1. Find the Taylor's series expansion of: f(x, y) = x^3 + xy^2 about point (2, 1). 2. Find the first six terms of the expansions of the function e^x log(1 + y) in a neighbourhood o... 1. Find the Taylor's series expansion of: f(x, y) = x^3 + xy^2 about point (2, 1). 2. Find the first six terms of the expansions of the function e^x log(1 + y) in a neighbourhood of the point (0, 0). 3. Prove that a^x = 1 + x log a + (x^2/(2!))(log a)^2 + (x^2/(3!))(log a)^3 + …… 4. Expand x^2y + 3y - 2 in powers of (x - 1) and (y + 2) using Taylor's Theorem. 5. Expand x^y in powers of (x - 1) and (y - 1) up to the third-degree terms and percentage change in the volume of the balloon. Read more

Steps to Solve

1. Taylor's Series Expansion of ( f(x, y) = x^3 + xy^2 )

To find the Taylor's series expansion of ( f(x, y) ) about the point ( (2, 1) ), we need to calculate the function's value, and its first few partial derivatives at that point.

Calculate ( f(2, 1) = 2^3 + 2(1^2) = 8 + 2 = 10 ).

Next, find the first-order partial derivatives:

• ( f_x(x, y) = 3x^2 + y^2 )
• ( f_y(x, y) = 2xy )

At ( (2, 1) ):

• ( f_x(2, 1) = 3(2^2) + 1^2 = 12 + 1 = 13 )
• ( f_y(2, 1) = 2(2)(1) = 4 )

Now, find the second-order derivatives:

• ( f_{xx}(x, y) = 6x )
• ( f_{yy}(x, y) = 2x )
• ( f_{xy}(x, y) = 2y )

At ( (2, 1) ):

• ( f_{xx}(2, 1) = 12 )
• ( f_{yy}(2, 1) = 4 )
• ( f_{xy}(2, 1) = 2 )

Using these values, we can write the Taylor series:

[ f(x, y) \approx f(2, 1) + f_x(2, 1)(x - 2) + f_y(2, 1)(y - 1) + \frac{1}{2!} f_{xx}(2, 1)(x - 2)^2 + \frac{1}{2!} f_{yy}(2, 1)(y - 1)^2 + f_{xy}(2, 1)(x - 2)(y - 1) ]

2. Expanding the Series

Substituting the values into the Taylor series formula:

[ f(x, y) \approx 10 + 13(x - 2) + 4(y - 1) + \frac{12}{2}(x - 2)^2 + 2(y - 1)^2 + 2(x - 2)(y - 1) ]

Simplifying this gives the expansion around ( (2, 1) ).

3. Finding the first six terms of ( e^x \log(1 + y) )

The function is ( h(x, y) = e^x \log(1 + y) ).

Calculate ( h(0, 0) = e^0 \log(1 + 0) = 0 ).

Find first-order derivatives:

• ( h_x = e^x \log(1 + y) )
• ( h_y = \frac{e^x}{1 + y} )

Evaluate at ( (0, 0) ):

• ( h_x(0, 0) = 1 \cdot 0 = 0 )
• ( h_y(0, 0) = 1 )

Continuing this process to get higher-order derivatives and their evaluations would yield the coefficients for the expansion, leading to the first six terms.

4. Proof of ( a^x )

To show ( a^x = 1 + x \log a + \frac{x^2}{2!} (\log a)^2 + \frac{x^3}{3!} (\log a)^3 + \dots ), we expand ( e^{x \log a} ) using the Taylor series around ( x = 0 ).

5. Power Series Expansion of ( x^2y + 3y - 2 )

Using ( x - 1 ) and ( y + 2 ) as the variables and applying Taylor's theorem would give the required expansion, collecting terms in powers of ( (x - 1) ) and ( (y + 2) ).

6. Expansion of ( x^y )

Using Taylor expansion gives ( x^y ) in terms of ( (x - 1) ) and ( (y - 1) ), retaining terms up to the third degree.