APM3715 Assignment 01 2026

UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology


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APM3715: Applied Mathematics

Assignment 01 — Semester 1, 2026

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APM3715
Module Code:
Applied Mathematics
Module Name:
Numerical Methods: Root Finding
Assignment Topic:
01
Assignment Number:
2026
Due Date:
30
Total Marks:




Submitted in partial fulfilment of the requirements for APM3715 — UNISA 2026

, UNISA | APM3715 Assignment 01 — 2026



Question 1: Secant Method applied to x = cos x

Question: Solve the following expression using the Secant method: x = cos x. Comment on
your solution, approximations and round off errors.

The Secant method is an iterative root-finding technique that approximates the derivative
using a finite difference, making it useful when symbolic differentiation is inconvenient. The
equation is first rewritten in residual form, two initial guesses are chosen, and the iteration
proceeds until convergence.


1.1 Reformulation and Setup


Rewriting the equation as a root-finding problem:



f (x) = cos x − x = 0


Choosing initial guesses: x0 = 0 and x1 = 1.

The Secant iteration formula is:


xn − xn−1
xn+1 = xn − f (xn )
f (xn ) − f (xn−1 )


1.2 Iteration Steps


Step 1


Evaluating f at the initial guesses:



f (x0 ) = f (0) = cos(0) − 0 = 1 − 0 = 1




f (x1 ) = f (1) = cos(1) − 1 ≈ 0.5403 − 1 = −0.4597


Applying the Secant formula to obtain x2 :




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