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introduction to actuarial and financial

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

Perform the following indefinite integrals. a. f cos(5x) dx b. 10 sin (0.7+2) dy (e5z + ln(32)) dz C. d. f (cosec² (3p) - sec²(8p)) dp f sin(q) cot(q) dq e.
Perform the following indefinite integrals. a. fx³ (x4-2)³ dx b. f cos(y) esiny dy / dz S 2z 2²+5 C.
Determine the following indefinite integrals. a. f 4 dx b. f -2 dy f7 dz C.
Determine the following indefinite integrals. a. f (3x²+2x+1) dx b. /(2√x+3) dx c./ (+x++x) dx C. d. f (7x-1/2 + 5x+ + 3x³ + 2x+17) dx
Perform the following indefinite integrals. fx sin x dx b. Sy²er dy 6y a. fe²z sin z dz d. fp/1+pdp C. e. f sin(q) cos(q) dq
Demonstrate Eq. (6.5) with f (x) = x and g(x) = x2. [ f(x) > (x) × g(x) dx ‡ f f (x) dx × [ g(x) x dx (6.5)
Perform the following indefinite integrals. S b. S a. C. S d. / f e. 1+x² dx -2 y²+4y+5 dy 1 4-82-42² 6 25p²-5p+1 dz dp -2 S q²+49-5 dq
Determine the following indefinite integrals using a suitable change of variables. f exp(16x - 2) dx b. f(4y + 11)-² dy c. f cos(52 +0.1) dz d. f sec² (10u + 1) du a.
Using the methods stated, demonstrate that the following indefinite integral is correct.a. Integration by parts. You are given thatb. The substitution u = sin x. [² 1 kx=125 sec x tan x + ln | secx + tan x| + c 2 sec³ x dx =
Use an appropriate change of variable to determine the following indefinite integrals. a. √√√x√2+x5/4 dx b. f sin y cos5 y dy C. f secz tan z dz d. fu² exp (u³-5) du
Derive the standard expression for the roots of a quadratic polynomialBegin by obtaining the completed-square form of the polynomial. a₁x² + a2x + a3 for a; E R and a₁ # 0.
Derive the general result,Note that this was stated in Eq. (6.18). 1 a₁x² + 2x + az J dx = 2 4a1a3 - az arctan 2a1x + a₂ 4a1a3a²2 - +c
Use the change of variable approach to determine the indefinite integral of tan x.
Use integration by parts to perform the indefinite integral S 2xe -4x dx
Perform the following indefinite integral and comment on your result,Maclaurin series. k=0 (-1) ¹x2k (2k)! dx
Use integration by parts to compute the following indefinite integrals. a. fe²x (4x - 1) dx b. fy³e-dy C. . fe² (2²-1) dz d. f es sin s cos s ds
Use integration by parts to determine the following indefinite integrals of standard functions. a. f ln x dx b. arcsin dx C. farccos dx d. farctan dx
Determine the following indefinite functions. a. b. f c. f x + 4 x+2 dx 35x6+ 25x4 (x²+x5-π)5 2x + 4 3x² + 6x-24 dx dx
Determine the following indefinite integrals using the completing the square method. a. S b. f 1 x²8x + 18 1 y² + 2y +4 1 22²-82+20 C. S dx dy dz
Obtain the completed-square form of the following quadratics.a. x2 − 8x + 18b. x2 + 2x + 4c. x2 − 4x + 10
Determine the following indefinite integral -1 √ √7-2x = x2 ¹²x² dx -
Compute the following definite integrals. a. 30 √3 (x-³ C. d. b. f ² (³+2)5 dy Size-2² dz 10 - 2x + 1) dx e. f sin(p) cos pdp 2² 3. 4q³-1 dq
Evaluate and sketch an interpretation of each of the following definite integrals. a. (3x²+2x+2) dx b. f3 In x dx -2 3 1/2 dx C. d. fe* dx
Compute the following definite integrals and interpret your answer. a. b. fe-4y dy r2 е-лх dx
Explore Eq. (7.5) using the definite integral 10 [₁0 S (x² + 2) dx
Compute So 1 5+ (x - 2)² dx
Plot and evaluate the following definite integrals. a. S₁ (x² - 6x +9) dx b. S₂(2³-32² +32 − 1) dz -
You are given thata. Calculate the area Aa between the two curves bounded between x = −5 and 5.b. Calculate the area Ab bounded between the two curves. f(x) = |x and g(x)=x²-2
Use the change of variables approach to determine the following definite integrals. a. R² dx b. ¹/2 cos y sin² y dy x+1 x²+2x+1 π/2 f0.5 ze²² dz C.
Use a change of variables approach to prove Eq. (7.6) for odd and even functions. ff(x) dx = 0 f(x) is an odd function g(x) dx = 2 f g(x) dx g(x) is an even function (7.6)
Compute the area between the following curves. Illustrate your answers.a. f(x) = x3 and g(x) = x2 bounded between x = 0 and 0.5.b. f(x) = x3 and g(x) = x2 bounded between x = −1 and 5.
Evaluate 3 [²³. xe dr
Calculate the areas bounded between the following curves. Illustrate your answers.a. f(x) = 2x2 and g(x) = 4 − xb. h(γ) = γ2 and k(γ) = + √γc. l(z) = 1 and m(z) = 2 − z2
Evaluate 2π П et cos x dx
The equation x = γ2 − 3 implicitly defines a curve on the x-y plane. Calculate the area enclosed between this curve and γ = x − 1.
Determine the accumulated amount at t = 12 of an investment of $500 made at t = 0 under the action of the following force of interest.where t is measured in years. 8(t) = 4% 5% 0.6% per annum for t = [0,5) per annum for t = [5,9) per annum for t € [9,20]
Compute the following accumulations under the given δ(t) defined on t ∈ [0,∞] and measured in years.a. The accumulated value at time t = 1 of a unit investment made at t = 0 under δ(t) = δ per annum.b. The accumulated value at t = 10 of £100 invested at t = 0 under δ(t) = 0.005t per
Determine the present value of $100 due at time t ∈ [0, 20] under the force of interest stated in Question 7.8.Question 7.8Determine the accumulated amount at t = 12 of an investment of $500 made at t = 0 under the action of the following force of interest.where t is measured in years. 8(t)
In each case, determine the area between the stated curves for x bounded between 0 and π/2.a. f(x) = sin x and g(x) = cos xb. f(x) = −sin x and g(x) = −cos x
An n-year continuously paid unit annuity is a continuous payment stream paid at a constant rate such that a total of $1 is received each year. Determine an expression for the present value (at t = 0) of an n-year continuously paid unit annuity under a constant force of interest δ.
Determine the area between the curves f (x) = x2 +2 and g(x) = x + 4 bounded between x = −2 and 3.
Use Wolfram Alpha to compute the following. 10 x5 ²5 dx a. b. The area between f(x) = x² and g(x) = 2 - 4x³ bounded between x = -2 and 5. c. The area bounded between the two functions h(x) = 2x² and k(x) = 4 - x
Determine the area bounded by the functionsa. f(x) = x and g(x) = 4x2b. f(x) = 2 and g(x) = 1 − x2
Express the following in terms of i. a. √-m² for m € R b. i¹0 C. (ai)7 for a € R d. (-)101/2
Compute the following under the given δ(t) defined on t ∈ [0,∞] and measured in years.a. The present value at t = 0 of $550 due at time t = 2 under δ(t) = 0.02t3 per annum.b. The present value at t = 5 of $10,000 due at time t = 10 under δ(t) = 0.05 + 0.001t2 per annum.
Classify the following as real, imaginary, or complex. Plot each zi on a single Argand diagram.a. z1 = 3b. z2 = 2 + 4ic. z3 = 2id. z4 = −1 + 3ie. z5 = −2 − 4i
Classify the following numbers as either real, imaginary, or complex.a. 4b. 3ic. 10 + id. −5 − πi
Obtain the simplified polar form of each zi in Question 8.1.Question 8.1Classify the following as real, imaginary, or complex. Plot each zi on a single Argand diagram.a. z1 = 3b. z2 = 2 + 4ic. z3 = 2id. z4 = −1 + 3ie. z5 = −2 − 4i
Simplify the following expressions. 2-6i 1+5i b. 1-5i 1 a. C. d. 1+2i (3+2i)(4+4i) (2-3i)²
Determine the following for z3 = −2 + 6i, z4 = 20 + 0.3i, and z5 = −π + 15i.a. z3 + z4 + z5b. z3 + z4 − z5c. 4z3z4d. z3z4z5
Determine all roots of the following polynomials and express each root in simplified polar form.a. f(x) = x2 − 4b. g(y) = y3 + 3y2 − 6y − 8c. h(z) = (z2 + 1)(z2 + 2z + 2)d. m(k) = k5 + 32
Obtain and interpret the discriminant of the following quadratic functions. Determine the two roots in each case.a. h(z) = z2 + z + 1b. j(z) = z2 + 4z + 1c. k(z) = z2 − 5z + 1d. l(z) = 2z2 + z + 2e. m(z) = 4z2 − 8z + 4
If z = 1 + i, determine the following in simplified polar form of the following.a. zz*b. z10c. 1/zd. 1/z5
Use the Maclaurin series of the exponential function to prove Euler’s formula (Eq. 8.10). a + bir (cose + i sin 0) = rei (8.10)
Determine all roots of the function f (z) = z3 − 1 for z ∈ C.
Use your knowledge of quadratic functions to determine all roots in the complex plane of the quartic function g(z) = z4 − 1.
Derive expressions in terms of sin(x) and cos(x) for the following when x ∈ R.a. sin(4x)b. cos(4x)
Prove the following trigonometric identities for u, v ∈ R. 1+cos(2u) 2 a. cos² (u) = b. cos³ (u) = (3 cos(u) + cos(3u))
Write down the principal argument and modulus of the product z1z2 for a. 2₁ = 2 (cos+ i sin ) and 22 = 4 (cos+isin) b. 2₁ = √3 (cos+isin) and 22 = √5 (cos+isin 2) c. 2₁ = 2 + 1 and 22 = 21 d. 2₁ = 2 + 10i and 22 = −1+i
Express the following complex numbers in polar form. Illustrate each complex number on an Argand diagram.a. z = 4b. z = 2ic. z = 2 + 4id. z = −1 + 2ie. z = −1 − 3if. z = 2 − 5i
Write down the principal argument and modulus of the quotient z1/z2 for (cos+isin) and 22 = 4 (cos+isin) a. 2₁ = 2 b. 21 = √3 (cos+isin) and 22 = √5 (cos ² + i sin 2) c. 21 = 2+i and 22 = 21 d. 2₁ = 2 + 10i and 22 = −1+i
If z1 = 4, z2 = 3i, z3 = 4 + 4i, and z4 = −2 − 2i, write down the modulus and principal arguments of the following expressions. a. 212223 b. 21222324 C. 21 2224 d. 212224 23
User Euler’s formula to determine the following expressions in principal polar form.a. (5 + 5i)5b. (2 + 4i)12c. (−2 + 2i)6d. (−1 − i)102
Determine all distinct complex values of (3 + √3i)1/n fora. n = 3b. n = 5
Use Eqs. (8.13) and (8.14) to prove the following identities for real variables.a. sin2 u = 1/2(1 − cos(2u))b. cos u cos v = 1/2(cos(u + v) + cos(u − v)) sin x = 1 21 1 COS X=- 2 (eix - e-ix) eix + e¯ix) (eix (8.13) (8.14)
Use Wolfram Alpha to perform the following.a. Determine the three values of (πi)2/3.b. Determine the polar form ofc. Determine all roots of the cubic polynomial g(x) = x3 − 1 (1+2i)7 (1-21)³
Plot all values of the following on the complex plane.a. i3/2b. (1 + i)2/3c. 11/5d. (−2)7/3
Use De Moivre’s formula to prove the following double-angle identitiesa. cos(2x) ≡ cos2 x − sin2 xb. sin(2x) ≡ 2 sinx cos x
Where would 0 appear in the Venn diagram of Figure 1.2? Z N R Figure 1.2 Venn diagram of the real number systems. J
Identify the real number systems that the following belong to. a. 5 b. 6.48763 2 C.ㅠ d. 43 7 e. -6
Give three examples for each of the following number systems. a. R+ b. Z RNNOI c. N d. Q e. J
Translate the following mathematical statements into words. P a. Vx € (-∞,0), x² e R+ U {0} b. VpZ,qe Z\ {0}, £ € Q c. 3y = Z: y $a. Vx (-0, 0), x R+ U {0} b. VpZ,qe Z\ {0}, Q P c. y Z: y < 3 and y'is odd d. Vze R+, z>0 e. Va e R,$
Interpret the following mathematical statements in words and give two examples in each case. You should work in the set of real numbers, R. a. y > 5.4 b. z ≤ 10 c. x + 2 > 4 d. y = x and y=z⇒x=2 e.x~1/2 f. ql = 7
In the particular case that A = {0, 1, 2, 3, 4, 5}, B = {−2,−1, 1, 2}, and C = {2, 3, 4, 5, 6}, demonstrate that each of the following identities are true. a. AU (BN C) = (AUB) N (AUC) b. An (BUC) = (ANB)U (ANC) C\ (ANB) = (C\A) U (C\B) d. B\ Ø = B e. (B\A) nC = (BNC) \ A $a. AU (Bn C) = (AUB)n (AUC) b. An (BUC) = (ANB) U (ANC) C\ (ANB) = (C\A) U (C\B) d. B\ = B e. (B\A) nC =$
Give all possible values of x that would satisfy the following statements concerning the sets in Eq. (1.1).a. x ∈ Ab. x ∈ A and x ∈ Bc. x ∈ B and −x ∈ C A={-1, 1, 2} B = {0,2,3} and C = {-3, -2,4}. (1.1)
State whether each of the following expressions are identities or equations. Where appropriate, use any method to identify the values of γ such that the equations hold. ². y²-16 = (y-4) (y + 4) a. b. y + 9 = 10y² c. y³ + 25ay = (25 + a)y² where a € R is a constant d. -y + 5y³9y² + 7y= 2
Using the sets in Eq. (1.1), determine the relative complement of A in B. A = {-1, 1, 2} B = {0,2,3} and C = {-3, -2,4}. (1.1)
Interpret the following mathematical statements in words and give an example in each case. You assume that Ω = R. a. x ER b. y € Z EZ c. 2 € {0, 1, 2, 3} U {5, 6} d. y € ZnR+ e. R+ U{0}nZ = N f. B\B=0 $a. x ER b. y Z EZ c. 2 (0, 1, 2, 3} U {5, 6} d. y ZnR+ e. R+ U{0}nZ =N f. B\B=0$
Use an algebraic method to find intervals for z that solve the following inequalities.a. z − 1 > 0b. 2z + 1 ≤ −2c. 4z < 3z + 2d. z − 4 ≤ 2z + 1
Interpret the following mathematical statements in words and give two examples in each case. a. x € [100,00) b. y € (0, 10] c. p = [0, 1] d. € (-9.9, -9.8)
Translate the following mathematical statements into words. Give a numerical example in each case. a. Vx, y € R, xxy € R b. Vp, q € R,px q € R+ c. 32 € Z: 2 < 7 and 2 is odd. d. VpQ, 3q € Q: 3p = q
Demonstrate the intuitive fact that it is possible to find a rational number that approximates the value of π to any finite level of accuracy.Use concise mathematical notation to express that this is true for all real numbers.
Using an algebraic approach, classify the following expression as an equation or an identity: ²2² +1 y(2y² - 1)(y - 1) 1 Y 6y +3 2² - 1 + 2 y - 1 (1.4)
Use Wolfram Alpha to investigate whetheris an equation or an identity. (x - 2)² = x+8x³ +24x² - 32x + 15
Find the values of q such that the following inequality is true: 9³-39² - 4q+12 ≥ 0
Use plots of the functions identified in Question 2.2 to comment on the properties of each function. Your answer should include comment on the following properties.i. Domain.ii. Range.iii. Asymptotes.iv. Confirmation of the type of mapping identified in Question 2.1.Question 2.2Determine which of
Find the values of q such that the following inequality is true: q³ - 3q² - 4q+12 > 0
Determine 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 + 4z² +24z - 32 c. hy) = yx (0.5 - cos(y)) d. 1(p) = (p²4) x cos(2p) e. m(q) = √√√9-2q tan(q) 45
Determine 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 + 4z² +24z - 32 c. h(y) = yx (0.5 - cos(y)) d. 1(p) = (p²4) x cos(2p) e. m(q) =
State whether the following mappings are one-to-one, many-to-one, or one-to-many. Also state whether each is also a function.a. g(x) = x + 10b. f(z) = (z − 2)2c. h(p) = √p
Use an algebraic approach to find the roots of the function g(y) = y² - 4y +3
Determine the parity of the following functions.a. f(q) = q2b. g(y) = γ3c. h(z) = |z|d. l(x) = x − 1
Plot the following function using your preferred method, for example, using either Excel or the plot command in Wolfram Alpha. Use this plot to discuss the properties of the function. h(y) = y² + y² - 24y³ − 4y² + 80y
If g(x) = (x + 1)(x2 − 4x + 3) find the values of x such that g(x) > 0.
Determine the values of γ such that h(γ) ≤ 0, where h(γ) is stated in Example 2.5.Example 2.5.Plot the following function using your preferred method, for example, using either Excel or the plot command in Wolfram Alpha. Use this plot to discuss the properties of the function. h(y) = y³ +y¹
Determine the domain and range of the function f (x) = +√x on R.
Use an algebraic method to find intervals of z that solve the following inequalities. Confirm these with a computational plot. a. b. 23 ³ - 2z² - z > -2 ²+2z+5≥0 c.-2³-22²-2z> 0 d. ²-4≤0 C.
If f (x) = x2 +2 and g(x) = x3 − 2x2 − 3, derive expressions for the functions formed bya. f(x) + g(x)b. −2f (x) + 4g(x)
If f(x) and g(x) are as defined in Example 2.7, derive expressions for the functions formed byExample 2.7If f (x) = x2 +2 and g(x) = x3 − 2x2 − 3, derive expressions for the functions formed byf(x) + g(x)−2f (x) + 4g(x) a. f(x) = g(x) b. (f(x))² x (g(x))−¹ = ƒ²(x)
Explain in words why the following statements are true.a. A second-order polynomial must have either no real roots, two real roots, or one repeated real root.b. An odd-ordered polynomial must always have at least one real root.

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