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Taylor Polynomial, Meclaurin's Series, Relative Extrema

1. For the equation ?(x)= x^(1/2)
a) Find the Taylor polynomial of degree 4 of at c = 4
b) Determine the accuracy of the polynomial at x = 2.

2. Find the Maclaurin series in closed form of
a) ?(x)=((1) / ((x+1)^2)
b) ?(x)=ln ((x^2)+1)

3. Use the chain rule to find dw / dt, where
w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(e^t) sin t, z=(e^t), t=0.

4. Find the critical points and test for relative extrema:
?(x,y)=2(x^2)+2xy+(y^2)+2x-3

5. Maximize ?(x,y)= (6-(x^2)-(y^2))^(1/2) given the constraint
x+y-2=0.

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Solution Summary

For the expression f(x) = x^1/2, the Taylor series expansion is given step by step with explanation.
Also the Meclaurin series in closed form for other function is explained clearly how to compute it with clear explanation so that the students could work out other problems of the same type independently.
Maximize ?(x,y)= (6-(x^2)-(y^2))^(1/2) given the constraint x+y-2=0 is explained step by step in such a way that the students can solve other problems of the same type with ease.
Also the problems involving relative extrema and critical points are dealt with in an easy way.

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