Prove for r ≠ 1 that ∑_(k=0)^(n-1)(rk)=(1-rn)/(1-r). If ∑_(k=0)^(n-1)(rk), then compute s(1−r) = s−rs, and solve for s.

Q: Note that 1/4 ≠ 1/2; but (1/4)^(1/4) = (1/2)^(1/2). Use the hint "y = x^x" to prove that there…

A: Please check next step for details.

Q: Find f. f"(x) = −4+ 8x - 12x², f(0) = 4, f'(0) = 12 (Prob. 1,

A: Consider the derivative function and initial conditions, f''x=-4+8x-12x2,f0=4,f'0=12 The objective…

Q: Calculate s'(n) (8n + 5 s(n) = 2n-6/ 6

A: The given function is s(n)=8n+52n-66 To find the s'(n). Derivative formula:…

Q: If f(x) O _ x9 (4x +11)³' 99x8 (4x + 11)4 O 15x +97 4x + 11)4 0x8 x (32x + 109) (4x + 11)4 O x³ (24x…

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Q: dH f(5) = 8, f' (5) = 4 f(26) = 8, f'(26) = 6, g(5) = 1, g'(5) = -9, and H(x) = f(r² + g(x)). Find…

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Q: Solve the IVP, explicitly if possible. 3y x+9(-7)=1 (x– 7)* 8. y=(x+9)3 (x + 9|) yl = (x+9|)

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Q: Prove that for integers y=1 the equation 3h-n (2x₁ + x₂ + 4x₂) +3² (~1(2x₁+x₂+.+xn-1)+...+3^-1 (2²₁)…

A: We will use contradiction to prove it.

Q: {In(æ² + xy+y³)} Find

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Q: The auxiliary equation of  of  y'''-4y''-4y'+16y=0  is r^(3)-4r^(2)-4r+16=0  . The auxiliary…

A: Given that fr=r3-4r2-4r+16, r∈3,6

Q: b) y =1+ f(z)=2x-2 O 1ks f-(z)=-1 O 2 34s (2)=2x-1

A: Step 1 Replace y with x and x with y  Step 2 Then solve for y in this equation  Step 3 Replace y…

Q: Find f. f(x) = f'(x) = −2+ 24x 12x², -x² + 4x³-x² + 14x+8 x - f(0) = 8, f'(0) = 12

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Q: Suppose f(x) = N(h) + a,h + a,h² + azh³ + --. The value of N, (h) using Richardson's extrapolation…

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Q: 3. Divide x+4x-3x-2 by X-2) then divide if CX-2) isa factor of x² t4x-3y-2 why?

A: We have to divide (x3 + 4x2 - 3x - 2) by (x - 2)  Then ,

Q: How many solutions does x^2+bx=c have if c< -(b/2)^2?

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Q: f(t) = e^3t - cos(6t) - (e^3t)(cos(6t) * O F(s) = 1/(s-3) + s/(s*2 + 6*2) - (s-3)/(s*2 - 6x + 45) O…

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Q: X(s)=(s^2+6s+7)/(s^2+3s+2) and Re{s}>-1 what is x(t)?

A: We are given : X(s)=s2+6s+7s2+3s+2 ⇒X(s)=s2+3s+2+3s+5s2+3s+2 ⇒X(s)=1+3s+5s2+3s+2 ---(1) Now,…

Q: If f (x) 2e²x Then compute f" (0) + 2f' (0) + 2ƒ (0)

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Q: E (3x(9 ^ b) V Jaæ(p ^ 0)) = 3x(p A (4 V 0))

A: To prove ⊢(∃x(φ∧ψ)∨∃x(φ∧θ))≡∃x(φ∧(ψ∨θ)) Here we have to use existential quantifiers i.e. notation of…

Q: 8. Find all the roots of f(x) =x* - 4x' + 6x² – 4x+5 if 2-iis one root.

A: Topic- roots of polynomial

Q: A company determines that monthly sales S(t), in thousands of dollars, after t months of marketing a…

A: Find S'(t)?

Q: Use L and L to solue y' t 3y= f4) y)=5, where f4), (32)

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Q: If Ax) = 2x+ 1and g(x) = x, Find g0)) %3D

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Q: 8. Find all the roots of f(x) = x' - 4x' +6x² – 4x+5 if 2– i is one root.

A: We will find all the remaining roots as following.

Q: P(0) = {0} O P(A^) \ {0} C P(A)°

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Q: 18) Suppose P'(x) = 5ex - 6x³+ 4x and P(0) = -9. Find P(x).

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Q: I got to the step where I have (5h/h-1)/h. I think I can multimply (5h/h-1) by (h/1). This results…

A: We have to find actual expression for the given expression.

Q: Substitute the value of y' in 1 + (y)²

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Q: x) [in(3x²-36)]². Then f'(x) = A. 12x In (3x²-36) 3x²-36 O A. O B. O C. O D. O E. 2 In (3x²-36)…

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Q: Show that the f(x)=x-(In x)^{x} =0, has at least one solution in the interval [4, 5]. O f(4)=0.31,…

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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.