Numerical Methods · Unit 5
Numerical Differentiation
Exam-focused notes for Numerical Differentiation (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 4 solved past questions from this unit.
What this unit covers
- Derivatives of continuous functions
- Two point forward difference formula
- Two point backward difference formula
- Divided difference table for derivatives
- First and second order derivatives
- Algorithm and program for numerical differentiation
Two point forward difference formula
Derive the formula for two points forward difference. Derive the formula for two points backward difference. [2.5+2.5]
Note: The reference notes were not available for this topic. The derivation below follows the standard numerical methods approach as taught in BSc CSIT curriculum. --- (a) Two-Point Forward Difference Formula The forward difference formula approximates the ...
Full solved answer →Divided difference table for derivatives
Find the first and second derivative at $x = 2.5$ of the following data points.
$$\begin{array}{|c|cccccc|}\hline x & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \ \hline f(x) & 2.375 & 4.5 & 7.625 & 12 & 17.875 & 23 \ \hline \end{array}$$
[5]
$x$ 1.5 2.0 2.5 3.0 3.5 4.0 ----------------------------------- $f(x)$ 2.375 4.5 7.625 12 17.875 23 - Step size: $h = 0.5$ - Point of interest: $x = 2.5$ Take $x0 = 2.5$ so that $p = 0$. First compute all differences carefully. $x$ $f$ $\Delta f$ $\Delta^2 ...
Full solved answer →The table below gives the values of distance travelled by a car at various time intervals during the initial running. Estimate the velocity and acceleration at time t = 7 sec.
$$\begin{array}{|c|c|c|c|c|c|}\hline \text{Time (t sec)} & 5 & 6 & 7 & 8 & 9 \ \hline \text{Distance (s m)} & 10.0 & 14.5 & 19.5 & 25.5 & 32.0 \ \hline \end{array}$$
[5]
Time $t$ (sec) 5 6 7 8 9 ------------------ Distance $s$ 10.0 14.5 19.5 25.5 32.0 - Uniform spacing $h = 1$ - Required: velocity $\frac{ds}{dt}$ and acceleration $\frac{d^2s}{dt^2}$ at $t = 7$ (The problem header mislabels the second row as "Temp"; from con...
Full solved answer →Algorithm and program for numerical differentiation
How can we calculate derivatives of continuous functions? Write down algorithm and program for differentiating continuous function using two point forward difference formula. [5]
For a continuous function f(x), the derivative at a point x is defined as: $$f'(x) = \lim{h \to 0} \frac{f(x+h) - f(x)}{h}$$ In numerical methods, we cannot take the limit analytically for arbitrary functions, so we approximate the derivative by choosing a ...
Full solved answer →Make Unit 5 stick
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