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introduction to actuarial and financial

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

If f (x) = x2, g(x) = x4, h(x) = x, and l(x) = x3, determine the parity of the following functions.a. f (x) + g(x)b. h(x) + l(x)c. f (x) + l(x)
Determine the parity of all possible products of two odd or even functions.
If f (x) = 4x2 + x + 2 and g(x) = 2x2 + 2x − 1 calculate the output of the string f → g for input valuesa. x = 0b. x = 1c. x = 4
Use a substitution method to find all possible real roots of the following functions. a. f(x) = ²x + e* - 30 b. g(z) = cos² (5z) — 4 - c. h(y) = e6y + e4y - e²y - 1 d. n(q) = sin(29²) m(p) = ln (e²p+ 2e²) e. In
It is often convenient to define the following functions that are based on the standard circular functions.Explore each function and, in each case, comment on the following properties.i. Domainii. Rangeiii. Rootsiv. Symmetriesv. Asymptotes cosecx = 1 sin x sec x = 1 COS X cotc= 1 tan x
If f(x) = 4x2 + x + 2 and g(x) = 2x2 + 2x − 1, determine an explicit expression for the following composite functions.a. (g ◦ f )(x)b. (f ◦ g)(x)
If f (x) = 2x2 + 2 and g(x) = x3 + 2x + 1 demonstrate that functions resulting from the following are also polynomial. State the order of the resulting polynomial function in each case.a. f(x) + g(x)b. f(x) × g(x)c. (f ◦ g)(x)d. (g ◦ f )(x)
Use algebraic techniques to determine the domain and range of the following functions over R. Check your conclusions with a computational plot of each function. 2x-1 a. • f(x) = 2x²+x-1 b. g(z) = In : In (₂ 22-1 2z²+2-1 = sin (2²+4y-1 c. h(y) = sin( d. j(p) = exp 2p-1 2p²+p-1
Determine the horizontal asymptotes (if any) of the following functions. Use a computer to plot each function to check your answer and also to investigate the range. 3x²–2x+2 a. f(x) = 2x²+x+1 b. g(y) 5y² y+8y³+24y2+32y+16
If j(x) = 2x5 − x4 + 2x − 2 and k(x) = 4x5 + 3x3, determine values of a and b such that aj(x) + bk(x) is not fifth order.
Use Wolfram Alpha to confirm the horizontal and vertical asymptotes ofas illustrated in Figures 2.11(a) and 2.12(b).Figures 2.11(a)Figures 2.12(b) f(x) = 2x³2x²+2x+1 x³ + x + 2 and g(y) = 5y² y + 8y³ +24² +32y + 16
Obtain f−1(x) for each of the following functions. a. f(x) = (x − 2) 4/ b. f(x) = 2 x-2 3x+9 c. f(x) = 4x² d. f(x) = ln(x+6)
Demonstrate the respective left- and right-hand asymptotes of exp(x) and exp(−x) using numerical values.
State the accumulation function for investments made at a compounding interest rate of 10% per annum. Use this function to determine the following quantities at this interest rate.a. The accumulated value of $100 after 17 years.b. The accumulated value at t = 10 for a deposit of $40 made at t = 2
It is clear from Figure 2.16 that the function g(x) = e3x + e2x − e2 has a single root at x ≈ 0.5. Find the value of this root, accurate to four decimal places. (x)f e 6 0 -0.5 0 I 0.5 1 f(x) 20 10 0 -10 -1 -0.5 0 I 0.5 1
If f (z) = exp(z) and g(z) = exp(−z), use a computer to visually explore the range of each of the following composite functions.a. (f ◦ h)(z), where h(z) = z2 − 5z + 6.b. (g ◦ h)(z), where h(z) is as in a.c. (g ◦ m)(z), where m(z) = 1/ z2 − 4
An investment opportunity requires an initial investment of £12m and will return £10m and £5m after 3 years 6 months and 7 years, respectively.a. State the equation of value f (i) = 0 for this opportunity and determine all real roots of f (i).b. An investor is considering funding the investment
Find the single real root of the function g(x) = e2x − 2.
Find all real roots of the following functions, accurate to four decimal places.a. f(x) = e2x − 5ex + 6b. g(x) = e4x − 9e2x + 20
Demonstrate algebraically that h(x) = −l(x) as defined in Eqs. (2.18) and (2.19). (817) (t-zx) = (x)y
Evaluate the following logarithms and confirm that they are correct using the inverse operation.a. log(0.1)b. log(4)c. ln(e)d. ln(2)e. log(100)f. log(976)g. ln(e10)h. ln(22.3)
Verify the following identities using a calculator.a. log(12) = log(6) + log(2) = log(3) + log(4)b. ln(10) = ln(20) − ln(2) = ln(15) − ln(1.5)c. ln(100) = 2 ln(10)
Convert the following angles to radians. Confirm the numerical value of cos(θ) with your calculator in both degree and radian mode.a. 30°b. 45°c. 60.24°
Determine all real roots of the function m(x) = sin(x2 − 4).
Demonstrate algebraically that the following statements are true for f(x) = x2 − 5x + 6, g(x) = cos(x), and h(x) = tan(x).a. i. The composite function (f ◦ g)(x) has no real roots.ii. The composite function (f ◦ h)(x) has infinitely many real roots.b. Find all real roots in part a(ii).
Determine all real roots of the function q(x) = cos2(x) + cos(x) − 2.
Where possible, determine an explicit expression for the inverse of each of the following functions.a. f(x) = x + ab. g(x) = x2c. h(x) = (4x − 1)3d. l(x) = cos(x) + xe. p(x) = ln(x)
A bank account offers an interest rate of 4% per annum. Calculate the accumulated value of the following investments made into this account.a. $1 made 10 years ago.b. $200 made 4 years 3 months ago.c. $3000 invested at t = 4 and withdrawn at t = 6.5.
State the function A(t) that gives the accumulated value after time t of a unit investment made in an account that pays compound interest at 7% per annum. State and interpret the domain and range of this function.
An investment of £200 was made sometime ago into an account that pays compound interest at a rate of 5.6% per annum. If the accumulated amount is now £352.09, determine how long ago the investment was made.
An investor is looking to invest £2m for precisely 2 years. Calculate the accumulated value of his investment in the following situations.a. The money is invested in a deposit account that pays a fixed interest rate of i = 3% per annum. As before, A(i) applies to each £1 and so is used as a
State a function that gives the accumulated value of $5.6m after 5 years under a compounding rate of i% per annum. State and interpret the domain and range of this function.
An investment of $100, 000 was made 2 years ago and expected to earn a fixed annual rate of interest. If the accumulated value is now $345, 340, calculate the interest rate.
Calculate the present value of the following liabilities under an interest rate of i = 7% per annum.a. £1m due in 6.5 years.b. £3.50 due in 4 months.c. £10,000 due in 30 years.
Use the accumulating function A(t) = (1 + i)t with domain t ∈ R to calculate the following. You should assume that i = 5.5% per annum.a. The accumulated value of $500 after 15.5 years.b. The present value of $6867 due in 4 years.c. The present value of $700m due in 35 years.
A project requires an initial outlay of £20,000 and generates cash flows of £5000 at each of t = 1, 2, 3, 4, and 5 years. If the outlay is funded by a loan at rate 7% per annum and the inflows are invested in a deposit account earning 7% per annum, calculate the following.a. The net present value
A particular project requires a single investment of $50,000 which is to be funded by a loan. The project is then expected to return $20,000 and $40,000 2 and 4 years later, respectively, which will be invested in a deposit account. Determine the maximum interest rate such that the project is
Determine whether the following functions are continuous at the locations indicated.a. f (x) at x = 5 whereb. g(y) at γ = 2 wherec. h(z) at z = 0 where f(x) = 43 43 +x²+1 for x = (-∞, -1] +x²+2 for x € (-1,00)
A business venture consists of an initial investment of £1m, requires a further investment of £0.5m after 2 years, and leads to an income of £1.25m after a further 2 years. Calculate the internal rate of return for this project and interpret your answer.
Calculate the following limits. 555555555 a. lim cos x X X-T b. lim x-1 In(x) x→1 c. lim *²-2x+2 x-2 x-2
Determine the equation of the straight line through each of the following pairs of data points.a. (x, y) = (0, 0) and (3, 3)b. (x, y) = (−1,−5) and (4, 0)c. (x, y) = (π, 0) and (π2,−π)d. (x, y) = (−10,−56) and (−100,−129)
You are given that, ifthenUse this result to demonstrate thatfor n = 1, 2, 3, . . .. lim f(x) = L₁ and lim g(x) = L₂ x-a
Determine whether the following function is continuous on the domain R. g(x) x2 - 4 for x € (-00, 4] 5 for x = (4,0)
Determine whether the following function is continuous on the domain R. 4.3 x³ 2x+1 for x = (-∞0, 1) for x = 1 - - 2x + 2 for x = (1,00) h(x) = {2 43 -
Confirm the following limits using a tabular approach. lim 8-0 sin (8) 8 = 1 and lim 8 0 1 - cos(8) 8 = 0
Explore the potential existence of discontinuities in the following functions. x-11x2+38x-40 x-3 a. f(x) b. g(x)= tan(x) =
Determine the tangent lines to the following functions at the location stated. a. m(p) =p³ - 2p + sin(2p) at p: = 2π. b.n(q) = e 9 = e-9 (sin²(aq) + cos²(лq)) at q = 0.
Use Eq. (3.4) to determine the derivatives of the following functions from first principles. a. f(x) = x² - 2x + 1 b. g(y) =e²y c. h(z) = 1/1/2 d. m(p) = 4ep²
Demonstrate the existence of the discontinuities found in Example 3.4 using left-hand and right-hand limits.Example 3.4Explore the potential existence of discontinuities in the following functions. x-11x2+38x-40 x-3 a. f(x) b. g(x)= tan(x) =
Determine the derivatives of the following expressions. State which approach you have used in each case. a. f(x) = x²–2x+1 x++25 b. g(x) = exp(sin z) h(y) = cos² y 4 cos y + 2 - c. d. m(p) = cos(p) sin (el)
State the gradient and y-intercept of the linear function f (x) = 5x + 2. Confirm your value for the gradient by calculating it between the following values.a. x = 10 and x = 15.b. x = 1000 and x = 1250.c. x = −100 and x = −10.
Explain how one might approach the following derivative. Determine it. d d.x n (ex-²-4)) sin
State and interpret the gradient of the following straight-line functionsa. f(x) = 2x + 100b. g(x) = −5x + 12c. h(x) = −2
Determine the locations at which the instantaneous gradients of the following functions are zero. Interpret the meaning of these locations. a. f(x) = ³²-2+5 3 b. g(y) = cos(y) +2
Determine the linear functions with the following properties.a. Gradient 2 and f (0) = 3.b. Gradient −1 and f (2) = 6.c. Passes through f (1) = 2 and f (6) = 12.
Derive expressions for the following derivatives. a. arcsin(4x) b. 이름 이름 이름 C. dx dx dx arccos(x – 1) - arctan(e*)
Use the expression for the instantaneous gradient (3.3) to determine the gradient of the following expressions at general position x.a. f (x) = 4b. g(x) = 2x + 2c. h(x) = x2
A bank account is such that it pays the following continuously compounding rate of interestwhere t is measured in years. Calculate the amount that should be invested now to cover a liability of $5600 due in 7 years’ time. 8= 5% for t = [0, 4] 7% for t€ (4, 10]
Use the definition of a derivative stated in Eq. (3.4) to confirm that the following statements are true. d -f(x) = dx df dx = f'(x) = lim 8 0 f(x+8)-f(x) 8 (3.4)
Visually confirm the gradient function obtained in Example 3.9(c) by plotting h(x) = x2 and its tangent lines at x = −1, x = 0, and x = 2.Example 3.9(c)Use the expression for the instantaneous gradient (3.3) to determine the gradient of the following expressions at general position x.c. h(x) = x2
Use a tabular method to confirm that lim 8-0 28-1 8 = ln(2)
Deposits in a bank account are known to accumulate with factor A(0, t) = te0.05t3, where t is measured in years. Determine the underlying force of interest paid by this account.
State the derivative of each of the following functions.a. f(x) = x10b. g(y) = yπc. h(z) = za−1, where a ∈ R is a constant.
Write down expressions for the following derivatives. a. b. d4* dx dлx dë dx
Write down the derivatives of the following functions.a. f (y) = 2y3 − 4y2 + y − 1b. g(x) = 4 exp(x) − 5 sin(x)c. h(z) = 10 cos(z) − 4 exp(z) + sin(z) − z
Write down the derivatives of the following functions.a. f (x) = exp(x) cos(x)b. g(y) = y2 sin(y)c. h(z) = z4 exp(z) cos(z) sin(z)
Use the chain rule to derive an expression for d dx (cos x)" for neN
Using the polynomial functions f(x) = xn, g(x) = xm, and h(x) = xn+m, demonstrate the consistency between the standard formula for calculating the derivative of a polynomial and the product rule.
Use the chain rule to find the gradient of the following function at x = 1. f(x) = e(In x)²-In x+2
If f (x) = exp(x), g(x) = 4x, and h(x) = sin x, state the following composite functions and derive an expression for the derivative with respect to x in each case.a. (f ◦ g)(x)b. (g ◦ h)(x)c. (f ◦ g ◦ h)(x)d. (g ◦ h ◦ f )(x)
Derive expressions for the derivatives of the following functions. a. f(x) = b. g(x) = c. h(y): = 22²-2 2+3 3x³+2x²-x+1 2x³+x²+2 1 2²-4
Use the expressions in Eq. (3.16) to simplify the answers in Example 3.22 where appropriate.Example 3.22Derive expressions for the derivatives of the following functions.a. f (x) = tan(3x + 10)b. g(x) = 1/ sin xc. h(x) = 1/ cos xd. l(x) = tan x sin x cosec (x) = 1 sin x sec (x) = 1 COS X cot x
Use Eq. (3.17) to write down the following results directly. dx log10 (x) b.arcsin(x) a. C. : arctan(x) = d dx 1 x In (10) 1 1-2- 1 /1+x
Derive expressions for the derivatives of the following functions.a. f (x) = tan(3x + 10)b. g(x) = 1/ sin xc. h(x) = 1/ cos xd. l(x) = tan x sin x
Derive the general rule expressed in Eq. (3.19). d dx log(x) = 1 x ln(a) for a > 0 (3.19)
Determine the required quantities given the stated constant nominal rates of interest.a. The accumulated value of $200 after 1 year under a nominal rate i0.5 = 4% per annum.b. The accumulated value of $100 after 2 years under a nominal rate i0.25 = 5% per annum.c. The present value of $10,000 due
Compute the following under a fixed continuously compounding rate of interest of 8% per annum.a. The accumulated value of $10,000 after 4 years.b. The present value of a $100 due in 15 years.c. The accumulated value at t = 5 of $20,000 invested at t = 4. Time is measured in years.d. The present
Determine whether the following functions are smooth (by the restricted definition 4.2) over x ∈ R. a. f(x) = 2x² - 2x + 1 b. h(x) = c. g(x) d. k(x) = x 2 43 +4 10x + 4 13 for x ∈ (−o, 1] for x ∈ (1,0) x2 for x = (-0, 0] for x = (0, ∞) for x € (-∞0, 0] for x = (0,00)
Demonstrate that the following functions are strictly smooth over R. a. f(x) = cos(2x) b. h(y) = er² c. g(z) = d. k(p) = 2²-2z+2 p² p²+2
The growth of a unit deposit in a bank account is given by the accumulation factor e0.05t2+0.02, where t is measured in years from the initial deposit. Determine the underlying force of interest that is being paid by this account.
Identify the location of any turning points in the following functions. a. f(x) = cos(2x) b. h(y) = er² c. g(z) = 2² - 2x + 2 d. k(p)=²+2
State whether the following series have a finite or infinite value. Calculate the finite value where appropriate. 10 15 a. ΣΩ, 10 × 1.2-1 b. Σ (5 + 2i) Σ1 × (3) d. Σ 1 2η π C. 2
Determine expressions for the second derivatives of the following functions.a. g(x) = 2x2 + x − 2b. h(x) = sin(2x)c. l(x) = x exp(2x2 − x)
Determine the third- and fourth-order derivatives of the following functions. a. f(x) = cos(2x) b. h(y) = er² c. g(z) = 2²-2z+2 d. k(p) = p² p²+2
Determine expressions for the fifth-order derivatives of the functions stated in Example 4.2.Example 4.2Determine expressions for the second derivatives of the following functions.a. g(x) = 2x2 + x − 2b. h(x) = sin(2x)c. l(x) = x exp(2x2 − x)
Investigate the smoothness of the following function first considered in Example 4.1(d). k(x) x3 for x = (-00, 0] for x = (0, ∞) R
Determine the range of each of the following functions. a. f(x) = cos(2x) b. h(y) = er² c. g(z)=z² - 2z+2 d. k(p) = p² p²+2
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
Determine the range of each of the following functions.Comment on any differences between the ranges of these two very similar functions. a. f(x) = x+2x+4 5x4+2 x+ 2x+4 5x4-2 b. g(x) =
An investor purchases an asset at time t = 0 for $100. If the investor believes the market price of the asset will evolve according to the functiondetermine the optimal time to sell the asset and the profit that he is expected to make. Note that t is measured in years. $150.4891 (t — 7)²
Due to staffing issues, the daily output of a fresh sandwich manufacturer based in New York varies over time and is modeled by the functionNote that t ∈ [0, 365] is measured in days from 1 January. The manufacturer knows that the lunch-time market in lower Manhattan has a near insatiable appetite
Use second derivatives to confirm the classifications in Example 4.7.Example 4.7Classify 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.
Compute the maximum price at t = 0, the effective duration, and the duration of the following projects if the required rate of return is 6% per annum (expressed as a compounding rate).a. Annual proceeds received at the end of each of the next 5 years, starting at $50 and inflating by 5% per
Find and classify all stationary points of m(x) = x3 − 6x2 + 12x − 9.
Find and identify all stationary points in the following functions. Use this information to help determine each function’s range over R. a. p(x) = x³ b. q(x) = x - 4x² c. r(x) = d. t(x) = 4x² = x+4 - 4x4-2x+1 3x²+1 x²2²-2x+2 x-3
An investor is willing to pay $1010 for an investment with duration 10 years. If these values are based on a required return of 3% per annum (expressed as a compounding rate), estimate the new price in the following situations.a. The required return increases to 4% (expressed as a compounding
Use algebraic techniques to determine the range of f(x) = 3x² - 2x + 2 2x² + x + 1
You are given that x8+4 I - zx + ²x - gt f(x)
Demonstrate that the following series are convergent and determine their value. a. Σι(-0.99)k+10 b. Σ12(+)5 Σvo(−1)" (0.2)"-7 d. ΣΩ_100 10 × (0.95)2 C. 5j
For i = 7% per annum, an investment is known to have a price of $123 and an effective duration of 7.4. Estimate the new price if the required return changes toa. i = 6.5% per annum,b. i = 7.3% per annum.

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