Optimisation Problems
Optimisation problems involve finding the maximum or minimum value of a function within a given domain. These problems often require finding critical points using differentiation and determining whether they represent maxima, minima, or neither.
Steps to Solve Optimisation Problems
- Identify the quantity to be optimised and express it as a function of one variable.
- Determine the domain of the function based on the problem constraints.
- Find the critical points by setting the derivative equal to zero and solving.
- Evaluate the function at the critical points and at the endpoints of the domain.
- Compare the values to determine the optimal solution.
A farmer wants to fence off a rectangular field adjacent to a straight river. He has 100 meters of fencing and wants to maximise the area of the field. The side along the river doesn't need fencing. What dimensions will give the maximum area?
- Let $x$ be the width of the field and $y$ be the length. The area is $A = xy$.
- The perimeter constraint gives us: $x + 2y = 100$. Solving for y: $y = 50 - \frac{x}{2}$
- Substituting into the area function: $A(x) = x(50 - \frac{x}{2}) = 50x - \frac{x^2}{2}$
- Finding the critical point: $\frac{dA}{dx} = 50 - x$ Set this equal to zero: $50 - x = 0$ $\\x = 50$
- The domain is $0 \leq x \leq 100$. Evaluating at x = 0, 50, and 100:
$A(0) = 0$
$A(50) = 1250$
$A(100) = 0$ - The maximum area occurs when $x = 50$ meters and $y = 25$ meters.
In optimisation problems, always check the endpoints of the domain, even if they don't satisfy $\frac{dy}{dx} = 0$. Sometimes the optimal solution occurs at an endpoint.
NoteSolving optimization problems involving multiple variables is difficult, therefore we would like to write it using one single variable as done above.
Applications and Importance
The techniques of optimisation are crucial in various fields:
- Engineering: Optimisation is essential in designing efficient systems and structures.
- Economics: These methods are applied in cost minimization and profit maximization problems.
- Computer Science: Optimisation algorithms often use these calculus techniques at their core.
The ability to solve these types of problems demonstrates a deep understanding of calculus and its applications, which is highly valued in many academic and professional fields.
TipWhen solving these problems, always sketch a diagram if possible. Visual representation can greatly aid in understanding the problem and setting up the correct equations.
