Question:3 O + arctan) arcsin u = Discuss the continuity of function at (0,0)

Q: (a) Show that arctan z = log ( ) in the sense of multi-valued functions. Hint: show that w = log )…

A: To find - Show that arctan z = i2 log1 - iz1 + iz in the sense of multi-valued functions. Hint :…

Q: curve x= c sec 0 Show that the tangent at any point 0 on the curve x = c sec 0, y = c tane is y sin…

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Q: sinx xくT。 Discuss The Continuity of y= COSX

A: Continuity :     limx->af(x) = L            where, L is the limit value i.e. finite and will be…

Q: Derive the De Moivre's Theorem with detail Explanation?

A: Statement: For any real number x, we have (cos x + i sin x)n= cos(nx)  i sin(nx) Or, (eiθ)n=eiθn…

Q: {(x+² dx 1

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Q: Trigonometric limits Use Theorem 3.10 to evaluate the following limits.

A: We first try to find the limit by substitution

Q: sin 2z cot 8z lim z cot 4z

A: To evaluate: The limit limz→0sin 2z·cot 8z z·cot 4z . Formula used: limz→0sin azaz=1 limz→0tan…

Q: Verify the identity.

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Q: +siny+x

A: Side View

Q: sinat dt 2.1. Use Laplace transforms theory to compute

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Q: Find the limits

A: Our Aim is to find the limits of the following expression given below:-limx→0 sin (x) cot (2x) -(i)

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Q: y tan -- arccos y

A: In this question, we need to check whether the given function is homogeneous or not and the degree…

Q: A parametric curve is defined by : x =cos (r) and y =1-sin° (1) ; --sıs a) Find the Cartesian…

A: The parametric curve is defined by,x=cos(t) and y=1-sin2t,      -π2≤t≤π2

Q: How can De Moivre's theorem be described?

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Q: integral arc tan(4t)dt

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Q: State whether the following statement is true or false. Explain. The range of y=tan(x) is (-infinty,…

A: tanx will approach infinity if cosx approaches 0.

Q: Trigonometric limits Use Theorem 3.10 to evaluate the following limits.

A: Given: limx→0sinaxbx for finding this limit we will use standard limit given below limx→0sinkxkx=1,…

Q: Arc length Find the length of the curve y = In x on the interval [1, e*).

A: Given: The curve y=lnx,  for x∈1, e2.

Q: Length of a catenary Show that the arc length of the catenary y = cosh x over the interval [0, a] is…

A: Given  The curve is  y=cosh x over the interval [0,a] .     Formula  Arc length formula…

Q: Show that the function F= the fallowing canohical tranformatione %3D COS and %3D

A: Determine q from Q=ln1+qcosp and then express q and P in terms of p and Q.…

Q: Arc length Find the length of the curve y = In (sin x), 6 2

A: y=lnsin x, π6≤x≤π2

Q: Laplace equation is called a harmonic function. Let the functions u = u(x, y) and v = v(x, y) be of…

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Q: X-|x Show that f,given by f(x) = (x ±0) is continiuos on R-{0}

A: f is continuous at a  If at x= a L.H.L = R.H.L = f(a)

Q: sin z Prove that lim Cosz - = 1 and lim = 0. Z→0 Z Z→0 Z Hint: use the definition of the derivative,…

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Q: Arc Length, x=1/2(e^y+e^-y) from y=-1 to y=1

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Q: Question 6: Computing Limits 1- arctan z Define the function S(x) = sin(2) - cos(z)

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Q: ition of cont s continuous at the given point. a). f(x) = , a = 3 Use the 2z– -32

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: The of ab any tempseature of Satisfies ulxit) Semi - infinite bag equation point a the 2u (ocit) at…

A: The temperature of ux,t of any point of a semi infinite bar satisfies the equation ∂u∂tx,t=∂2u∂x2x,t…

Q: Question 33 1 Point Let f and g be continuous on the closed interval [0, 5]. Suppose that / f(x)dx =…

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Q: ‚arcsin(1+x²)] y'of y = In cscya

A: Given y=lnearcsin1+x2cscxWe need to find y'

Q: Find the limit

A: Given limit can be written as Dividing both numerator and denominator by , we get

Q: Fine arc length of AB in terms of n and using 3.14

A: Use angle formula for arc Convert angle in radian

Q: tan x Evaluate the integral / da 2 tan x -

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Q: Write an integral that represents the area of the surface generated by revolving the curve about the…

A: the formula used to find the area of the surface generated by revolving the curve about the x-axis…

Q: Example : Find the area of the region that lies inside the circle r = 3 sine and outside the…

A: A circle is a special type of ellipse in which both the focal points are same. A circle is 2-D shape…

Q: I ― √co COS X lim x 0 1 cos x

A:

Q: Using the Cauchy-Riemann Equation. u=x^2-y^2-2y show that is harmonic. Find the function v whose…

A: We have ,  u = x2 - y2 - 2y We need to show that , u is harmonic. We know that , A function  u(x ,…

Q: tan 3. lim 2t t→0 ´tan 3x' lim ( ) x→0 csc 3x 5. lim X-0 \3cotx,

A: Topic = Limits

Q: Excrcise 9. Find the arca bounded by the graphs of functions Y = y = x°, y = 8, x = -1.

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Q: cot (2x) Evaluate the integral - * xp- V cor (2x) - 15 csc (2x) (A) -Arcsec +C 4 (B -- In cot (2x) +…

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Q: lim tar Ө+ ЗП 3 tan 2

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Q: QI Find lim,o+ sin(x)tan(x) O 0.3 O 0.6 O O O O OO

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Q: -"/13) 3sin(4x) dx "/13) x2020+1

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Q: By the Fundame 2+x? dx. 1

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Q: (a) Show that if a projectile is launched straight upward from the surface of the earth with initial…

A: At surface of Earth: Velocity of projectile = v0 Hence Kinetic energy at surface of Earth will be:…

Q: sin 0 (1) Lim %3D

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Q: sin(ln x) r→0+ In(sin x) Use Squeeze Theorem to evaluate lim

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Q: using the Cauchy- Riemann equations in polar Coordi'nates to find out whether the funchons are…

A: Let us denote the function by fz i.e.  fz=z2                     (i) Suppose, z=reiθ  , r>0,…

Q: Parametrize the intersection of the surfaces y- z = x - 5, y + z? = 4 using trigonometric functions.…

A: We have to find the intersection of given function using trigonometric function.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.