Q5: a\. Let f(x) = x³ - 3x² -1, x ≥ 2. State and prove the theorem of derivative of inverse functions and use them to find the value of df-1/dx at the point x = -1 = f(3). b\, Write the Pascal's triangle for values of n = 6, then use it to determine the binomial series expansion of (3-2x)6, and simplify the result.

Q: Write the power series for (1 + x)* in terms of binomial coefficients. 00 (1 + x) .

A: Concept Used: Let p be any number and x<1then Binomial series expansion is given by…

Q: Find the Taylor Series for the function f(x) = ax + bx? + cx + d about the point x = 2. What is the…

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Q: Use the power series X f(x) = (1-x)² Select one: a. c. x e. Ση·(-1)*·(x)" n = 1 x 1 1-X Ση·(x)+1 n =…

A: Given power series is Noted that here given series is geometric series and converges to Now let…

Q: Use the power series expansion of ƒ(x) = 1 x)²· 1- x for x < 1 to find a power series ex

A: Remember power series expansion for f(x)=11-x; x&lt;1 is  f(x)=11-x=∑k=0∞xk

Q: Q/ Prove that fca)=G has a Power Series (z+2) (z-1) as the form ()" (1-4") litd 3

A: We have to show that fz=1z+2z-1 has a power series as the form 13∑n=0∞-1n1-4nz-2n+1

Q: Use the Taylor series for 1/ (1 - x2) to obtain a series for 2x/ (1 - x2)2.

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Q: Let f(x)=e(x-1)^2-1/(x-1)2  for x ≠ 1 and f(1)=1 a. Write the first four nonzero terms and the…

A: NOTE: Since we can solve at most three subparts in a question, we have solved the first three…

Q: 3. Find Taylor series for tan-1x about x =1 .

A: We find the Taylor series for tan-1x about x=1.

Q: Determine the value of f(2) when X 2x3 3x5 + 4² 46 f(x) 1. f(2) (Hint: differentiate the power…

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Q: 1- Find &, and Ea of sin (x) for x = 2 with 1-8 terms of the series. sin(x) - Σ k-0 (-1)* (2k + 1)!…

A: Given : sinx=∑k=0∞-1k2k+1!x2k+1  =x-x33!+x55!-. . . . . . To Find : et   ,  ea Note : According to…

Q: 00 Write the Taylor series for f(x) = e at x =-3 as en (x + 3)". n=0 first five coefficients.

A:

Q: Let f be a function whose Maclaurin series expansion is given below. f(x) = 3+ 12x + 24x² + 32x³ +..…

A: A Macluarin series is a Taylor series expansion 0 a function about 0…

Q: 2 11 { 6 1 if - <a < 0 if 0 < x < n

A:

Q: Prove. Show all steps to get to ln (2^(-2n))

A: Step 1: Given data :The expression is represented as, ln(BA​)=ln(((2N)![(N−n)!]2)(N!)2(2N−n)!​).…

Q: Š (-1)"(5n)² 6n4 E(-1)"an Consider the alternating series n=1 El-1"bm = § (-1)"(7n)² 7n3 And n=1 If…

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Q: find the sum of 1X ¹x (2-6) X (2-60²) + 2X (3 - w) x ( 2² w²) + -+ (-1x (m-w) x (m-1²);

A: A

Q: Find a Taylor series expansion around x = 1 for f(x) = 2/x2

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Q: 9 Write the Taylor Series for f(x) = sin(x) at x = 12/12/2 as & Cn (x-7) n Find the first 5…

A: Taylor serues

Q: the angle of rotation for the matrix transformation 2 2

A: Given the matrix of transformation is  f(xy)=32-121232xy To find the angle of rotation.

Q: Which if the following is a power series representation of the function - 1+x? f(x) = = +∞ Σ(-x" -…

A: Given: function f(x)=-1+x1-xwe know that11-x=1+x+x2+x3+..........=∑xn0∞where…