4. (a) Use Gauss's lemma to evaluate (), showing all working.

Q: 8. Theorem 1.7.4 x -le* dx is convergent if and only if n > 0.

A:

Q: Determine the convergence or divergence of serie sin Σ 6n2 n +4 n=1

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Q: Example: Examine the convergence of COs x dx (ii) / x'e*² dx

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Q: State and prove the convergence order of Newton's approach:

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Q: Explain fermat's little theorem.

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Q: if 0 < x < 1 Problem 1. Prove that f (x) is integrable using Riemann's 1 if x = 0 criterion…

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Q: If n = 18, then the interval of the form (- t(n-1), alpha/2 , t(n-1), alpha/2), that contains 99% of…

A: From the given information, number of observations n = 18. Confidence level  Level of significance

Q: 2. Find the principal value using De Moivre's Theorem: |20 3 2ez

A: We know that,  eiθ=cosθ+isinθ De-Moivure's theorem:  cosθ+isinθn=cosnθ+isinnθ

Q: @) 2) Fs of (x) = -PFC (Psy) -} [ If f (x,y) → as 6004X proof. (H-w.) then в

A: Introduction: The Fourier sine and cosine transforms are variations of the Fourier rework that do…

Q: 04: A) Discuss the convergence of f,(x) =, , on A-(0, 0).

A: Since solve the first question for you. If youwant any specific question to be solved then please…

Q: 5. Let fn(x) ". Show that z"0 uniformly on (0, ).

A: We have to use squeeze theorem here.

Q: 3, Re call Eudlid's proof of In finitude * primes! Shouw that T Cx): = 1{ ps X: p-pinae }12 logz…

A:

Q: f fn → ƒ uniformly and the functions fn have only jump discon- sinuities, prove that f has only jump…

A: Here we use epsilon delta proof

Q: Limit n---infinity for the function nln((1-(3/5n))/4 using L'Hopitals

A: Evaluate the given function  as follows.

Q: i) Discuss the pros and cons of the various numerical methods such as the bisection method, linear…

A: The Bisection method is a method used in mathematics that helps an individual find the square root…

Q: If X1, Y1, are two positive unequal numbers and Xn = (xn– 1+ Yn – 1) and y,= Vx – 1 Yn –1 V n 2 2.…

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Q: Find the absolute min/marx of te Aunction.

A:

Q: 4 A function fx) = 1- x² + xr is defined in the closed interval [-1, 1]. The value of x in i onen…

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Q: Show that the sequence of functions on the domain [-1, 1] defined by if -1<r<0, 0, if 0 <x < 1/n, 1,…

A:

Q: converges, then C„-)" " con COr n=0

A: We have to choose the correct answer from the given option according to the given series

Q: Convergence can be achieved faster in Newton-Raphson Method. O True O False compared to the Secant

A: Convergence can be achieved faster in Newton Raphson compared to secant? True or False.

Q: 2. Use the Intermediate Value Theorem to show that 2.x 3 - if – 3 0 has a zero on the interval [-2,…

A:

Q: plot Y funetion of as over Hhe Peenge ☺.0l <*< l00.

A:

Q: Use a linear approximation to estimate the value of sin 0.06. Verify that f(x) = satisfies the…

A: Use formula of linear approximation

Q: Number 8

A: first find the derivative.

Q: For the part highlighted in green, why doesn't negative infinitiy cancel with the negative near the…

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Q: Briefly discuss the origin/history of De Moivre's Theorem.

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Q: Hiwiy Express the ballowsing functions by Palar equationft 2.

A: To change a rectangular equation to a polar equation just replace x with (r cos θ ) and y with  (r…

Q: Suppose that we roll a balanced four-sided die twice. Let Y be the maximum of the two numbers. Type…

A: If a four sided die is rolled then total possible outcomes = 4 and each outcomes are equally likely…

Q: Discuss the two versions of the fundamental theorem of calculus

A: This the problem of Real Analysis, Theory of Reimann Integration.

Q: 4n2 Tantel show whether (an )Cconverge diverge or

A: Given : The given series is :  ∑n=1∞un= ∑n=1∞4n29n4+1. To find : We have to find whether the given…

Q: None of These cosh(4t) 7. 2. 7. - sinh( 21) sinh(4t) 7. cosh(21) 7. Sて 1/2 1/2 1/2 1/2

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Q: What is the convergence for Newton's method for systems?

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Q: 4. Letf(x)=x22/5 - 19x². Determine the largest n for which fis n-times continuously differentiable.

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Q: 2. Determine the positive real root of the function f(x) = In(x²) – 2 with initial guess [5/2, 3]…

A:

Q: 5. Show that cos .3 G) do not converge.

A: We need to prove that: cosnπ3 does not converge This can be proved as follows:

Q: 5. Does f(x) = In(x) satisfy the hypotheses of the Mean Value Theorem on [1,4]? If it satisfies the…

A:

Q: Try to prove the convergence of Newton’s Method

A: The given problem is to prove the convergence of the Newton's method, We have to derive the…

Q: 7). but you have not said if the diverges test applies or not . please xeplain the final solution ?

A: This question can be solved using the concept of divergence test.

Q: 13 By use of De Moivre's Theorem, evaluate (-+i) .

A: We have to evaluate -12+32i13 by using De Mover's Theorem.

Q: (1) Use Ito's formula to compute this stochastic integral (Riemann integeral are considered done

A: Given: Let Xt is defined by stochastic integral, And its obtained by using the Reimann integral.

Q: ENRI Green's theorem to

A:

Q: 04: A) Discuss the convergence of f,(x)=x+- -.on A [0,0).

A:

Q: 5.If the tangent 1ine to the functlen fx)= axt-6x at X=- has the equatfon. y= 3x+2then 6=?

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.