Question

Consider the function f(x)=2e2sin(x). In this question, we will first use a linear approximation to estimate the value of f(0.1). Then, we will use a Taylor polynomial of degree three to estimate de value of f(0.1).

a) What is a good choice for the base point a of the linear approximation and the Taylor polynomial?

Answer: a=

b) Compute the derivative of f and evaluate it at x=a.

Answer: f′(x)=

               f′(a)=

c) The linear approximation L(x) of f(x) based at a is:

Answer: L(x)=

d) Use the linear approximation that you have found in (c) to estimate f(0.1).

Answer: f(0.1)≈    

e) Compute the second and third derivatives of f.

f′′(x)=

f′′′(x)=

f) Compute f′′(a) and f′′′(a).

f′′(a)=

f′′′(a)=

g) Find the Taylor polynomial of order three of f at the base point a.

T3(x)=

h) Use the Taylor polynomial that you found in (g) to estimate f(0.1).

Answer: f(0.1)≈   

i) Compute the error (in absolute value) of the two approximations of f(0.1) that you have found. Give your answer with an accuracy of five decimal places.

Answer: Error for the linear approximation:    

Error for the Taylor polynomial of order three:   

To appreciate the improvement in the approximation provided by the Taylor polynomial of degree three over the linear approximation, you should sketch the graph of f, L, and T3 near a.

Answer 1