Numerical Method · Unit 3 · 8 hrs
Numerical Differentiation and Integration
Exam-focused notes for Numerical Differentiation and Integration (Numerical Method, CSC212): what the TU syllabus asks and how it has actually been tested, with 11 solved past questions from this unit.
What this unit covers
- Differentiating Continuous Functions (Two-Point and Three-Point Formula)
- Differentiating Tabulated Functions by using Newton's Differences
- Maxima and minima of Tabulated Functions
- Newton-Cote's Quadrature Formulas
- Trapezoidal rule
- Multi-Segment Trapezoidal rule
- Simpson's 1/3 rule
- Multi-Segment Simpson's 1/3 rule
- Simpson's 3/8 rule
- Multi-Segment Simpson's 3/8 rule
- Gaussian integration algorithm
- Romberg integration
Differentiating Tabulated Functions by using Newton's Differences
What is numerical differentiation? The table below gives the values of distance travelled by a vehicle at various time intervals. Estimate the velocity and acceleration at $x=4$.
| Time $(x)$ | 1 | 2 | 4 | 8 | 10 |
|---|---|---|---|---|---|
| Distance $(y)$ | 0 | 1 | 5 | 21 | 27 |
[5]
Numerical differentiation is the process of estimating the derivative (rate of change) of a function from a set of tabulated/discrete data values, instead of differentiating a closed-form expression analytically. It is used when the function is either unkno...
Full solved answer →How can we calculate derivatives of discrete (tabulated) functions? Write down its algorithm. [5]
When the function values are known only at some discrete points (tabulated data) but the function itself is unknown, we cannot differentiate it directly. In such cases, we use numerical differentiation for tabulated functions. The approach is to fit an inte...
Full solved answer →Simpson's 3/8 rule
What is an application of numerical integration? Find the value of the integral using Simpson's 3/8 rule with n=6. $$\int_{1}^{2} \frac{e^x}{x} dx$$ [5]
Numerical integration computes the approximate value of a definite integral from numerical/tabular values of the integrand. Key application: - Evaluating definite integrals when no closed-form (analytical) antiderivative exists (such as $\int e^{x}/x\,dx$)....
Full solved answer →Multi-Segment Trapezoidal rule
Find the following integral using composite trapezoidal rule for using 2 segments (k=2) and 4 segments (k=4). $$\int_{2}^{4} (x^3 + 2) dx$$ [5]
- Integrand: $f(x) = x^3 + 2$ - Lower limit: $a = 2$ - Upper limit: $b = 4$ - Segment counts: $k = 2$ and $k = 4$ Formula: $$I \approx \frac{h}{2}\left[f(x0) + 2\sum{i=1}^{k-1}f(xi) + f(xk)\right], \quad h = \frac{b-a}{k}$$ $$\int2^4 (x^3+2)\,dx = \left[\fr...
Full solved answer →Multi-Segment Simpson's 1/3 rule
Simpson's 1/3 Rule Integration
Estimate the integral value of the following function from $x = 1.2$ to $x = 2.4$ using Simpson's 1/3 rule.
$$\begin{array}{c|cccccccc} x & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 & 2.6 \ \hline f(x) & 1.53 & 2.25 & 3.18 & 4.32 & 5.67 & 7.23 & 8.98 & 10.94 & 13.08 \end{array}$$
[5]
$x$ 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 -------------------------------------------------- $f(x)$ 1.53 2.25 3.18 4.32 5.67 7.23 8.98 10.94 13.08 Integration limits: $x = 1.2$ to $x = 2.4$ Step size: $h = 0.2$ $$h = 0.2, \quad n = \frac{2.4 - 1.2}{0.2} = 6 \...
Full solved answer →Gaussian integration algorithm
What is Gaussian integration formula? Evaluate the following integration using Gaussian integration three ordinate formula. $$\int_{0}^{1} \frac{\sin x}{x} dx$$ [5]
- Integral: $\int0^1 \frac{\sin x}{x}\,dx$ - Limits: $a=0$, $b=1$ - Method: Gauss-Legendre three-ordinate (n = 3) formula Three-point Gauss-Legendre nodes and weights: - $t1 = -\sqrt{3/5} = -0.7745967,\ w1 = 5/9$ - $t2 = 0,\ w2 = 8/9$ - $t3 = +\sqrt{3/5} = ...
Full solved answer →Romberg integration
Evaluate the following integration using Romberg integration. $$\int_{0}^{1} \frac{\sin^2 x}{x} dx$$ [5]
- Integrand: $f(x) = \dfrac{\sin^2 x}{x}$ - Limits: $a = 0$, $b = 1$ - Interval length: $b - a = 1$ Removable singularity at $x=0$: $\lim{x\to 0}\frac{\sin^2 x}{x} = \lim{x\to0}\frac{x^2}{x} = 0$, so $f(0)=0$. $x$ $\sin x$ $\sin^2 x$ $f(x)=\sin^2 x/x$ -----...
Full solved answer →Evaluate the following integration using Romberg integration. $$\int_{0}^{1} \frac{\sin x}{x} dx$$ [5]
- Integrand: $f(x) = \dfrac{\sin x}{x}$ - Limits: $a = 0$, $b = 1$ - Special value at $x=0$: $\lim{x\to 0}\dfrac{\sin x}{x} = 1$, so $f(0)=1$ - Method: Romberg integration - Interval width: $b-a = 1$ I compute the required function values (arguments in radi...
Full solved answer →Differentiating Continuous Functions
Write down algorithm and program for the differentiating continuous function using three point formula. [5]
Differentiating continuous functions is the process of approximating the derivative f'(x) of a function f(x) when the function itself is available. The Three-Point (Central Difference) Formula is: $$f'(xp) = \frac{f(xp + h) - f(xp - h)}{2h}$$ This formula u...
Full solved answer →Simpson's 1/3 rule
How Simpson's 1/3 rule differs from Trapezoidal rule? Drive the formula for Simpson's 1/3 rule. [5]
Aspect Trapezoidal Rule Simpson's 1/3 Rule --------- Approximation Approximates the integrand using straight lines (linear polynomials) between points Approximates the integrand using parabolas (quadratic polynomials) through three points Accuracy Less accu...
Full solved answer →Calculate the integral value of the function given below from $x = 1.8$ to $x = 3.4$ using Simpson's 1/3 rule.
$$\begin{array}{c|cccccccc} x & 1.8 & 2.0 & 2.2 & 2.4 & 2.6 & 2.8 & 3.0 & 3.4 \ \hline f(x) & 0.003 & 0.778 & 1.632 & 2.566 & 3.579 & 4.672 & 7.097 & 8.429 \end{array}$$
[5]
X 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.4 -------------------------------------------------------------- F(x) 0.003 0.778 1.632 2.566 3.579 4.672 7.097 8.429 Limits: $x = 1.8$ to $x = 3.4$ Spacing check: - 1.8 to 3.0: step $h = 0.2$ (equally spaced, 6 intervals) - ...
Full solved answer →Make Unit 3 stick
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