(a) Suppose z1, z2 € C and z satisfies the conditions Re(2) > 0 and Im(2) = 0. Show %3D then that Ln(z12) = Ln(z1) + Ln(22). (b) Consider following statement: Ln(-1+ i)" = n Ln(-1+i) holds for all n e N. Determine now whether the statement is true or not, by either proving it for all neN, or by exposing a value for n EN for which it is not true.

Q: log(-x) +log(2 – x) = log( 3)

A: Since you have posted multiple questions, as per our policy we will answer first question. Please…

Q: If E (X) = 1 and Var (X) = 4 then E (X² + 2X – 1) = :- 1)

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Q: Let T(x) and T(y) be the complete future lifetimes for the lives x and yrespectively. If T(x) and…

A: Given: The whole future lifetimes for the lives x and y, respectively, are functionsT(x) and…

Q: Suppose f= f(x) satisfies f(1) = 7 and f(1.002) 7.008. Estimate the value of at x = 1. dt %3D

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Q: dx J x. In5 (x)

A: Given that: ∫dxx·ln5x…

Q: ii) 2 log, (x)– log2 (3x – 4) = log, (x – 1) – 1

A: We make the coefficient 2, exponent inside the log.  And for 1 we replace it by log22 because…

Q: Show that E(X-µ)³ = E(x³) −3 [E (x²) ][E(X)] + 2[E(X)]³

A: The linearity of expectation property: With the assumption that, Xi's (i=1,2,...n) with n positive…

Q: Find a formula for f (z) if f(z) = I+1 f '(z) =O ( )'(2) =D-

A: We will use concept of inverse function and differentiation of a function.

Q: Find f"(x) for the functions x² +5x f(x) = ex b. f(x) = (x² – 1)e* a.

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Q: If g(x) = max {x, 1/x} what is g(x) when x ϵ N2 (.5)?

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Q: If f(x) = (2x – 1)(4x + 5), then a. f( – 1) = b. f'(– 1) = | c. f''(– 1) =

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Find the d. c. of T from definition.

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Q: Dat

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Q: Suppose our company is selling some product that has been on the market t years.The total number of…

A: So given that, the total number of products sold over those t years in thousands of units is…

Q: By using Rodrigue's formula, prove the following relation Ph+1(x) – Ph-1(x) = (2n + 1) P,(x) 4- %3D

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Q: If f(x) = an(x – 1)²n for all æ, then f'(3) is equal to n=0 OA. 2nan-122n-1 n=1 В. 2 2na,22n n=1 OC.…

A: According to the problem, we have

Q: Given function G(x)= 3√2t² - 2 dt. Which V one of the following is true? A) G(3) <G'(3) <G"(3) B)…

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Q: Solve for F(s) L[(1+ e2t)²] a. 1 2 1 s's-2's- 4 Ob. 1 2 s - 2 s's+2 1 + s's+2' s + 2 d. 1 + s's-2' s…

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Q: (a) Let f(x) = In x and g(x) = Va. Show that f(x) and g(x) are both increasing when x > 0. (b) Let…

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Q: b. 8(x) = E ()" k = 1

A: domain: where function is defined

Q: If f(x) = (2x – 5)(3x + 5), then - a. f(1) = b. f'(1) = %3D

A: Solution -  Given function is as -  f(x) = (2x-5)(3x+5)we can expand it further - f(x) =…

Q: 2. The equation of the horizontal asymptote of the exponential function f(x)=7(0.5*4)–-1 is: A) y =…

A: The equation of the horizontal asymptotes of exponential equation f(x)= 7(0.5x-4)-1

Q: a) Given a surjective function f : [-1, 1] → [-1, 1], prove or disprove: f-l (f{0}) = {0}. b) Find…

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Q: 5. Let X1, X2, .. be iid Exponential (0), and Y1, Y2,....be iid Expomential (20), = min {n :I (X, >…

A: Given X1, X2, .... be iid Exponentialθ, and Y1, Y2, .... be iid Exponential2θ and X's and Y's are…

Q: AR.7.38 Find f(x) and g(x) such that h(x) = (f o g)(x). h(x) = (1 – 5x)? Suppose that g(x)% = 1-5x.…

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Q: Which of these functions have a domain of (-*, 0)? Select all that apply. A f(x)= e*+2 B g(x)=…

A: We will find such Fucntions.

Q: If ln x + ln(x − 3) = ln(7x), then z

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Q: Suppose f(z) = In z and g(z) = ecos. Let h(x) = f(f(q(=))). (a) Find a simplified formula for h(z)…

A: Solution of given question is followed by:

Q: (ミ) (20) %D %3D

A: Given R(z) = 2z2+2S(z) = 3z Then (R∘R)(z) = R(R(z))= R(2z2+2)= 2(2z2+2)2+2 = 24z4+4+8z2+2=…

Q: If g(t) = 2r +9t2-7 find. %3D g(0) = -7 g'(0) = -7 g" (0) = g"(0) = %3D g(4) (0) = g(5)(0) =

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Q: g (x) = V2x-1- 1. Which function is the inverse of of (x) = 2(x+ 1)° + 1 of (x) = 2 (z+1)3 +1 ° f…

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Q: Find ds ldx if y=3) X(X+1)(x-2) (x² +1) (2x+3)

A: Note:-Since you asked multiple questions, according to our policy we can answer only first question…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.