that were derived in part (a). Make a plot of i versus t for 0 t 2 s. (Overdamped response.) (d) Use the subs command to substitute L = 3 H, R = 201 Q, and C = 300 jlF into the expression that were derived in part (a). Make a plot of i versus t for 0 t 2 s. (Critically damped response.)

Damped free vibrations can be modeled by a block of mass m that is attached to a spring and a dashpot as shown. From Newton's second law of motion, the displacement x of the mass as a =.!LJ--t function of time can be determined by solving the differential equation m-d2x +cd-x +kx dt2 dt = 0 where k is the spring constant and c is the damping coefficient of the dashpot. If the mass is displaced from its equilibrium position and then released, it will start oscillating back and forth. The nature of the oscillations depends on the size of the mass and the values of k and c. For the system shown in the figure, m = 10 kg and k = 28 N/m. At time t = 0 the mass is displaced to x = 0.18 m and then released from rest. Derive expressions for the displacement x and the velocity v of the mass, as a function of time. Consider the following two cases: (a) c = 3 (N s)/m. (b) c = 50(N s)/m. For each case, plot the position x and the velocity v versus time (two plots on one page). For case (a) take 0 t 20 s, and for case (b) take 0 t 10 s.