Explore y=a^x including calculus
- Richard Trimble
Investigate the shape of the graph by varying the constant below. This curve is typical of many types of exponential growth or decay in nature such as early stage bacterial growth or radioactive decay. It also comes up in financial contexts with, say, a specific rate of interest applied to an account. If we wish to differentiate this function we need to return to first principles. This expression tells us that the gradient of the tangent is the limiting value of a sequence of ever smaller chords from to . In our case this gives us: [factorising by the power laws] [taking outside the limit] This shows us that the gradient function of is itself but multiplied by a constant equal to the limiting value of as . Choose different values of and use for example to estimate this constant. What do you notice? There are two other geometric ways of understanding this limit. - Firstly if we divide the value of the gradient at a point by the value of the function at that point the will cancel leaving the limit. - secondly the value of the limit is the same as the gradient of the curve at . Use the interactivity below to explore the value of this limit for different values of using either or both of these geometric understandings.
Is there a value of a for which the limit is equal to one? Stop and think for a moment what that would mean, if there was such a value then it would mean that the derivative of that exponential function was equal to itself! To determine such a value we can cheat slightly and require the limit to equal one for a particular value of . i.e. It is then easy to rearrange this to solve for in terms of . Try small values of h in this expression. What sort of answers do you get for this special value of ? In fact we can find an explicit expression for this value just by using the binomial expansion. Remember that if then so we can re-write this expression for using the sequence of integers if we set and hence . This gives us: And if we denote our special value of by the letter we can then say: Try this expression with some large values of and see what you get.
If you are familiar with the binomial expansion we can imagine what we would get if we binomially expanded this expression. Some of the s cancel and we are left with expressions like . What happens to an expression like or as ? Does the same thing happen for ? It turns out that any expression like this does tend to one because we can always make as large as we please and overwhelm the deficit on the top line. What does this mean for our expansion? If all the expressions like then all we are left with is the coefficients. i.e. is the sum of the reciprocals of all the factorials.
Properties of e
is known as Euler's number, not because it was first found by Euler but because he derived so many of its properties and secured it's place as central in mathematics. The basic properties we have derived are that it is a special exponential curve whose gradient at is and hence it is its own derivative. Conversely we also have . other results that can be shown: