**I Taylor series approximation [25 points]**

1. [5 points] If you are given a function f(x) and you seek its nth order Taylor
series approximation

[and remainder
] around
and in the interval [a, b], how do you go about doing this?

Qualitatively, explain what conditions you think the function f(x) should
satisfy and the different

forms of the remainder at your disposal. Do you think there is a relationship
between the two forms

of remainder?

[2 points] for the first part if you give the formulation of Taylor series and
the remainder.

[1 point] for the conditions. It is important to state that f(x) as well as its
n derivatives are contin-

uou and differentiable over [a, b]. For most of you I took off 0.5 points if you didn't talk about the

differentiabilty of the derivatives.

[1 point] different forms of remainder: Cauchy form and the Lagrange form. Also
give the mathematical

form. Just stating the names is not sufficient.

[1 point] The relationship is that both are equivalent and one can be derived
from the other using the

Integral Mean Value theorem.

2. [5 points] Work out the second-order Taylor-series approximation
for the function f(x) =

exp at
and in the interval [a, b]. What is the remainder
? Write down the

condition that must satisfy for
.

[2 points] The taylor series expansion is:

[2 points] The remainder is: R

[1 point] . For the condition to be satisfied
, state that we need to find
.

Some people also solved it by using the series of e^{x} and then replacing xin the
expansion with -x^{2 }which

is perfectly okay . If I took off your points for doing this bring it back to me
and you "might" get your

points back.

3. [5 points] What is the first order Taylor series approximation
for the
function

[where log is the natural logarithm] in the interval [0, 1] and for x_{0} = 0?
[Hint: You'll have to take the

limits and
and use the
form.] What is the remainder R_{1}(x) and what

is its worst (largest absolute) value in the interval [0, 1]? What is the error
? Is

there a discrepancy? Discuss.

[2 points] : Note that here as
.So you were required
to tale the limit of

the function at x=0 using the L'Hopital's rule.

[1 point]

[1 point] l R_{1}(x) l is maximum when
then

[1 point] There is no discrepancy because the error
:and hence

that the max abs remainder is infinite is perfectly okay.

4. [10 points] Given a function where m is
an arbitrary integer greater than zero, find the

order n for which the Taylor series approximation
at
is non-zero.
Is there a fundamental

problem with finding such a Taylor series approximation? Discuss.

[8 points] Keep differentiating and you will find that each derivative term of f(x)
contains term

and term. So for the second term again we have to take the limit
at x = 0. Otherwise

there is this fundamental difficulty that f(x) becomes infinity at x = 0 [2
points].

Now take first derivative of x. We see that . In this the
first term that is

will become non zero at x = 0 if we differentiate it m-1 times , i.e., we
have to have which

means n = m and hence the number of terms in the taylor expansion = m+1.

**II Derivative approximations [25 points overall]**

1. [5 points] What is the fundamental goal of the derivative approximation?
Explain clearly the steps

involved in beginning with a function f(x) and ending up with a first derivative
approximation for the

function.

Almost every one got 5 points.

2. [5 points] If you are given the Taylor series approximation
for a
function f(x) at x_{0}, how does this

change into an equivalent approximation for f(x + h) in terms of f(x)? Write
down the equivalences

between (x, x_{0}) in and (x + h, x) in the approximation for f(x + h).

x is same as x+h and x_{0} is same as x.

3. [5 points] Is the approximation a valid
first derivative approximation? [For a

derivative approximation to be valid, should
equal zero.]

You have to evaluate the above limit and show that
.This can be

easily done by expanding the two taylor series for
and showing that the higher

order terms tend to zero as x tends to zero.

4. [10 points] Begin with . Construct an
approximation to

f''(x) by ensuring that i) the term involving f(x) is zero, ii) the term
involving f'(x) is zero, and iii)

that the coefficient of f''(x) is one. Write down the resulting constraints
involving A, C, a, b, and h.

Pick a set of possible values for (A, C, a. b) that satisfy the constraints.

Applying the conditions we get

choose A = 1,C = -2,a = 1and b = -1. Now our A and C in the above equations
are different than

those in the question. So A = 1/h^{2} and C = -2/h^{2}

**III linear interpolation and Trapezoid Rule [25 points overall]**

1. [5 points] What is the relationship between the linear interpolation of a
function f(x) in the interval

[a, b] and the trapezoid rule. Conceptually relate the two.

The trapezoid rule approximates the integral of the function by integrating over
the linear interpolant

of the function rather than the actual function itself.

2. [5 points] Construct a linear interpolation for f(x) = sin(x) for
. What is the worst case error

for this linear interpolation if you use the error bound?

The worst case error is:

3. [5 points] Construct a trapezoidal rule integ ration of
.

The trapezoid rule integration is the area under the linear interpolant:

So,

4. [10 points] Given four points A = (1, 5), B, C and D = (2, 3) and the linear
interpolation between B

and C to be 3x + 4y = 5 can you find the coordinates for B and C if you also know
that the slope of

the line joining A and B is 0.5 and that the slope of the line joining C and D
is 2. (This kind of a

problem arises in the spline literature.) Note that we know the equation of the
line between B and C.

[If you have difficulty solving simultaneous equations , just write down the
conditions satisfied by the

points B and C and you'll get a lot of partial credit.]

Find the equation of AB:

y = 0.5x + c_{1}This will satisfy point A(1,5). So,

5 = 0.5 *1 + c_{1}So, c_{1} = 4.5.

Hence, AB: y = 0.5x + 4.5

Similarly find CD:

y = 2x + c_{2} This satisfies point D(2,3). So,

3 = 2 *2 + c_{2} So, c_{2} = -1

Hence, CD: y = 2x - 1

Now you get B by solving AB and BC. And you get C by solving BC and CD. the
equation of BC is

given to be : 3x + 4y = 5

It is uptill this point that you get 9/10 points. This is also what was
mentioned in the question that

if you have difficulty solving simultaneous equations then you can leave upto here
and you get a lot of

partial credit. If you also solve further then you get 10/10.

**IV. Newton's method [25 points overall]**

1. [5 points] Give a conceptual level explanation of Newton's method. Pick a
function f(x) and clearly

explain what it is that the algorithm is trying to achieve.

see the textbook.

2. [5 points] Apply Newton's method to find the root at the origin for the
function with an

initial condition x_{0} > 0. Does the process converge?

: The process does not converge. Note that it
is also not oscillatory

because -x_{0} because x_{0} > 0 is negative and does not lie in the domain making
f(x) imaginary there.

If you do not specify this or write it to be oscillatory instead you lose 2
points.

3. [5 points] Given a function g(x) which is continuous and twice differentiable
in the interval (-∞,∞),

use Newton's method to find the location α where the function attains its minimum.
[You may assume

if you wish that the function has only one minimum occurring at x = 0.]

The Newton's method finds the roots of the equation. Now a function g(x) attains
minima when

g^{1}(x) = 0. So, here we have to apply Newton's method on g^{1}(x) and not g(x).

So, Newton's method will be:

4. [10 points] Sketch a function that satisfies the fol lowing properties . i) The
function f(x) is continuous

and differentiable in (-∞,∞), ii) the function f(x) = 0 at x = 0, iii)
for where

can be made arbitrarily small, iv) for
. What happens when you run Newton's

method on such a function? Qualitatively explain the different scenarios that
unfold for different initial

conditions x_{0}.

See the scanned image that I will post tonight.