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Define the vector v = [8 6 4 2]. Then use the vector in a mathematical expression to create the following vectors: (a) a = [1 1 1 1] (b) b = [1/82 1/62 1/42 1/22] (c) c = [1/√8 1/√6 1/√4
Define x and y as the vectors x = [1, 2, 3, 4, 5] and y = [2, 4, 6, 8, 10]. Then use them in the following expressions to calculate z using element-by-element calculations. (a) z = (x + y)/x-y (b) w
Define r and s as scalars r = 1.6 x 103 and s = 14.2, and, t, x, and y as vectors t = [1, 2, 3, 4, 5], x = [0, 2, 4, 6, 8], and y = [3, 6, 9, 12, 15]. Then use these variables to calculate the
The area of a triangle ABC can be calculated by |rAB Ã— rAC/2, where rAB and rAC are vectors connecting the vertices A and B and A and C, respectively. Determine the area of the triangle
The volume of the parallelepiped shown can be calculated by r0B · (rOA x rAC). Use the following steps in a script file to calculate the area. Define the vectors rOA , rAC, and rOB from
Define the vectors: u = 5l - 2j + 4k, v = -21 + 7J + 3k, and w = 81 + lj - 3k Use the vectors to verify the identity: (u + v)·[(v + w)x(w + u)] = 2u·(v × w) Use MATLAB's built-in functions cross
The dot product can be used for determining the angle between two vectors:Use MATLAB's built-in functions acosd, sqrt, and dot to find the angle (in degrees) between r1 = 6i - 3j + 2k and r2 = 21 +
For the function y = (x + 25)3/x2, calculate the value of y for the following values of x using element-by-element operations: 1, 2, 3, 4, 5, 6.
Use MAILAB to show that the angle inscribed in a semi-circle is a right angle. Use the following steps in a script file to calculate the angle. Define a variable with the value of the x coordinate of
The position as a function of time (x(t),y(t)) of a projectile fired with a speed of v0 at an angle a is given byx(t) = v0cosa·t y(t) = v0sinÎ±·t €“ 1/2 gt2where
Use MATLAB to show that the sum of the infinite series converges to e2. Do this by computing the sum for: (a) n = 5, (b) n = 10, (c) n = 50 For each part create a vector n in which the first
Use MATLAB to show that the sum of the infinite series verges to 1n10. Do this by computing the sum for (a) n = 10, (b) n = 50, (c) n = 100 For each part, create a vector n in which
According to Zeno's paradox any object in motion must arrive at the halfway point before it can arrive at its destination. Once arriving at the halfway point, the remaining distance is once again
Show that cos(2x) -1/cosx - 1 = 4 . Do this by first creating a vector x that has the elements 1.0, 0.5, 0.1, 0.01, 0.001, and 0.0001. Then, create a new vector y in which each element is
Show that = 4/3 Do this by first creating a vector x that has the elements 2.0, 1.5, 1.1, 1.01, 1.001, 1.00001, and 1.0000001. Then, create a new vector yin which each element is
The demand for water during a fire is often the most important factor in the design of distribution storage tanks and pumps. For communities with populations less than 200,000, the demand Q (in
The ideal gas equation states that P = nRT/V, where P is the pressure, V is the volume, T is the temperature, R = 0.08206 (L atm)/(mol K) is the gas constant, and n is the number of moles. Real
Create the following three matrices:(a) Calculate A + B and B + A to show that addition of matrices is commutative.(b) Calculate A + (B + C) and (A + B) + C to show that addition of matrices is
For the function y = (x + 74/(x + 1) √x, calculate the value of y for the following values of x using element-by-element operations: 1.5, 2.5, 3.5, 4.5, 5.5, 6.6.
Use the matrices A, B, and C from the previous problem to answer the following: (a) Does A*B = B*A? (b) Does A*(B*C) = (A*B)*C? (c) Does (A*B)t = At*Bt? (t means transpose) (d) Does (A+B)t = At + Bt?
Create a 4 × 4 matrix A having random integer values between 1 and 10. Call the matrix A and, using MATLAB, perform the following operations. For each part explain the operation.(a) A* A(b) A.*A(c)
The magic square is an arrangement of numbers in a square grid in such a way that the sum of the numbers in each row, and in each column, and in each diagonal is the same. MATLAB has a built-in
Solve the following system of three linear equations: - 4x + 3y + z = -18.2 5x + 6y - 2z = -48.8 2x - 5y + 4.5z = 92.5
Solve the following system of five linear equations: 2.5a – b + 3c + 1.5d - 2e = 57. 1 3a + 4b - 2c + 2.5d - e = 27.6 -4a + 3b + c - 6d + 2e = -81.2 2a + 3b + c - 2.5d + 4e = -22.2 a + 2b + 5c - 3d
A food company manufactures five types of 8 oz Trail mix packages using different mixtures of peanuts, almonds, walnuts, raisins, and M&Ms. The mixtures have the following compositions:How many
The electrical circuit shown consists of resistors and voltage sources. Determine i1, i2, i3, and i4, using the mesh current method based on Kirchhoff's voltage law (see Sample Problem 3-4).
The electrical circuit shown consists of resistors and voltage sources. Determine i1, i2, i3, i4 and i5, using the mesh current method based on Kirchhoff's voltage law (see Sample Problem 3-4).V1, =
For the function y = 2 sin x + cos2x/sin2x, calculate the value of y for the following values of x using element-by-element operations: 20°, 30°, 40°, 50°, 60°, 70°.
The radius, r, of a sphere can be calculated from its surface area, s, by: r = √s/x/2 The volume, V, is given by: V = 4πr3/3 Determine the volume of spheres with surface area of 50, 100, 150, 200,
The electric field intensity, E(z), due to a ring of radius R at any point z along the axis of thering is given by:where Î» is the charge density, Îµ0 = 8.85 Ã— I0-12 is the electric constant,
The voltage Vc(t) (in V) and the current i(t) (in Amp) t seconds after closing the switch in the circuit shown are given by:Vc(t) = V0(1- e-t/τo )i(t) = v0/R e-t/τowhere τo = RC is the time
The length |u| (magnitude) of a vector u = xi+ yj + zk is given by |u| = √x2 + y2 + z2. Given the vector u = 23.5i - 17j + 6k, determine its length in the following two ways: (a) Define the vector
A vector wL of length Lin the direction of a vector u = xi + yj + zk can determined by wL = Lun (multiplying a unit vector in the direction of u by L). The unit vector un in the direction of the
The Heat Index HI, calculated from the air temperature and relative humidity, is the apparent temperature felt by the body. An equation used by the National Weather Service for calculating the HI is
The balance of a loan, B, after n monthly payments is given bywhere A is the loan amount, P is the amount of a monthly payment, and r is the yearly interest rate entered in % (e.g., 7.5% entered as
Early explorers often estimated altitude by measuring the temperature of boiling water. Use the following two equations to make a table that modem-day hikers could use for the same purpose.p =
An isosceles triangle sign is designed to have a triangular printed area of 600 in.2 (shaded area with a base length of a and height of h in the figure). As shown in the figure, there is a 2 in. gap
A round billboard with radius R = 55 in. is designed to have a rectangular picture placed inside a rectangle with sides a and b. The margins between the rectangle and the picture are 10 in. at the
A student has a summer job as a lifeguard at the beach. After spotting a swimmer in trouble, he tries to deduce the path by which he can reach the swimmer in the shortest time. The path of shortest
An airplane is flying at a height of h = 900 ft while watching a target that is 70 ft tall (H = 70 ft) as shown in the figure. The best view of the target is when e is maximum. Write a MA TLAB
The stress intensity factor K at a crack in a beam exposed to pure M bending M is given by: K = CÏƒˆšÏ€awhere Ïƒ = 6M/tb2. a is the crack length. b is the width t is the thickness, and C is
The airplane shown is flying at a constant speed of v = 50 m/s in a circular path of radius P = 2000 m and is being tracked by a radar station positioned a distance h = 500 m below the bottom of the
The intrinsic electrical conductivity cr of a semiconductor can be approximated by:where Ïƒ is measured in („¦-m)-1, Eg is the band gap energy, k is Boltzmann's constant (8.62 x I0-5 ev/K), and
The pressure drop Î”p in Pa for a fluid flowing in a pipe with a sudden increase in diameter is given by:Where p is the density of the fluid, v, the velocity of the flow, and d and D are defined
The monthly saving P that has to be deposit in a saving account that pays an annual interest rate of r in order to save a total amount of F in N years can be calculated by the formula: p =
The net heat exchange by radiation from plate 1 with radius b to plate 2 with radius a that are separated by a distance c is given by:q = ÏƒÏ€b2F1€“2(T41 €“ T42)Where T1 and T2 are the
Given the coordinates of three points (x1,y1), (x2,y2) , and (x3,y3) it is possible to find the coordinates of the center of the circle (Cx, Cy) that passes through the three points by solving the
A truss is a structure made of members joined at their ends. For the truss shown in the figure, the forces in the nine members are determined by solving the following system of nine equations:F2 +
A truss is a structure made of members joined at their ends. For the truss shown in the figure, the forces in the 13 members are determined by solving the following system of 13 equations.F2 +
The graph of the function f{x) = ax3 + bx2 + cx + d passes through the points (-2.6 , --68), (0.5, 5.7), (1 .5, 4.9), and (3.5, 88). Determine the constants a, b, c, and d. (Write a system of four
The surface of many airfoils can be described with an equation of the formy = ˆ“ tc/0.2 [a0ˆšx/c + a1 (x/c) ++ a2(x/c)2 + a3(x/ c)3 + a4(x/c)4]where t is the maximum thickness as a fraction of
During a golf match, a certain number of points are awarded for each eagle and a different number for each birdie. No points are awarded for par, and a certain number of points are deducted for each
The dissolution of copper sulfide in aqueous nitric acid is described by the following chemical equation:where the coefficients a, b, c, d, e,J, and g are the numbers of the various molecule
The wind chill temperature, Twc, is the air temperature felt on exposed skin due to wind. In U.S. customary units it is calculated by:Twc = 35.74 + 0.6215T- 35.75v0.16 + 0.4275Tv0.16where T is the
The stress intensity factor K at a crack is given by K = CÏƒˆšÏ€Î± where Ïƒ is the far-field stress, a is the crack length, and C is a parameter that depends on the geometry of the
The growth of some bacteria populations can be described by N = Noekt where N is the number of individuals at time t, N0 is the number at time t = 0 , and k is a constant. Assuming the number of
The volume V and the surface area S of a torus-shaped water tube are given by:V = ¼ Ï€2(r1 + r2)(r2 - r1)2 and S = Ï€2(ri - rf)If r1 = 0.7 r2, determine V and S for r2 = 12, 16, 20, 24, and 28
A beam with a length L is attached to the wall with a cable as shown. A load W = 500 lb is attached to the beam. The tension force, T, in the cable is given by:T = WLˆšh2 + x2/hxFor a beam with L =
Write a MATLAB program in a script file that calculate the average, standard deviation, and median of a list of grades as well as the number of grades on the list. The program asks the user (input
A rocket flying straight up measures the angle e with the horizon at different heights h. Write a MATLAB program in a script file that calculates the radius of the earth R (assuming the earth is a
Decay of radioactive materials can be modeled by the equation A = A0ekt, where A is the amount at time t, A0 is the amount at t = 0, and k is the decay constant ( k < 0 ). Iodine-132 is a
The monthly payment, P, of a N years mortgage of an amount L that with a yearly interest rate of r is given by:where r is in % (e.g., 7.5% entered as 7.5). Write a MATLAB program in a script file
Plot the function f(x) = x2- 3x + 7/√2x + 5 for –l < x < 5.
Two parametric equations are given by: x = cos3(t) , y = sin3(t) u = sin(t), v = cos(t) In one figure, make plots of y versus x and v versus u for 0 < t < 2π. Format the plot such that the both axes
Plot the function f(x) = x2 - 5x – 12/x2 – x – 6 in the domain –1 < x < 7. Notice that the function has a vertical asymptote at x = 3. Plot the function by creating two vectors for the
Plot the function f(x) = x2 + 3x – 5/x2 – 3x – 10 for -4 < x < 9. Notice that the function has two vertical asymptotes. Plot the function by dividing the domain of x into three parts: one
A parametric equation is given by: x = 3t/1 + t3, y = 3t2/1 + t3 (The denominator approaches 0 when t approaches -1.) Plot the function (the plot is called the Folium of Descartes) by plotting two
An epicycloid is a curve (shown partly in the figure) obtained by tracing a point on a circle that rolls around a fixed circle. The parametric equation of a cycloid is given by:x = 13cos(t)-
The shape of the pretzel shown is given by the following parametric equations:x = (3.3 - 0. 4t2) sin(t) y = (2.5- 0.3t2) cos(t) where -4
Make a polar plot of the function r = 2sin(3θ)sinθ for 0 < θ < 2π.
Plot an ellipse with major axes of a = 10 and b = 4 and a center at x = 2 and y = 3.
The following data gives the approximate population of the world for selected years from 1850 unti12000.The population P, since 1900 can be modeled by the logistic function:P = 11.55/1 +
The force F (in N) acting between a particle with a charge q and a round disk with a radius R and a charge Q is given by the equation:F= Qqz/2Îµ0(1- z2 +R2)where Îµo = 0.885
Plot the function f(x) = (3 cosx - sinx)e-02x for -4 < x < 9 .
The position as a function of time of a squirrel running on a grass field is given in polar coordinates by:r(t) = 25 + 30[1- esin(0.07t)] mÎ¸(t) = 2Ï€(l - e-0.2t)(a) Plot dle
Consider the motion of the squirrel in the previous problem. The components of the velocity vector of the squirrel are given by vr, = dr/dt and vθ = rdθ/dt. The speed of the squirrel is v = √v2r
The curvilinear motion of a particle is defined by the following parametric equations:x = 52t-9t2 m and y = 125-5t2 mThe velocity of the particle is given byv = ˆšv2x + v2y, Where vx = dx/dt and
The demand for water during a fire is often the most important factor in the design of distribution storage tanks and pumps. For communities with populations less than 200,000, the demand Q (in
The position x as a function of time of a particle that moves along a straight line is given by: x(t) = (-3 + 4t)e-0.4t ft The velocity v(t) of the particle is determined by the derivative of x(t)
The area of the aortic valve, Av in cm2, can be estimated by the equation (Hakki Formula): Av = Q√PG where Q is the cardiac output in L/min, and PG is the difference between the left ventricular
A bandpass filter passes signals with frequencies that are within a certain range. In this filter the ratio of the magnitudes of the voltages is given byRV = Vo/Vi = Ï‰RC/ˆš(l - Ï‰2LC)2 +
A resistor, R = 4„¦, and an inductor, L = 1.3 H, are connected in a circuit to a voltage source as shown in Figure (a) (an RL circuit). When the voltagesource applies a rectangular voltage pulse
In a typical tension test a dog bone shaped specimen is pulled in a machine. During the test, the force F needed to pull the specimen and the length L of a gauge section are measured. This data is
According to special relativity, a rod of length L moving at velocity v will shorten by an amount 8, given by: δ = L(1 - √1 – v2/c2 where c is the speed of light (about 300 x 106 m/s). Consider
Plot the function f(x) = x2/2 + sinx + x4 for -4 < x < 4.
The shape of a symmetrical four-digit NACA airfoil is described by the equationy = ± tc/0.2[0.2969 ˆšx/c €“ 0.1260x/c -0.3516(x/c)2 + 0.2843(x/c)2 €“
The ideal gas law relates the pressure P, volime V, and temperature T of an ideal gas: PV= nRT where " is the number of moles and R = 8.3145 J/(K. mol). Plots of pressure versus volume at constant
The vibrations of the body of a helicopter due to the periodic force applied by the rotation of the rotor can be modeled by a frictionless spiring-mass-da system subjected to an external periodic
A railroad bumper is designed to slow down a rapidly moving railroad car. After a 20,000 kg railroad car traveling at 20 m/s engages the bumper, its displacement x (in meters) and velocity v (in m/s)
Consider the diode circuit shown in the figure. The Current iD and the voltage vD can be determined from the solution of the following system of equations:iD = Io(eqvp/kT - 1), iD = Vs - VD/RThe
When monochromatic light passes through a narrow slit it produces on a screen a diffraction pattern consisting of bright and dark fringes. The intensity of the bright fringes. I, as a function of 9
A simply supported beam is subjected to distributed loads w1 and w2 as shown. The bending moment as a function of x is given by the following equations:M(x) = RAx €“W1x2/2 for 0 M(x) = RAx €“
Biological oxygen demand (BOD) is a measure of the relative oxygen depletion effect of a waste contaminant and is widely used to assess the amount of pollution in a water source. The BOD in the
The temperature dependence of the diffusion coefficient D (cm2/s) is given by an Arrhenius type equation: D = D0e(- Ea/RT) where D0 (cm2/s) is pre-exponential constant, Ea (J/mol) is activation
The resonant frequency f(in Hz) for the circuit shown is given by:f = 1/2Ï€ ˆšLC R21C €“ L/R22C €“ LGiven L = 0.2 H, C = 2 Ã— 10-6 (a) f versus R2
Plot the function fix) = x3- 2x2 - 10sin2x - e0·9x and its derivative for -2 < x < 4 in one figure. Plot the function with a solid line, and the derivative with a dashed line. Add a legend and label
The Taylor series for cos(x) is:x2/2! + x4/4! €“ x6/6! + x8/8! €“ x10/10! + €¦€¦Plot the figure on the right, which shows, for -2Ï€
Make two separate plots of the function f(x)= -3x4 + 10x2 - 3, one plot for -4 < x < 3 and one for -4 < x < 4.
Use the f plot command to plot the function f(x) = (sin2x+ cos25x)e-0.2x in the domain -6 < x < 6.
Plot the function f(x) = sin2(x)cos(2x) and its derivative, both on the same plot, for 0 < x < 2π. Plot the function with a solid line, and the derivative with a dashed line. Add a legend and label

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