Math Real Analysis

undergraduate math real analysis 2

find taylor series at point 1 and radius of convergence

MATH3132 Winter 2021

Final Exam

April 22, 2021

Name:_____________________

Instructor: Dr. Olga Vasilyeva

• You have 4 hours to complete and submit the test

(by 1pm, Thursday, April 22)

• You can use your own lecture notes only; any other sources

are not permitted

• Absolutely no collaboration is allowed (if there is an

evidence of copying or using online resources, double

points will be deducted).

• To receive full credit you must write your answers in the

space provided, perform all of the steps, and submit the

exam as a single pdf file with all the pages in the original

order.

• In your solutions you can refer to the theorems or results

covered in the course.

1. (10 points) Let 𝑓(𝑥) = 𝑥3 − 7.

(a) Show that there is a unique root of 𝑓(𝑥) = 0

on the interval [1, 2].

(b) Use the Bisection method to approximate the root (perform four
iterations).

(c) Derive a formula for the minimal number of steps required to perform in
the Bisection method on the given interval [𝑎0, 𝑏0] to achieve accuracy 𝜀.

(d) Use the formula in (c) to find the minimal number of steps required to
perform in the Bisection method to achieve accuracy 0.001.

2. (10 points) Let 𝑓(𝑥) = 𝑥3 − 7.
(a) Explain when is it appropriate to use Newton’s Method.

(b) What are the main steps of the Newton’s Method?

(c) Starting from 𝑥0 = 1, approximate the root of 𝑓(𝑥) = 0
using Newton’s Method (perform five iterations).

3. (2 points) Give a statement of Existence/Uniqueness Theorem for polynomial

interpolation.

4. (17 points) Let 𝑓(𝑥) =
1

𝑥+2
, 𝑥0 = 0, 𝑥1 = 1, 𝑥2 = 2.

(a) Construct the interpolating polynomial 𝑃2(𝑥) for 𝑓(𝑥)in the Lagrange form

(do not simplify your answer).

(b) Interpolate 𝑓(0.8) using the polynomial obtained in (a).

(c) Find the actual value of 𝑓(0.8).

(d) Evaluate the absolute value of the interpolation error of the above
approximation (𝑖. 𝑒. | 𝑓(0.8) − 𝑃2(0.8)|) .

(e) Construct the interpolating polynomial 𝑄2(𝑥) in the Newton form (do not
simplify your answer).

(f) Interpolate 𝑓(0.8) using the polynomial in (e).

(g) Evaluate the absolute value of the interpolation error of the approximation
in (e) (𝑖. 𝑒. | 𝑓(0.8) − 𝑄2(0.8)|).

(h) State the Interpolation Error Theorem (including a formula for the upper
bound on the error on interval [𝑎, 𝑏])

(i) Evaluate an upper bound on the error resulting from the interpolation at
�̅�=0.8.

5. (6 points) Given data points (𝑥0, 𝑦0), (𝑥1, 𝑦1), (𝑥2, 𝑦2), (𝑥3, 𝑦3), write down the

interpolating and matching conditions used in construction of a linear spline.

6. (5 points) Let 𝑔(𝑥) be a continuous function on [𝑎, 𝑏] such that

𝑔([𝑎, 𝑏]) ⊂ [𝑎, 𝑏]. Show that there is at least one fixed point on [𝑎, 𝑏] generated
by 𝑔(𝑥).

7. (10 points) (a) Derive an upper bound for the error resulting from

approximating 𝑓′(�̅�) using the forward difference scheme with a small ℎ > 0.

(b) ) Derive an upper bound for the error resulting from approximating 𝑓′(�̅�)
using the centered difference scheme with a small ℎ > 0.

8. (10 points) Consider 𝑓(𝑥) = ln(𝑥 + 3), 𝑥0 = 2, 𝑥1 = 2.1, 𝑥2 = 2.2.

(a) Use the forward difference method to approximate 𝑓 ′(𝑥0).

(b) Use Problem 7 to determine the upper bound on the error resulting from
approximating 𝑓′(𝑥0) using the forward difference method.

(c) Use the centered difference scheme to approximate 𝑓 ′(𝑥1).

(d) Use Problem 7 to determine the upper bound on the error resulting from
approximating 𝑓′(𝑥1) using the centered difference method.

(e) Use the backward difference scheme to approximate 𝑓 ′(𝑥2).

(f) Find the exact values of 𝑓 ′(𝑥0), 𝑓

′(𝑥1), 𝑓
′(𝑥2) and the actual truncation

errors in (a), (b) and (c).

𝑓 ′(𝑥0) = __________________________, the actual truncation error=_______________________

𝑓 ′(𝑥1) = __________________________, the actual truncation error=_______________________

𝑓 ′(𝑥2) = __________________________, the actual truncation error=_______________________

9. (4 points) When is it appropriate to use Euler’s Method?

10. (5 points) What are the main steps of Euler’s Method?

11. (8 points) Consider the following initial value problem:

{

𝑑𝑦

𝑑𝑥
= 𝑥(𝑦 + 1)

𝑦(0) = 2
.

(a) Check that the exact solution of the IVP is given by 𝑦(𝑥) = 3𝑒
𝑥2

2 − 1

(b) Find an approximate solution �̅�(𝑥) of the IVP using the Euler’s Method with
ℎ = 0.1 (perform 4 iterations).

(c) Evaluate �̅�(0.3).

(d) What is the error of approximation at �̅� = 0.3
(𝑖. 𝑒. |𝑦(0.3) − �̅�(0.3)|)?

12. (30 points) Approximate

∫
𝑑𝑥

1 + 𝑥

3

0

using the following methods with 𝑛 = 6.

(a) The Midpoint method.

(b) The Trapezoidal method.

(c) Simpson’s method.

(d) Find the exact value of the integral.

(e) Evaluate the exact truncation errors for each method.

Midpoint rule truncation error: _________________

Trapezoidal rule truncation error: _________________

Simpson’s method truncation error: _________________

(f) Derive the formula for the upper bound of the global truncation error for the
Midpoint method in the general case.