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Customer Question

a) Use the graph-sketching strategy...

a) Use the graph-sketching strategy to sketch the graph of the fuction:

f(x)=5-24x/x^2 +5x

c)Consider the function f(x) = 2xln x − 1.

(i) Show that the equation f(x) = 0 has a solution in the

interval (1, 2).

b) f(x)=e^(arcsinx) / sqrt 1-x^2 (-1<x<1)

2) In this question, f and g are the functions given by

f(x) =3/sqrt(9+4x) and g(x)=3/sqrt(1+4x)

(a) By writing

3/sqrt(9+4x)=1/sqrt(1+4x)

and using substitution in one of the standard Taylor series given in

the module, find the Taylor series about 0 for f. Give explicitly all

terms up to the term in x3. Determine a range of validity for this

Taylor series. [

(b) Use your answer to part (a), and the fact that 1 + 4x = 9+ 4(x−2),

to find the Taylor series about 2 for g. Give explicitly the same

number of terms as in part (a). Determine a range of validity for this

Taylor series.

(c) Check the first four terms in the Taylor series that you found in

part (b) by finding the first, second and third derivatives of g, and

using these to find the cubic Taylor polynomial about 2 for g.

f(x)=5-24x/x^2 +5x

c)Consider the function f(x) = 2xln x − 1.

(i) Show that the equation f(x) = 0 has a solution in the

interval (1, 2).

b) f(x)=e^(arcsinx) / sqrt 1-x^2 (-1<x<1)

2) In this question, f and g are the functions given by

f(x) =3/sqrt(9+4x) and g(x)=3/sqrt(1+4x)

(a) By writing

3/sqrt(9+4x)=1/sqrt(1+4x)

and using substitution in one of the standard Taylor series given in

the module, find the Taylor series about 0 for f. Give explicitly all

terms up to the term in x3. Determine a range of validity for this

Taylor series. [

(b) Use your answer to part (a), and the fact that 1 + 4x = 9+ 4(x−2),

to find the Taylor series about 2 for g. Give explicitly the same

number of terms as in part (a). Determine a range of validity for this

Taylor series.

(c) Check the first four terms in the Taylor series that you found in

part (b) by finding the first, second and third derivatives of g, and

using these to find the cubic Taylor polynomial about 2 for g.

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