Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Chapter 8.8, Problem 13P
Find the inverse transforms of the functions
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Chapter 8 Solutions
Mathematical Methods in the Physical Sciences
Ch. 8.1 - Verify the statement of Example 2. Also verify...Ch. 8.1 - Solve Example 4 using the general solution...Ch. 8.1 - Verify that y=sinx,y=cosx,y=eix, and y=eix are all...Ch. 8.1 - Find the distance which an object moves in time t...Ch. 8.1 - Find the position x of a particle at time t if its...Ch. 8.1 - A substance evaporates at a rate proportional to...Ch. 8.1 - The momentum p of an electron at speed v near the...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...
Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - In Problems 13 to 15, find a solution (or...Ch. 8.2 - In Problems 13 to 15, find a solution (or...Ch. 8.2 - In Problems 13 to 15, find a solution (or...Ch. 8.2 - By separation of variables, find a solution of the...Ch. 8.2 - The speed of a particle on the x axis, x0, is...Ch. 8.2 - Let the rate of growth dN/dt of a colony of...Ch. 8.2 - (a) Consider a light beam traveling downward into...Ch. 8.2 - Consider the following special cases of the simple...Ch. 8.2 - Suppose the rate at which bacteria in a culture...Ch. 8.2 - Solve the equation for the rate of growth of...Ch. 8.2 - Heat is escaping at a constant rate [dQ/dtin(1.1)...Ch. 8.2 - Do Problem 23 for a spherical cavity containing a...Ch. 8.2 - Show that the thickness of the ice on a lake...Ch. 8.2 - An object of mass m falls from rest under gravity...Ch. 8.2 - According to Newtons law of cooling, the rate at...Ch. 8.2 - A glass of milk at 38 is removed from the...Ch. 8.2 - A solution containing 90 by volume of alcohol (in...Ch. 8.2 - If P dollars are left in the bank at interest I...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.2 - Find the orthogonal trajectories of each of the...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Using (3.9), find the general solution of each of...Ch. 8.3 - Water with a small salt content (5 lb in 1000 gal)...Ch. 8.3 - Find the general solution of (1.2) for an RL...Ch. 8.3 - Find the general solution of (1.3) for an RC...Ch. 8.3 - Prob. 18PCh. 8.3 - If 1=2= in (3.10), then e21tdt=dt. Find N2 for...Ch. 8.3 - Extend the radioactive decay problem (Example 2)...Ch. 8.3 - Generalize Problem 20 to any number of stages.Ch. 8.3 - Find the orthogonal trajectories of the family of...Ch. 8.3 - Find the orthogonal trajectories of the family of...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Use the methods of this section to solve the...Ch. 8.4 - Solve the differential equation yy2+2xyy=0 by...Ch. 8.4 - If an incompressible fluid flows in a corner...Ch. 8.4 - Find the family of orthogonal trajectories of the...Ch. 8.4 - Find the family of curves satisfying the...Ch. 8.4 - Find the shape of a mirror which has the property...Ch. 8.4 - As in text just before (4.11), show that (a)...Ch. 8.4 - Show that the change of variables (4.13) in (4.11)...Ch. 8.4 - Show that (xP+yQ)1 is an integrating factor for...Ch. 8.4 - Solve Problems 9 and 10 by using an integrating...Ch. 8.4 - An equation of the form y=f(x)y2+g(x)y+h(x) is...Ch. 8.4 - Show that the substitution given in Problem 25...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Solve the following differential equations by the...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation (8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation ( 8.5), that a set...Ch. 8.5 - Recall from Chapter 3, equation (8.5), that a set...Ch. 8.5 - Solve the algebraic equation D2+(1+2i)D+i1=0 (note...Ch. 8.5 - As in Problem 19, solve y+(1i)yiy=0. Hint: See...Ch. 8.5 - By the method used in solving (5.4) to get (5.9),...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Use the results of Problem 21 to find the general...Ch. 8.5 - Let D stand for d/dx, that is, Dy=dy/dx; then...Ch. 8.5 - In Example 3, we used the second solution in...Ch. 8.5 - A particle moves along the x axis subject to a...Ch. 8.5 - Find the equation of motion of a simple pendulum...Ch. 8.5 - The gravitational force on a particle of mass m...Ch. 8.5 - Find (in terms of L and C) the frequency of...Ch. 8.5 - A block of wood is floating in water; it is...Ch. 8.5 - Solve the RLC circuit equation [(5.33)or(5.34)]...Ch. 8.5 - (a) Find numerical values of the constants and...Ch. 8.5 - The natural period of an undamped system is 3 sec,...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Find the general solution of the following...Ch. 8.6 - Verify that (6.4) is a particular solution of...Ch. 8.6 - Solve (6.16) by the method used in solving (...Ch. 8.6 - Consider the differential equation...Ch. 8.6 - (a) Show that (Da)ecx=(ca)ecx;...Ch. 8.6 - (a) Show that Deaxy=eax(D+a)y, D2eaxy=eax(D+a)2y,...Ch. 8.6 - Using Problems 29 and 31b, show that equation...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - In Problem 33 to 38, solve the given differential...Ch. 8.6 - Find the solutions of (1.2) (put I=dq/dt ) and...Ch. 8.6 - In (6.38), show that for a given forcing frequency...Ch. 8.6 - Solve Problems 41 and 42 by use of Fourier series....Ch. 8.6 - Solve Problems 41 and 42 by use of Fourier series....Ch. 8.6 - Consider an equation for damped forced vibrations...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - Solve the following differential equations by...Ch. 8.7 - The differential equation of a hanging chain...Ch. 8.7 - The curvature of a curve in the (x,y) plane is...Ch. 8.7 - Solve y+2y=0 by method (c) above and compare with...Ch. 8.7 - The force of gravitational attraction on a mass m...Ch. 8.7 - Show that (7.15) is a separable equation. [You may...Ch. 8.7 - In Problems 10 and 11, solve (7.14) to find v(x)...Ch. 8.7 - In Problems 10 and 11, solve (7.14) to find v(x)...Ch. 8.7 - In Problem 11, find v(x) if v=0,x=1, at t=0. Then...Ch. 8.7 - The exact equation of motion of a simple pendulum...Ch. 8.7 - Verify (7.19) and (7.20). Hint:...Ch. 8.7 - If you solve (7.17) when f(x)=0 by assuming a...Ch. 8.7 - Solve the following equations either by method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the following equations using method (d)...Ch. 8.7 - Solve the two differential equations in Problem...Ch. 8.7 - Substitute (7.22) into (7.21) to obtain the...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.7 - For the following problems, verify the given...Ch. 8.8 - For integral k, verify L5 and L6 in the Laplace...Ch. 8.8 - By using L2, verify L7 and L8 in the Laplace...Ch. 8.8 - Using either L2, or L3 and L4, verify L9 and L10.Ch. 8.8 - By differentiating the appropriate formula with...Ch. 8.8 - By integrating the appropriate formula with...Ch. 8.8 - By replacing a in L2 by a+ib and then by aib, and...Ch. 8.8 - Verify L15 to L18, by combining appropriate...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Find the inverse transforms of the functions F(p)...Ch. 8.8 - Show that a combination of entries L3 to L10, L13,...Ch. 8.8 - Prove L32 for n=1. Hint: Differentiate equation...Ch. 8.8 - Use L32 and L3 to obtain L11.Ch. 8.8 - Use L32 and L11 to obtain Lt2sinat.Ch. 8.8 - Use L31 to derive L21Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.8 - Table entries L28 and L29 are known as translation...Ch. 8.9 - Continuing the method used in deriving (9.1) and...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - By using Laplace transforms, solve the following...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Solve the following sets of equations by the...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.9 - Evaluate each of the following definite integrals...Ch. 8.10 - Show that g*h=h*g as claimed in I34. Hint: Let u=t...Ch. 8.10 - Use L34 and L2 to find the inverse transform of...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the convolution integral to find the inverse...Ch. 8.10 - Use the Laplace transform table to find...Ch. 8.10 - Use the convolution integral (see Example 2) to...Ch. 8.10 - Use the convolution integral (see Example 2) to...Ch. 8.10 - Consider solving an equation like (10.1) but with...Ch. 8.10 - Solve the differential equation ya2y=f(t), where...Ch. 8.10 - A mechanical or electrical system is described by...Ch. 8.10 - Following the method of equations (10.8) to...Ch. 8.11 - Find the inverse Laplace transform of e2p/p2 in...Ch. 8.11 - Verify L24 in the table by using L1, L27, and the...Ch. 8.11 - Verify L28 in the table by using L27 and the...Ch. 8.11 - Show that fn(t)dt=1 for the functions fn(t) in...Ch. 8.11 - Solve the differential equation y+2y=f(t),y0=y0=0,...Ch. 8.11 - (a) Let a mechanical or electrical system be...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Using the function method, find the response (see...Ch. 8.11 - Evaluate the functions fn(xa) defined by the...Ch. 8.11 - Using functions, write the following mass or...Ch. 8.11 - Integrate by parts as we did for (11.14) to obtain...Ch. 8.11 - Use (11.6) and (11.14) to (11.16) to evaluate the...Ch. 8.11 - Verify the operator equation ddxsgnx=2(x) where...Ch. 8.11 - Verify (11.18a) and (11.18c) by multiplying by a...Ch. 8.11 - Use equation (11.16) to generalize the operator...Ch. 8.11 - (a) Show that you can differentiate a generalized...Ch. 8.11 - Verify the operator equations in (11.19) not done...Ch. 8.11 - Make use of the operator equations (11.19) and...Ch. 8.11 - You may find the spherical coordinate function...Ch. 8.11 - Write a formula in rectangular coordinates, in...Ch. 8.11 - Prob. 24PCh. 8.11 - Let F(x)=x2,x0,0,x0. Show that F(x)=0 for all x0,...Ch. 8.12 - Solve (12.3) if G=0 and dG/dt=0 at t=0 to obtain...Ch. 8.12 - In Problems 2 and 3, use (12.6) to solve (12.1)...Ch. 8.12 - In Problems 2 and 3, use (12.6) to solve (12.1)...Ch. 8.12 - Use equation (12.6) to solve Problem 10.18.Ch. 8.12 - Obtain ( 12.6 ) by using the convolution integral...Ch. 8.12 - For Problem 10.17, show (as in Problem 1) that the...Ch. 8.12 - Use the Green function of Problem 6 to solve...Ch. 8.12 - Solve the differential equation...Ch. 8.12 - Following the proof of (12.4), show that (12.9)...Ch. 8.12 - Solve (12.12) and (12.14) to get (12.15). Hint:...Ch. 8.12 - In Problems 11 to 13, use (12.17) to find the...Ch. 8.12 - In Problems 11 to 13, use (12.17) to find the...Ch. 8.12 - In Problems 11 to 13, use (12.17) to find the...Ch. 8.12 - (a) Given that y1(x) and y2(x) are solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.12 - In Problems 15 to 18, use the given solutions of...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - Identify each of the differential equations in...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - In Problems 25 to 28, find a particular solution...Ch. 8.13 - If 10kg of rock salt is placed in water, it...Ch. 8.13 - A mass m falls under gravity (force mg ) through a...Ch. 8.13 - The acceleration of an electron in the electric...Ch. 8.13 - Suppose that the rate at which you work on a hot...Ch. 8.13 - Compare the temperatures of your cup of coffee at...Ch. 8.13 - A flexible chain of length l is hung over a peg...Ch. 8.13 - A raindrop falls through a cloud, increasing in...Ch. 8.13 - (a) A rocket of (variable) mass m is propelled by...Ch. 8.13 - The differential equation for the path of a planet...Ch. 8.13 - Use L15 and L31 to find the Laplace transform of...Ch. 8.13 - Use L32 and L9 to find the Laplace transform of t...Ch. 8.13 - Use the Laplace transform table to evaluate:...Ch. 8.13 - Use the Laplace transform table to evaluate:...Ch. 8.13 - Find the inverse Laplace transform of: p(p+a)3Ch. 8.13 - Find the inverse Laplace transform of: p2p2+a22Ch. 8.13 - Find the inverse Laplace transform of: 1p2+a23Ch. 8.13 - Prove the following shifting or translation...Ch. 8.13 - Use the table of Laplace transforms to find the...Ch. 8.13 - Solve Problems 47 and 48 either by Laplace...Ch. 8.13 - Solve Problems 47 and 48 either by Laplace...
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- Sand and clay studies were conducted at a site in California. Twelve consecutive depths, each about 15 cm deep, were studied and the following percentages of sand in the soil were recorded. 34.4 27.1 30.8 28.0 32.2 27.6 32.8 25.2 31.4 33.5 24.7 28.4 Converting this sequence of numbers to a sequence of symbols A and B, where A indicates a value above the median and B denotes a value below the median gives ABABABABAABB. Test the sequence for randomness about the median with a 5% level of significance. Verify that the number of runs is 10. What is the upper critical value c2? arrow_forwardSand and clay studies were conducted at a site in California. Twelve consecutive depths, each about 15 cm deep, were studied and the following percentages of sand in the soil were recorded. 34.4 27.1 30.8 28.0 32.2 27.6 32.8 25.2 31.4 33.5 24.7 28.4 Converting this sequence of numbers to a sequence of symbols A and B, where A indicates a value above the median and B denotes a value below the median gives ABABABABAABB. Test the sequence for randomness about the median with a 5% level of significance. Verify that the number of runs is 10. What is the upper critical value c2?arrow_forward29% of all college students major in STEM (Science, Technology, Engineering, and Math). If 46 college students are randomly selected, find the probability thata. Exactly 11 of them major in STEM. b. At most 12 of them major in STEM. c. At least 11 of them major in STEM. d. Between 11 and 15 (including 11 and 15) of them major in STEM.arrow_forward
- Sand and clay studies were conducted at a site in California. Twelve consecutive depths, each about 15 cm deep, were studied and the following percentages of sand in the soil were recorded. 27.3 34.6 30.6 27.8 33.4 31.5 27.3 31.2 32.0 24.7 24.4 28.2 Test this sequence for randomness about the median. Converting this sequence of numbers to a sequence of symbols A and B, where A indicates a value above the median and B denotes a value below the median gives BAABAABAABBB. Verify that the number of runs is 7, the lower critical number is 3, and the upper critical number is 11. Use a 5% level of significance. State the conclusion of the test and interpret your results.arrow_forward29% of all college students major in STEM (Science, Technology, Engineering, and Math). If 46 college students are randomly selected, find the probability thata. Exactly 11 of them major in STEM. b. At most 12 of them major in STEM. c. At least 11 of them major in STEM. d. Between 11 and 15 (including 11 and 15) of them major in STEM.arrow_forward4. Assume that a risk-free money market account is added to the market described in Q3. The continuously compounded rate of return on the money market account is log (1.1). (i) For each given μ, use Lagrange multipliers to determine the proportions (as a function of μ) of wealth invested in the three assets available for the minimum variance portfolio with expected return μ. (ii) Determine the market portfolio in this market and calculate its Sharp ratio.arrow_forward
- 3. A market consists of two risky assets with rates of return R₁ and R2 and no risk-free asset. From market data the following have been estimated: ER₁ = 0.25, ER2 = 0.05, Var R₁ = 0.01, Var R2 = 0.04 and the correlation between R1 and R2 is p = -0.75. (i) Given that an investor is targeting a total expected return of μ = 0.2. What portfolio weights should they choose to meet this goal with minimum portfolio variance? Correct all your calculations up to 4 decimal points. (ii) Determine the global minimum-variance portfolio and the expected return and variance of return of this portfolio (4 d.p.). (iii) Sketch the minimum-variance frontier in the μ-σ² plane and indicate the efficient frontier. (iv) Without further calculation, explain how the minimum variance of the investor's portfolio return will change if the two risky assets were independent.arrow_forward2. A landlord is about to write a rental contract for a tenant which lasts T months. The landlord first decides the length T > 0 (need not be an integer) of the contract, the tenant then signs it and pays an initial handling fee of £100 before moving in. The landlord collects the total amount of rent erT at the end of the contract at a continuously compounded rate r> 0, but the contract stipulates that the tenant may leave before T, in which case the landlord only collects the total rent up until the tenant's departure time 7. Assume that 7 is exponentially distributed with rate > 0, λ‡r. (i) Calculate the expected total payment EW the landlord will receive in terms of T. (ii) Assume that the landlord has logarithmic utility U(w) = log(w - 100) and decides that the rental rate r should depend on the contract length T by r(T) = λ √T 1 For each given λ, what T (as a function of X) should the landlord choose so as to maximise their expected utility? Justify your answer. Hint. It might be…arrow_forwardPlease solving problem2 Problem1 We consider a two-period binomial model with the following properties: each period lastsone (1) year and the current stock price is S0 = 4. On each period, the stock price doubleswhen it moves up and is reduced by half when it moves down. The annual interest rateon the money market is 25%. (This model is the same as in Prob. 1 of HW#2).We consider four options on this market: A European call option with maturity T = 2 years and strike price K = 5; A European put option with maturity T = 2 years and strike price K = 5; An American call option with maturity T = 2 years and strike price K = 5; An American put option with maturity T = 2 years and strike price K = 5.(a) Find the price at time 0 of both European options.(b) Find the price at time 0 of both American options. Compare your results with (a)and comment.(c) For each of the American options, describe the optimal exercising strategy.arrow_forward
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