# Discovering "e"

- Author:
- Marc Azara

- Topic:
- Exponential Functions

## The coordinates for Points B and C on the function, and the gradient of chord BC, are shown. You can move points A and B, and also change the exponential function by using the slider.

Use the sliders to create the function to y =
Move point B closer to point A, from both sides of point A.
Note the "limiting value" that the gradient of AB approaches.

## Use your calculator to estimate the value using gradient = rise/run. (Use x= 0.0001, and x=0)

Gradient of tangent to y = at the point (0,1) The limiting value of the gradient of the chord represents the gradient of the tangent to the curve. What is value of the gradient of the tangent to y = at (0,1), correct to 3 decimal places?

Use the slider to change the function to y =
Move point B closer to point A, from both sides of point A.
Note the "limiting value" that the gradient of AB approaches.

## Use your calculator to estimate the value using gradient = rise/run. (Use x= 0.0001, and x=0)

Gradient of tangent to y = at the point (0,1) The limiting value of the gradient of the chord represents the gradient of the tangent to the curve. What is value of the gradient of the tangent to y = at (0,1), correct to 3 decimal places?

## Can you make the gradient at (0,1) equal to 1?

It seems logical that somewhere between y = 2^x and y = 3^x, the gradient at A(0,1) will be exactly 1.
Use the slider to change the function y = , and experiment by moving point B closer to point A, from both sides of point A, until you find the function that has a gradient of 1 at (0,1).

What is value of "a" for y = , to have a gradient of 1 at (0,1), correct to 3 decimal places?

## You have discovered a very special number, known as "Euler's Number", and denoted by "e".

Now try moving point A to different locations on this special exponential function.
as well as point A(0,1) will be exactly 1.
Move point B closer to point A, from both sides of point A, and note the gradient of the tangent at various positions of point A.
What does this tell you about the gradient of the tangent anywhere on the function y = e^x