x'- Excrcése l Let k be a field. Prove that the set of irreducible pelynomials over k is infinite. Exercise 2. (a)Prove that a"1 = ñ (x-3') in %3D where 3- (6) Let F be a finite field and IF(=. Prove that ax% _c 7 (x-a) in F[x]. %3D KEF

Q: (a) Find an approximation to the integral (x2 - 2x) dx using a Riemann sum with right endpoints and…

A: We have to find (a) using a Riemann sum with right endpoints and n = 8 the value of the R8 and (b)…

Q: (a) Find an approximation to the integral | (x2 - 3x) dx using a Riemann sum with right endpoints…

A:

Q: 1. One can construct Riemann integrable functions on [0,1] that have a dense set of discontinuities…

A:

Q: Prove that the piece-wise function defined by 1 3x if r -2 is integrable on [-4, 7]. (Hint: do not…

A:

Q: Expand f(2) = rin powers of (z– 12). Also clearly mention the domain in which this expansion is…

A:

Q: Prove that sec x dx = In sec x + tan x + C. (Hint: Multiply the numerator and denominator of the…

A: To prove  ∫secxdx=lnsecx+tanx+C

Q: f(x)= 10* %3D

A:

Q: 25-26 Express the integral as a limit of Riemann sums using right endpoints. Do not evaluate the…

A:

Q: The Taylor series for f(x) = In(sec(x)) at a = 0 is en(æ)". %3D n=0 Find the first few coefficients.…

A: solution :- the given series  f(x)= ln(sec(x)) at a=0 is ∑n=0∞cnxn

Q: Expand f(z) = in powers of (z – 12) . Also %3D (z- 6) clearly mention the domain in which this…

A: Power series exaption, interval of converges

Q: (3) Prove that the piece-wise function defined by 1 3x if x -2 is integrable on [-4, 7]. (Hint: do…

A: The given question is related with integrability. Given a piece -wise function…

Q: Express the integral as a limit of Riemann sums. Do not evaluate the limit. 5 + x2 dx lim

A: Given: integral ∫245+x2dx The Reimann sum of the midpoint of the subinterval,…

Q: Let f be a function with second derivative f"(x) = Vi + 3x. The coefficient of x' in the Taylor…

A:

Q: Using Riemann sums, prove that Lrdx = b² - a² 2.

A:

Q: (a) Find an approximation to the integral (x2 - 3x) dx using a Riemann sum with right endpoints and…

A: The given function is,

Q: The n-th term in a sequence an is given by an = 3-eol (a) Is this sequence increasing, decreasing or…

A: Given:           an = 3-e-3n (a) Here:      a1 = 3-e-3(1) =3-e-3        a2 = 3-e-3(2) =3-e-6…

Q: Expand f (z) = in powers of (z – 12). Also clearly mention the domain in which this (z–15) expansion…

A:

Q: tan-1n Determine whether E is convergent using integral test. Zn=1 n2+1

A:

Q: 6. Express the Riemann sum with n = 3 terms and the left endpoint rule corre- b=2 Ax= b-a sponding…

A:

Q: sin( 4. Classify the singularity of f(z) = at z = 0 as one of {removable singularity, essential…

A: Given function is  fz=sinz3-z3z13 The given function is analytic everywhere except at z=0 Therefore,…

Q: Use integration by parts to obtain asymptotic expansion of e-zt -dt. Prove tn that the expansion is…

A: As per the question we have to find an asymptotic expression of the given improper integral using…

Q: 7. Prove that for every x E (-1,1) 1 1– 2x + 3x2 – 4x³ E(-1)" (n + 1)x" : - (1+x)² ` n=0 Make sure…

A:

Q: Approximate the area under f(x)=3x^2 +1 on [-5,0] using RIGHT Riemann sum with 5 approximating…

A: According to question,Given Function,f(x)=3x2+1,In the interval -5, 0

Q: Evaluate the definite integral: x²)dr using limit of the Riemann sum.

A: Given: Let I=∫022-x2 dx Here, fx=2-x2a, b=0, 2

Q: The integral in this exercise converges. Evaluate the integral without using a table. ee de .....…

A: The objective is to evaluate the integration.

Q: Q4) Evaluate both sides of divergence theorem for the field D-2r ar +(4/sine)a and a sphere of r-3…

A:

Q: Expand f (z) = (z-11)" in powers of (z- 11). Also clearly mention the domain in which this expansion…

A:

Q: What is the interval of convergence of Taylor Polynomial of f(x)=cos x at a=pi/3

A: Given function: fx=cosx a=π3 The Taylor polynomial of the function f(x) at x=a is given by…

Q: ek is convergent or Use the Integral Test to determine whether the scries 1+e2k k=1 divergent.…

A: For integral test, f(x)=ex1+e2x. Now,               ∫ex1+e2xdxLet  u=ex.=∫11+u2du=arctanuSubstitute…

Q: 6. Give a complete demonstration to determine the convergence or divergence of the im- proper…

A:

Q: Express the Pollowing defnite integral as alimit of the Riemann Sum .Then evaluate the in tegral by…

A: Note that, the definite integral as a limit of Riemann sum is,

Q: Evaluate the definite integral: | (2-)dz using limit of the Riemann sum.

A:

Q: A heavy rope, 60 ft long, weighs 0.7 lb/ft and hangs over the edge of a building 130 ft high. (Let x…

A:

Q: 1 Use the Integral Test to determine whether the scries is convergent or V3k + 5 k=1 divergent.…

A:

Q: 13. Evaluate the definite integral ((x²- a) using the limit of a Riemann sum b) using the…

A:

Q: Express the Pollowing definite integal as a limit of the Riemann Sum .Then evaluate the in tegral by…

A:

Q: Expand f(z) in powers of (z – 12). Also (z- 15) clearly mention the domain in which this expansion…

A:

Q: The integral in this exercise converges. Evaluate the integral without using a table. 4dx 4dx…

A: =?

Q: Express the funetions the Form on W = utiv W- そ - そ え - Wニ x -iy e e 3 W =

A: Here we express w in the form of u+iv

Q: Express the integral as a limit of Riemann sums. Do not evaluate the limit. 7 + x2 dx 7 ln 7 - 7 ln…

A: According to question given that∫137+x2dxWe have to express the integral as a limit of Riemann sums.

Q: Prove convergence or divergence.

A:

Q: 3

A: It is required to find the value of integral ∫01x3dx. Recall the fact that…

Q: 2. Write the Riemann sum to find the area under the graph of the function f(x) = x* from x = 5 tox =…

A:

Q: Expand f (z) = in powers of (z – 12). Also clearly mention the domain in which this expansion is…

A: We have to expand the function and also find the region of convergence.

Q: 3. Let (9n} be a sequence of integrable functions such that 9,-9 a.e. with g integrable. Show that…

A: In the given question we have to show that limn→∞∫gn-g=0 iff limn→∞∫gn=∫g wher gn is a sequence of…

Q: A. Use intermediate value theorom (IVT) to show that yatx=3 has a soluhon on the intenval [1,2] %3D…

A:

Q: Direction: Prove by mathematical induction that k3²(+1)² for all n E N. Use the definition of the…

A:

Q: Express the integral as a limit of Riemann sums. Do not evaluate the limit. 4 5 + x2 dx lim n- co i…

A:

Q: 2. If f is integrable on [a, b], then f² is integrable on [a, b]. This statement however is false…

A: