List of Notes - Category: Applications of Derivatives
Suppose we are given one quantity `x` that depends on another quantity `y`. If rate of change of quantity `x` with respect to time `t` is given, how do we find rate of change of `y` with respect to time? We need to do it because in real-worlds problems it is often easier to calculate rate of change of `x` then rate of change of `y`.
Perhaps, the most important application of derivatives is solving optimization problems.
With the help of derivatives we can find minimum and maximum values. That's exactly what we need in optimization problems.
Suppose that some factory produces `x` units of good. Let's denote cost of producing `x` units of good by `C(x)`.
Cost function `C(x)` is the cost of producing `x` units of certain product.
Marginal cost is the rate of change of cost with respect to `x`. In other words marginal cost function is the derivative of cost function.
Sometimes function is defined parametrically, but we still need to find equation of tangent line.
So, let parametric curve is defined by equations `x=f(t)` and `y=g(t)`.
Suppose `f` and `g` are differentiable functions and we want to find the tangent line at a point on the curve where `y` is also a differentiable function of `x`.
Let's see how to derive equation of tangent line when we are given equation of curve `r=f(theta)` in polar coordinates.
We will proceed in the same fashion as with tangent lines to parametric curves, because polar coordinates in some sense similar to parametric curves.