Math Real Analysis
MATH 3132 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.