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introduction to actuarial and financial

Introduction To Actuarial And Financial Mathematical Methods 1st Edition Stephen Garrett - Solutions

Confirm, in each case, that the point stated is on the curve defined by the implicit expression. Determine the equation of the tangent at that point.a. γ2 + x2 = 4 at (x, γ) = (0, 2)b. γ = cos(3x + 7γ) at (x, γ) = (2, 0.2315)
Demonstrate that the function f (t) = Gek(t−t0) satisfies Eq. (11.4) and the associated condition f (t0) = G. d dif(t) = kf (t) (11.4)
Solve the following boundary value problems. P²-1 = 0 such that y(5) = 0 b. = such that y(10) = 0 d 3 2² cos y • g = (5t + 1)g such that g(0) = 1 d. k' = 5(ky +2y+k+ 2) such that k(12) = 3 a. y -
Classify the following ODEs in terms of their order, linearity, and coefficients. a. h" +5h - 5=0 b. gg" + xg sin x = 2 (dr) - 4y= 2 C. d. xf' + xf = x
Solve the following boundary value problems. a. f' = x + such that f(1) = π b./ = 5 + 2h such that h(−4) = 2 c. g = 10- 2g such that g(0) = 0 d. y + y=sin(x) such that y(л) = 1
The GDP of two different countries is thought to be modeled by Eq. (11.4).a. If, at time t0, the GDPs of countries A and B are GA and GB, respectively, write downa boundary value problem in each case. You should assume the value of k is the same for each country.b. Use the result of Example 11.3 to
Solve the following boundary value problems. a. (2xy+1)y=-(²2x) such that y(0) = 1 cos y-cos f-f y-y sinf such that f(7) = 1 df b. dy = c. (g +1)e²=(2g - eg such that g→ 4 as z→→∞ (1+2²) arctan(2) such that z(1) = 1 12 d. 1 dz t dt =
Determine whether the following ODEs are separable. a. g(x) + g(x) = x² b. eg'(x) = e-g(x) x C. y(x) dy(x) sin x dx d. eydz(y) dy = π z(y)+1 y-2 +1²
Solve the following first-order separable boundary value problems. 3f such that f(0) = 1. dx dy = x3 + 2 such that y(1) = π. a. b. xy dx c. g'(x) = g(x) sin(x) such that g(0.5) = 3 = d. e2x dy = ey such that y(10) = 2. dx
While flying between San Francisco and Honolulu, a trainee actuary notices a new island 500 miles off the coast of California. He alerts the authorities and, soon later, the island is subject to a detailed study. Scientists discover that a new species of beetle inhabits the island and a sample of
Solve the following second-order initial value problemsuch that y′(0) = 1 and y(0) = 0. d² d dx2y+3y-10y = 0
Use an integrating factor to solve the boundary value problem dy dx + 3x²y = 0 such that y(0) = 2
Solve the boundary value problem dy dx X- + y = 4xsin x such that y(0) = 0
In standard actuarial notation, the quantity tp50 denotes the probability that, on his 50th birthday, an individual will survive for a further t years. Under the Weibull model of mortality, tp50 is determined bywhere α and β ≠ 1 are constants. You are given that the probability that the
Determine whether the following ODEs are exact. 1 dy a. 1/2+ = 0 x dx b. (f² cos z – sin 2) + (2ƒ sin ≈ + 2)ƒ' = 0 c. 2xg2 +4+2(x²g – 3)g' = 0 d. (2x+x²y³) (x²³y² +4y³) + ddy = 0 dx
Find the solution to the boundary value problem given by the exact ODE Eq. (11.15) subject to the boundary condition y(1) = 10. You should begin by using an appropriate integral of Q(x, y). 이용 + 1 dy = 0 x dx —— (11.15)
Solve the following boundary value problems formed from exact ODEs and a single boundary condition a. (f² cos z sin 2) + (2f sin 2 +2)f' = 0 such that ƒ(7) = 0. b. 2xg² + 4 +2(x²g - 3)g' = 0 such that g(1) = 1. (2x+x²y³) (x³y² +4y³) + ddy = 0 such that y(0) = 1. dx C.
You are given thatObtain the following partial derivatives. f(x1, x2) = x1 sin x2 x1 + x2 and g(x, y, z) = x³y²z lnx
Use Wolfram Alpha to solve the following ODEs. a. f(x)f'(x) = ƒ² (x) - x such that f(0) = 10. b. g' (t) = 5t² + sint such that g(10) = 1.
You are given the multivariate functionsDetermine the following partial derivatives. f(x, y, x) = x²yz + sin(xy + x) g(x, y) = e+²+cos(y)
You are given that g(x1, x2, x3, x4) = x21x42 sin x3 + x53 ln x4. Obtain the following mixed partial derivatives. a. gx1x2 b. gx2x1 c. 8x1x2x3 d. gx3x1x2 e. gx1x1x1x4 f. 8x1x4x4x1
Confirm the location of the three critical points illustrated in Figure 12.3 and use the second partial derivative test to classify them. f (x1x2) @ 20 -20 -2 I1 0 2 4-4 0 x2 f(x1x2) (b) 20 -2 0 2 4-4 2 x2 4
Where they exist, determine, and classify all stationary points of the following bivariate functions. a. . f(x, y) = 5x - 2x² +10y-y² b. g(x,z) = (1+z²) e²-x c. h(x,z) = x+z x² +2₂² +6 d. k(a, b) = sin²(a) + b²
Solve the following constrained optimization problems, where possible. a. Maximize x² + 2y² such that x² + y² = 4. b. Minimize x + 2y² such that x + y² = 1. c. Minimize 2x² + y² - ² such that 3xy - 42² = 0. d. Maximize -y +z-x+6 such that x - y-z = 2 and x² + y² = 1.
Find and classify any critical points of the bivariate function g(y, z) = 5y2 − 8y − 2yz − 6z + 4z2.
Demonstrate that the function f (x, y, z) = x2 has only one critical point on the surface x2 + y2 = z. Classify the critical point.
Find and classify any critical points of the bivariate function h(x, y)=2x2 − 4xy+y4 +2.
Maximize the volume of a rectangular box with surface area 2m2 and height 0.1 m.
Evaluate the following definite integrals over the rectangular domains stated.Attempt to express the integrand in a separable form. a. ff 2xy dx dy for Da = {x, ylx = [1, 2], y € [2, 4]}. Da b. ff(x²+10) dx dz for Db = C. {x, xlx [0, 10], z = [-1,1]}. Db ff 0.1p cos(q) dp da for D₁ = {p, q\p
Find and classify any critical points of the bivariate function k(x1, x2) = x31 + x32 − 2x1 + 6x2.
Evaluate the following definite integrals over the domains stated. ff 2xy dx dy for Da = {x, ylx = [1, 2], y = [-x, x]}. Da b. ff(x²+10) dx dz for Db = {x, z|x € [2z, 4z], z € [1,2]}. Dh c. ff 0.1p sin(q) dp dq for Dr = {p, q\p € [-q, 9²], q € [-π,^]}. Dc d. ff e/Y dx dy for D₁ = {x,
Maximize x1 + x2 subject to the constraint x21 + x22 = 4. Give a geometrical interpretation of this problem.
Working in units of $1000, the minimum return from a portfolio consisting of an investment of x1 and x2 in Asset 1 and Asset 2, respectively, is thought to be given byYou decide to invest $10,000 in the portfolio. Determine the optimal holding of each asset. f(x1, x2) = 65-1.5x-x
An insurance company provides buildings and contents insurance to its policyholders.We use X to denote the random variable for annual claims under buildings insurance and Y to denote annual claims under contents insurance. Working in units of $10M, X and Y are known to have a joint density function
You are aware that your happiness in the early afternoon depends on what you have eaten for lunch. After many years of research, you are reasonably sure that your happiness, H, is a function of the number of sandwiches, s, and chocolate bars, c, consumed and is given byIf each sandwich costs $3.50,
Determine and classify the extreme values of x1 + 3x2 + 2x3 subject to the constraint x21 + x22 + x23 = 9.
A factory manufactures three products and has an annual profit given bywhere x, y, and z represent the number of units manufactured (in units of 10,000) of products X, Y, and Z, respectively, in the year. In order to meet the demands of its current contracts, the factory is required to manufacture
Find the shortest distance between the origin and the plane defined by x1 + 2x2 − 2x3 = 9.
Find and classify the extreme points of f (x1, x2, x3) = x1 + 4x2 − 2x3 subject to the constraints that 2x1 − x2 − x3 = 2 and x21 + x22 = 4.
Perform the following integrals in the order indicated by the brackets. a. I₁ = f (x² + y² dx) dy b. 12 = f (x²+² dy) dx
Locate and classify the extremes of the function f (x, y, z) = 2xyz subject to the constraints x + y + z = 2 and 2x + 2y + z = 1.
Where possible, apply Eq. (12.9) to the integrals in Example 12.14.Example 12.14Evaluate the following iterative integrals. ·b d x = [ f(x) dx [ ^_x(y) 8 dy a S. [" ["f(x)g(y) dy dx = (12.9)
Evaluate the following iterative integrals. a. fi f²₁ ye* dy dx b. f f cos(x) sin(y) dx dy •2π J-27 10 re c. f,º f (x²+) dy dx C. d. f f (x²y³ - 2y cos x) dx dy -5
Evaluatewhere D3 is a triangular region on the x1-x2 plane with vertices at (−1, 0), (1, 0), and (0, 4). V3 = ff (x² − 2) dxj dx2 D3
Use Wolfram Alpha to answer the following questions. a. Locate and classify stationary points of the function f(x₁, x2) = x₁x² + x² + 2x² - 2x1x2. b. Find the location of the minimum value of g(x, y) = 5x² + 10xy such that x and y sit on a circle of radius 3 and centered at (x, y) = (3, 1)
Use the bisection method to estimate all roots of the following function with an error tolerance of ϵ = 0.002, 7-x+ εx = (x)8
You are given that f (x) = x2 − 2 ex − 2x + 5 has a single real root xr on [0, 1]. Use the following methods to determine the root to within an error tolerance defined by |f (xr)| ≤ 0.001. Comment on your answers.a. Bisection methodb. Regula falsi method
Use the bisection method to obtain the roots required of the following functions to an error tolerance of ϵ = 0.01.a. The single real positive root of f (x) = x2 + 2.011x − 5.04795.b. The single real negative root of f (x) in part a.c. The single positive root of g(x) = sin(x2) − x + 1.
Repeat Question 13.1 using the Newton-Raphson and Secant methods.Comment on your answers.Question 13.1You are given that f (x) = x2 − 2 ex − 2x + 5 has a single real root xr on [0, 1]. Use the following methods to determine the root to within an error tolerance defined by |f (xr)| ≤ 0.001.
Implement an appropriate numerical method to determine all real roots of g(x) = -0.67 × 2.5x¹.3 +2.5x¹.3 Xx 0.2 +1.34x5.3 - 2x-5.5
Compare the theoretical number of iterations required under the bisection method against the actual number of iterations for each part of the solution to Example 13.2.Example 13.2Use the bisection method to obtain the roots required of the following functions to an error tolerance of ϵ = 0.01.a.
Implement the elementary definition of a derivative to determine a converged estimate of f′(−1) for f(x) = sin(x3). Table 13.20 Samples of an unknown function f(x) at various node points {Xn} Xn f(xn) -
Repeat Example 13.1 using the regula falsi method on the interval [0, 1.5]. You should use an error tolerance of 0.002.Example 13.1Use the bisection method to estimate all roots of the following function with an error tolerance of ϵ = 0.002, 7-x+ εx = (x)8
Investigate the sensitivity of the required number of iterations required under the bisection method toa. The initial interval width for fixed ϵ = 0.001b. The error tolerance for a fixed initial width of b − a = 1.
A saver has been depositing money in a fixed-interest bank account for the last 10 years. If the balance of his account is $15,432 at time t = 10, calculate an approximate value of the annual interest rate earned. You are given that the investor deposited $2000 at time t = 0, $3000 at time t = 3.5,
Repeat Example 13.2 using the regula falsimethod on the following intervals. You should use a fixed error tolerance of ϵ = 0.008 in each case.Example 13.2Use the bisection method to obtain the roots required of the following functions to an error tolerance of ϵ = 0.01. a. [a, b] = [0, 2] b. [a,
Repeat Question 13.5 using the central-difference methods of orders 2 and 4. Comment on your answers.Question 13.5Implement the elementary definition of a derivative to determine a converged estimate of f′(−1) for f(x) = sin(x3). Table 13.20 Samples of an unknown function f(x) at various node
Use the data given in Table 13.20 to estimate the gradient of the unknown function f (x) at each node point using the following approaches.a. Elementary approximation.b. Central-difference method of order 2.c. Central-difference method of order 4.You are given that f (x) = 1 + x2 cos x. Comment on
A business project requires an initial outlay of £10,000 and a further outlay of £20,000 after 2 years in order to generate an income of £40,000 after 5 years. Determine the approximate yield obtained from the project.
Determine all real roots of the following function over the interval [−5, 5]. You should use the regula falsi method with error tolerance ϵ = 0.00005. g(x) = x sin(x) - 1.2
For each of the following methods, determine the node points that should be used to evaluate the integralto within a theoretical error of 0.001.a. Trapezoidal approachb. Simpson’s approachImplement each approximation. - [15 sin(x²) d dx -1.5 I=
Use the Newton-Raphson method to determine an improvement on the initial estimate of the root in the following cases.a. f (x) = ex − 4 from initial estimate x0 = 1.5.b. g(x) = x3 − 2x − 4 from initial estimate x0 = 2.5.c. h(x) = ln(x) − 2 from initial estimate x0 = 8.d. k(x) = ln(x3) − 5
An asset, currently priced at $200, is expected to have a market price 1 year from now given by $200S. Here, S is a random variable with density functionfor some fixed constant a. Estimate the probability that the price 1 year from now will have increased by between 0% and 300%. You should use
If Z is a random variable with Standard normal distribution, confirm that the probability that Z is between 0 and 4 is 49.997% to within 0.001%. You should use an appropriate implementation of Simpson’s approach.
Repeat Example 13.8 using the Newton-Raphson method with error tolerance ϵ = 0.0001.Example 13.8Determine all real roots of the following function over the interval [−5, 5]. You should use the regula falsi method with error tolerance ϵ = 0.00005. g(x)=xsin(x) - 1.2
Use the Newton-Raphson method to find the repeated root of f (x) = (x − 1)3 with an error tolerance ϵ = 0.0001.
Explore the use of the Newton-Raphson method to determine all real roots of the function g(x) = (x − 2)2 − 1. Use an error tolerance of ϵ = 0.0002.
Use the secant method to determine an improvement on the estimated root in the following cases.a. f (x) = ex − 4 from initial estimates x0 = 1.2 and x1 = 1.5.b. g(x) = x3 − 2x − 4 from initial estimates x0 = 2.3 and x1 = 2.5.c. h(x) = ln(x) − 2 from initial estimates x0 = 7 and x1 = 8.d.
Repeat Example 13.12 using the secant method with error tolerance ϵ = 0.00001.Example 13.12Repeat Example 13.8 using the Newton-Raphson method with error tolerance ϵ = 0.0001.Example 13.8Determine all real roots of the following function over the interval [−5, 5]. You should use the regula
Use the secant method to determine the single root of h(x) = ln(x) − 2 with an error tolerance ϵ = 0.00001.
Use Taylor expansions of f(x ± h) to show that the theoretical error in using the central-difference formula (Eq. 13.11) to approximate f(x) is O(h2). f'(x) ~ D²f(x) = = f(x+h)-f(xh) 2h (13.11)
Compute {Dhk f(x)} at x = 2 for the following functions. You should use stepsize hk = 10−k for k = 1, 2, . . . until an apparent convergence to five decimal places is observed.a. f(x) = 0.7xb. g(x) = ln(x)c. k(x) = exd. m(x) = cos(x)
Use Eqs. (13.8), (13.11), and (13.12) to obtain the numerical derivative of f(x) = ex2−x+2 at x = 0.75 with stepsize hk = 10−k for k = 1, . . . , 10. [2] f'(x) ~ D²f(x): h = f(x+h)-f(x- h) 2h (13.11)
Table 13.16 gives { f(xn)} and {xn} for 21 regularly spaced node points over [0, 3] for an unknown function. Use the O(h2) and O(h4) central-difference formula to approximate {f′(xn)}. Plot the results and comment on your answer. Table 13.16 Numerical derivatives using the O(h²) and O(ht)
Repeat Example 13.18 using the central-difference formula of O(h2).Example 13.18Compute {Dhk f(x)} at x = 2 for the following functions. You should use stepsize hk = 10−k for k = 1, 2, . . . until an apparent convergence to five decimal places is observed.f(x) = 0.7xg(x) = ln(x)k(x) = exm(x) =
Repeat Example 13.18 using the central-difference formula of O(h4).Example 13.18Compute {Dhk f(x)} at x = 2 for the following functions. You should use step size hk = 10−k for k = 1, 2, . . . until an apparent convergence to five decimal places is observed.f(x) = 0.7xg(x) = ln(x)k(x) = exm(x) =
Compute the trapezoidal approximation of order 1 for the integral 1 = [ (20-0.5x²) dx 2
Repeat Example 13.24 using trapezoidal approximations of order (a) 2, (b) 3, and (c) 4.Examples 13.24Compute the trapezoidal approximation of order 1 for the integral 1 = [ (20-0.5x²) dx 2
Write down the trapezoidal rule of order k for the integral of f (x) over [−4, 6] for (a) k = 1, (b) k = 2, and (c) k = 5.
Determine the actual and theoretical errors of the approximations used in Examples 13.24 and 13.26 at various k.Examples 13.24Compute the trapezoidal approximation of order 1 for the integralExamples 13.26Repeat Example 13.24 using trapezoidal approximations of order (a) 2, (b) 3, and (c) 4. 1 =
Use the trapezoidal rule to approximate the following integral to within an error of ETk ≤ 0.001. = [₁ ' si 0 I= sin(x²) dx
Demonstrate that Eq. (13.18) is a quadratic that passes through (xo, f (x0)), (x1, f (x1)), and (x2, f (x2)). Pb1,2(f(x)) = (x − x₂) (x − x1) - (xox₂) (xox1) + -f(x0) + (x-x₁)(x - x0) (x2-x₁) (x2-xo) f (x₂) (x-x₂)(x - x0) (x₁ - x₂)(x₁ - xo) -f(x₁) (13.18)
Write down Simpson’s rule of order k for the integral of f (x) over [−4, 6] for (a) k = 2, (b) k = 4, and (c) k = 6.
Approximate the following definite integral using Simpson’s rule for (a) k = 2, (b) k = 4, and (c) k = 6. I = = L ²₁ ( ₁²06 + e²(²+²) e-(x+2) dx
Determine the actual and theoretical errors of the approximations used in Example 13.32 at various k.Example 13.32Approximate the following definite integral using Simpson’s rule for k = 2, k = 4, and k = 6. I = = L ²₁ ( ₁²6 + e²(²+²) e-(x+2) dx
Repeat Example 13.26 using Simpson’s approximation of order 2 to approximate the integrand. Comment on the result.Examples 13.26Repeat Example 13.24 using trapezoidal approximations of order 2, 3, and 4.
Repeat Example 13.28 using Simpson’s rule.Example 13.28Use the trapezoidal rule to approximate the following integral to within an error ofETk ≤ 0.001. - ² si I = sin(x²) dx
Approximate the following integral using both the trapezoidal rule and Simpson’s rule at order k = 2, 4, 6, 8, and 10. Comment on the relative behavior of the errors with k. 1 = f ² e e* cos(x) dx
State whether the following are valid probability density functions. a. fx (x) = sin(x) for x = [-∞,00]. b. gy (y) = for y € [a, b] and b> a are real constants. c. hz(2) = λ ez for z = [0,00) and λ some positive constant.
You are studying a newly discovered species of rodent. Observations suggest that a rodent selected at random with a tail of length described by a random variable X cm with probability density functionwhere α is some constant.a. Determine the value of α such that fX(x) is a valid probability
Given that, Φ (0.50) = 0.69146 and Φ(1.00) = 0.84134 and X ∼ N(0, 1), determine the following probabilities.a. P(X > 0.50)b. P(−1 < X < 0.50)
Repeat Example 13.38(b) using a numerical integration approach. You should use Simpson’s rule of order 6.Example 13.38(b)Given that, Φ (0.50) = 0.69146 and Φ(1.00) = 0.84134 and X ∼ N(0, 1), determine the following probabilities.b. P(−1 < X < 0.50)
A random variable X has probability density function given byfor −∞ a. Demonstrate that gX(x) is a valid probability density function.b. Calculate P(1.5 c. Determine a transformation that takes X to a Standard normal distribution and use statistical tables to verify your answer to part b.You
A random variable Y has probability density function given byfor y > 0.a. Demonstrate that hY(y) is a valid probability density function.b. Calculate P(1 c. Given that this distribution is actually Log-normal, that is lnY ∼ N(0, 1), use published statistical tables to verify your answer to
Determine which of the functions identified in Question 2.2 demonstrate either odd or even symmetry.Question 2.2Determine which of the mappings in Question 2.1 are functions.Question 2.1Determine whether the following mappings are one-to-one, many-to-one, or one-to-many mappings. a. f(x) = x5-29x³
Determine all real roots of the functions identified in Question 2.2.Question 2.2Determine which of the mappings in Question 2.1 are functions.Question 2.1Determine whether the following mappings are one-to-one, many-to-one, or one-to-many mappings. a. f(x) = x5-29x³ + 100x b. g(z) = 24 - 6x³ +
Derive expressions for the following derivatives. a. b. C. d dx d dx d dx :arcsin(x) arccos(x) :arctan(x)
Classify all stationary points found in Example 4.5.Example 4.5.Determine all stationary points of the following functions. Write down the equation of the tangent in each case. a. h(x) = 3x + 2 b. k(x) = e²-2x+2 c. 1(x) = sin x d. m(x) = x³ +2
Evaluate the following expressions using Properties i.-iii. 100 a. Σ-10 2. b. Σ! (2x; - 4yi) if Σ! Σ5 (10 – 5x;) if Σ C. 100 x = 100 and Σ γ = 20. di=1 5 5x; = 6. ==5
Determine the following indefinite integrals. a. f x² dx b. fy⁹.2 dy c. f 2-5 dz Y

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