3: Applications of derivatives

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In Section 2.2 we defined the derivative at \(x=a\text{,}\) \(f'(a)\text{,}\) of an abstract function \(f(x)\text{,}\) to be its instantaneous rate of change at \(x=a\text{:}\)

\begin{align*} f'(a) &= \lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a} \end{align*}

This abstract definition, and the whole theory that we have developed to deal with it, turns out be extremely useful simply because “instantaneous rate of change” appears in a huge number of settings. Here are a few examples.

If you are moving along a line and \(x(t)\) is your position on the line at time \(t\text{,}\) then your rate of change of position, \(x'(t)\text{,}\) is your velocity. If, instead, \(v(t)\) is your velocity at time \(t\text{,}\) then your rate of change of velocity, \(v'(t)\text{,}\) is your acceleration. We shall explore this further in Section 3.1.
If \(P(t)\) is the size of some population (say the number of humans on the earth) at time \(t\text{,}\) then \(P'(t)\) is the rate at which the size of that population is changing. It is called the net birth rate. We shall explore it further in Section 3.3.3.
Radiocarbon dating, a procedure used to determine the age of, for example, archaeological materials, is based on an understanding of the rate at which an unstable isotope of carbon decays. We shall look at this procedure in Section 3.3.1
A capacitor is an electrical component that is used to repeatedly store and release electrical charge (say electrons) in an electronic circuit. If \(Q(t)\) is the charge on a capacitor at time \(t\text{,}\) then \(Q'(t)\) is the instantaneous rate at which charge is flowing into the capacitor. That's called the current. The standard unit of charge is the coulomb. One coulomb is the magnitude of the charge of approximately \(6.241 \times 10^{18}\) electrons. The standard unit for current is the amp. One amp represents one coulomb per second.

3.1: Velocity and Acceleration
If you are moving along the \(x\)–axis and your position at time \(t\) is \(x(t)\text{,}\) then your velocity at time \(t\) is \(v(t)=x'(t)\) and your acceleration at time \(t\) is \(a(t)=v'(t) = x''(t)\text{.}\)
3.2: Related Rates
Consider the following problem: A spherical balloon is being inflated at a rate of \(13cm^3/sec\text{.}\) How fast is the radius changing when the balloon has radius \(15cm\text{?}\)
3.3: Exponential Growth and Decay — a First Look at Differential Equations
A differential equation is an equation for an unknown function that involves the derivative of the unknown function. For example, Newton's law of cooling says: The rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings.
3.4: Approximating Functions Near a Specified Point — Taylor Polynomials
Suppose that you are interested in the values of some function \(f(x)\) for \(x\) near some fixed point \(a\text{.}\) When the function is a polynomial or a rational function we can use some arithmetic (and maybe some hard work) to write down the answer. For example:
3.5: Optimisation
One important application of differential calculus is to find the maximum (or minimum) value of a function. This often finds real world applications in problems such as the following.
3.6: Sketching Graphs
One of the most obvious applications of derivatives is to help us understand the shape of the graph of a function. In this section we will use our accumulated knowledge of derivatives to identify the most important qualitative features of graphs \(y=f(x)\text{.}\) The goal of this section is to highlight features of the graph \(y=f(x)\) that are easily
3.7: L'Hôpital's Rule and Indeterminate Forms
Let us return to limits (Chapter 1) and see how we can use derivatives to simplify certain families of limits called indeterminate forms. We know, from Theorem 1.4.3 on the arithmetic of limits, that if