1) Prove that 0|a - a=0

Q: b. Prove that coth² x-1=csch?x

A: Sinh(x)= (e^x-e^-x)/2. Cosh(x)=(e^x+e^-x)/2

Q: Solve for r: In(x) + In(x + 2) = 3

A: Use the properties of lograthim and simplify.

Q: 3) Prove that ae*+be-x a) b b > 0 ex+e~x

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Q: 5. Find the on the interval [0, 4]

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Q: ) Prove that sech¯¹(x) = In ( ¹+ √¹-x²) X

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Q: 4 to onabans ! (0)0 4. Prove that

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Q: 4. Prove that (À x B) - (B x C) x (C x A) = [AB %3D

A: The solution is given below.

Q: Prove that fVf · dr = 0.

A:

Q: Use direct prove to prove that for any two positive numbers x and y, x+y Vxy s 2

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Q: 1. • If r + 1| < o, prove that |r³ + 1| < 0.331.

A: Here we use the fact  | x |&lt; a implies  -a&lt; x &lt; a .

Q: 1S prove that (a + b) = a² + b².

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Q: Prove that 1. 1. 1 + 1x 2'2x 3 1 = 1 – (п — 1) хп + 3 x 4

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Q: 4. Prove that 3i² + i = n(n+ 1)?

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Q: Prove that Σ = 2" . a. x=0 b. = 0. x=0

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Q: Prove Theorem 11.3

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Q: Verify Young's Theorem z = In(2x + y) -

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Q: PROVE that 2a + b = 0.

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Q: To prove that dx In /1+ (In x)² + In x + C x/1+ (In x)2 we use the substitutions

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Q: U prove that In (l+x)) 2.

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Q: Prove that 223 a E lo,1) く 22 + 1영 ( 1-계 3(1-x2)

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Q: Prove that if x+y=%30 then y=-x

A: See the attachment

Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

Q: : Prove that [ X" Ju Cx]=x* Ju-l X)

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Q: 2 16 v2 6. Prove that x V(2 – x) dr = 15

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Q: 34. Prove parts (a) and (c) of Theorem 3.2.3.

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Q: prove that esch'x - In (* + x2 %3D

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Q: Prove that: (1+ x)*** (n+ 1)x C. + + 2 3 n+1

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Q) prove that (1) -2P xSinx 2.

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: Suppose x, n ∈ N and n not equal to  0. Then, show that xn is positive if and only if x is positive

A: Given that x, n∈N and n≠0. According to question, N=0,1,2,3,4,5........ First suppose xn is positive…

Q: Prove That Cn) =nsh ツールがー

A:

Q: Find a number z that satisfies: |z| - z = 2 + i

A: Complex number which satisfy |z|-z= 2+i

Q: Compute a (х + 2) dx. O x3 + x² + C O x² (x² + 2x)+C x2 + 2x ) + C 2 - + C 2 + x2 + C 3

A:

Q: 5) Prove that d tanh ) |-x2

A: Prove that  d/dx[tanh-1x] =1/(1-x2)

Q: Prove that 0<1.

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Q: "(2) is also analytic, and hence its

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Q: Show that If |z| = 1, then: -1 = Z

A: z is a complex number. z=a+ib |z|=(a+ib)(a-ib)     =a2-b2 if |z|=1, then a2+b2=1

Q: Prove that where e, > (1,0)"

A: Prove that     conve1, e2, -e1, -e2=x∈ℝ2 : x1+x2≤1 where e1=1, 0T , e2=0, 1T.

Q: Prove that the number V7 lies between 1 and 2.

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Q: 17. Prove thatsin x = =x 1-x2 dx

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Q: 6) Prove mathematically that (1 grad = 0.9 deg)

A: We know, that 1grad= 0.9 deg.  we can solve any other similar problem by multiply the number of…

Q: Prove that R (D-1 and R)%=(-)

A: The objective is to prove that  Pn1=1Pn-1=-1n

Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

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Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: Prove: le* a. ſ csch x = In - |+c lex+1

A: As per Bartleby guidelines rule I am answering only first question as Both question is not subpart…

Q: 2. Let A be non singular. Show that if Ax = 0, then x = 0.

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Q: 2.2 Use algebraic manipulation to prove that (x +y) · (x + y) = x.

A: As 2 Questions are given, I will solve only the first, We need to Prove that: x+y(x+y¯)=x Solving…

Q: -x+y<y. 2 If x < y, prove that x <

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Q: Prove that le-2z| 0.

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Q: By expanding both sides of the identity (1+ x)"+4 = (1+x)"(1+x)* prove that ("**) - (") + «(,",) +…

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Q: Prove the following:  if x^2 = y^2, then x = y or x = -y.

A: Subtract y^2 from both sides.

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.