Determine x[2] and x[3] given (a) 2x[n+2] – 5x[n+ 1] = 4n, x[1] = 2 (b) 6x[n] – x[n – 1]+ 2x[n – 2] = n² – n, x[0] = 1, x[1] = 2 (c) 3x[n – 1] + x[n – 2] – 9x[n – 3] = (n – 1)², x[0] = 3, x[1] = 2 %3D %3D -

Q: 1) Use reduction of order method with y₁ (x) = e³x to find a second solution y2 for the DE: y" - 9y…

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Q: 1 1. Solve the rational equation x - x - 6 x- 3 x +2 and state any NPV's.

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Q: The polynomial of degree 5, P(x)P(x), has leading coefficient 1, has roots of multiplicity 2 at…

A: Definition used -             A polynomial with the given degree, roots, and multiplicity of roots…

Q: 4-The equation x^2+2x-3=O has only one root in the open interval O a. (-2,0). O b. (-1,1). O c.…

A: Thus is a quadratic equation we can find the roots of the equation by solving them

Q: 1. Find the Wronskian for y1 = x, y2 = x² and y3 = x³. 2. Find the Wronskian for y1 = sin 3x , and…

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Q: Residue of at z = -3 is (z+3)² Select one: a. -2 b. 2 C. 3 d. -1 е.

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Q: %3D 1. Find the critical points forf (x) = 3x +

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Q: X1 - x2 +1 = 0) 1. Solve : x,? + x,² = 4 } with seed values x1(0) = 0.8 and x2(0) = 1.8. Compute up…

A: x1 - x2 + 1 = 0  =>  x1 = x2 - 1  x12 + x22 = 4 =>  x2 = (4 - x12)1/2    x1(0) = 0.8, x2(0) =…

Q: ¤1 + ag – r3 = 0, -2r1 – 12 + 5r3 = 0, -#1 + 4.x3 = 0 find the Rank(A), where 1 1. -1 A = -2 -1 -1…

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Q: 5. Find the domain of the rational function f(x) = 3x 2 - 8x - 3 2x 2 + 7x- 4. D a. D(-co, -4) U…

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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: 2. Find all the critical numbers a. f(x) = 2secx (0,2n) b. g(x) = (x² – 4)3

A: Critical Numbers : Critical numbers are the numbers where the function is defined and its derivative…

Q: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: Given: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

Q: Q4: Use the Gauss-Seidel iterative technique to find approximate solutions to 10x -x +2x3 -x +11x2-…

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Q: 2.2 Solve y = x – y + 1 for y(0) = 1 and x = 0(0.1)0.3 , using the Runge-Kutta method of order four.…

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Q: Find a formula for the polynomial P(x)P(x) with degree 3 a root of multiplicity 2 at x=1x=1 a root…

A: As we know that the equation for a polynomial when the roots for the polynomial is at x=a,x=b,x=c is…

Q: -n+1 Xn 4 Xn = A Lyn -1 6 . Pn+1 –1 by using the method of elimination. II

A: Given:- xn+1yn+1=Axnyn=24-16xnyn

Q: &x +4 OI. Express as partial fractions. *(1 – x)(2 + x) b Hence or otherwise expand; 8x +4 in…

A: To express as a partial sum and state the set of values of x for which the expression is valid.

Q: n-1 n-1 k n-2 n-2 - = P k X6n-10 X6n-9 = 4 - n-1 п-1 n-1 h k k n-1 X'6n-8 = r X'6n-7 = P h - . k - 4…

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Q: 2. Find the inverse of A = 1 1 1 1 2 -1 3 1 -1 2 3 H2H2 1 using Gauss-Jordan method.

A: The solution is given as

Q: x2n+1 El-1)"; 2n +1 n=0

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Q: 1. max z = 2x1 + 3x2 X1 + 2x2 < 10 3x1 + 4x2 S 25 s.t. X1, X2 2 0; x1, X2 integer

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Q: One solution of ay" + (3x + x²)y' + (1 – x)y = 0 about o = 0 for a > 0 has the form Y1 %3D -3+v5 An…

A: Given ODE is x2y''+(3x+x2)y'+(1-x)y=0        (1) We can write it as y''+3x+x2x2y'+1-xx2y=0…

Q: Find all values of n such that y = xn satisfies x2y" - 12y = 0

A: To solve the equation x2y''-12y=0, we first need to find y''. y'' can be obtained by differentiating…

Q: Consider three systems s1 s2 s3 as responses to a complex exponential input e ^ j5t are specified…

A: Given the three systems s1,s2 ,s3 as a response to a complex exponential input ej5t. Such a that…

Q: The coefficient recursion relation for the solution Y1 = E an x"+1 of the %3D n=0 equation x2y" –…

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Q: A polynomial of degree n 2 1 has exactly zero(s) if a zero of multiplicity m is counted m times.

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Q: :Find the first 3 terms for (1 (-1)*1 (2n – 1)!J n+1

A: We have to find the first three term of the given series.

Q: X+5x+1 After decomposing the rational function x³-6x²+9x into partial fractions without evaluating…

A: Given problem:-

Q: 3. f(x) = x2- 4x- 5 4. f(x) = X2 - 9x - 36 (x-12)(x+3) roots are x=12 x=-3 5. f(x) = x2+ 3x- 40 6.…

A: Given3)f(x)=x2-4x-55)f(x)=x2+3x-406)f(x)=x2+22x+21 To find Roots of above

Q: a(1 – rª ) n³(1-0.5'g(n) ; (1 – r) (1-0.5) (n³ – n³+0.5!g(n)) 0.5 =(2n3 – – 2n² ) 2 * n³ * 0.5'g(n))…

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Q: 5. (a) Find the real constants P, Q, R, for which 3 P Qx + R x2 – x + 1° x3 + 1 x + 1 | Show that V3…

A: 5(a)  3 / (x3+1) = P/(x+1) + (Qx+R)/(x2-x+1) ......(1) Now, 3/ (x+1)(x2-x+1) = P/(x+1) +…

Q: f(x) = xsinx (-1)"x2n (2n+2)! .2n+2 OA. (-1)"r2n+2 (2n + 1)! O B. n=1 (-1)"x2n+1 (2n+ 1)! C. 1.…

A: Maclaurain series

Q: A fundamental set of solutions of x' = 3 2 x is: 1 (w) x. - “(;). x-() (F). (1). *--(;) (a) x1 = et…

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Q: 1 9. A fundamental set of solutions of x' = | -3 -1 3 x is: 3 2 -2 2 (a) x1 = e-2#| -3 , x2 = e-t X3…

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Q: 15-18. Find the extrema of f(x) = 3x - 4x³ on the interval [0, 2].

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Q: -3-2-1 -1 543-2 -2 -3 -3 O g(z) = (x + 1)² – 2 O g(2) = (x – 2)² – 1 O g(z) = (x – 1)² + 2 O g(x) =…

A: We have in first figure that function f(x) of y=x2  which translated one unit downwards and two…

Q: Differentiating (2x – 1)" with respect to x leads to - n=0 а. 2 2n(2x + 2)"-1 n=1 O b. ∞ С. (2x –…

A: We have to find Derivatives of given Summation.

Q: 7. Find a fourth-degree polynomial with integer coefficients that has zeros x=3i and x=-1. The root…

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Q: -1 7 24 41-69 - [ ³ 4] = -2 9 B 07 02 00.0 ↓ 1 e dimension of the result. (If not possible, enter…

A: Topic:- matrix algebra

Q: 5. Show that the formula for the Secant method can be written in the form xn+1 = [f(xn)xn-1…

A: Secant Method: Consider, y = f (x). Let, x0 and x1 are two initial approximations for the roots of f…

Q: (2) Determine k such that kx – 4x+=0 has real and different solutions. 2

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Q: the polynomial of degree 4, p(x) has leading coefficient 1 had a zero of multiplicity 2 at x=2, a…

A: We are given that the polynomial p(x) has a zero of multiplicity 2 at x=2, i.e.; the polynomial has…

Q: Simplification of In x + In(x + 1) = 0 is Ox2 + x-1= 0; x > 0 Ox2 +x +1%= 0; x > 0 %3D x2 +x-e = 0;…

A: Given: ln(x)+ln(x+1)=0

Q: Find a formula for the polynomial P(x) with degree 3 leading coefficient 1 zeros at 8, 5, and 0…

A: polynomial P(x) with degree 3 leading coefficient 1 zeros at 8, 5, and 0

Q: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: Given Zeroes= 4+3i, -2+i therefore conjugate of above zeroes are also zeroes of f(x) Therefore 4-3i,…

Q: 334 In attempting to solve a simple PDE, a system of finite-difference equations of the form u+1 =…

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Q: 1. Using Gauss-Seidel and Jacobi Method, find the values of x1, x2 and x3 with ek < 0.001. 2x — 6х,…

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.