Q1. null
- (a)) State two real-world problems where you think that mathematical modelling is the only approach to find the solution of the problem. Give 4 essentials for each of the problems. Why do you think that there is no other scientific alternative for the treatment of these problems. (200 words)
- (b)) Classify the following into linear and non-linear models, justifying your classification. i) Simple harmonic motion for small amplitude of oscillation. ii) Population growth model given by dN/dt = aN (B-kN), a, B, k are constants. iii) Equation for velocity v of a particle at any time t, moving with a constant acceleration a, and initial velocity u. iv) Equation describing dynamic stability of market equilibrium price given by p_t = [1 + k(a - A)p_t-1 + k (b − B)], a, b, A, B, k are constants and p_t is the price in period t. (300 words)
- Mathematical modeling is essential for problems too complex, large, ethical-sensitive, or long-term for direct study.
- Climate change prediction relies solely on mathematical models due to its vast scale and temporal scope.
- Drug dosage optimization employs modeling for ethical, safety, and efficiency reasons in pharmacology.
- Linear models have dependent variables and derivatives appearing only to the first power, without products.
Answer: Mathematical modeling serves as an indispensable tool for understanding and solving complex real-world problems that are otherwise intractable through direct experimentation or observation. It involves representing a system using mathematical concepts and language, allowing for analysis, prediction, and optimization. This approach is particularly crucial when systems are too large, too small, too dangerous, too slow, too fast, or too complex to study directly. Models can be broadly classified b...