Calculus From First Principles

Finding the Gradient of a Straight Line We know how to find the gradient of a straight line. Gradient = rise run ( x 1 , y 1 ) ( x 2 , y 2 ) rise run

We know how to find the gradient of a straight line.

Gradient = rise

run

Finding the Gradient of a Curve We have a problem if we would like to find the gradient of a curve. If we look at which way each part of the curve slopes, we can see that the gradient is not the same at each part of the curve.

We have a problem if we would like to find the gradient of a curve.

If we look at which way each part of the curve slopes, we can see that the gradient is not the same at each part of the curve.

Negative, Zero and Positive Gradients Recall the following: Negative gradients slope downhill Positive gradients slope uphill Horizontal lines have a gradient of 0

Recall the following:

Finding the Gradient of a Curve Going back to y = x 2 , we can see that the gradient changes from negative, to zero, to positive. - - - - - - - + + + + + + + 0

Going back to y = x 2 , we can see that the gradient changes from negative, to zero, to positive.

Finding the Gradient of a Curve Since finding the gradient of a curve is difficult, we can consider the following: The gradient of a point on the curve is the same as the gradient of the tangent to the curve at that point.

Since finding the gradient of a curve is difficult, we can consider the following:

The gradient of a point on the curve is the same as the gradient of the tangent to the curve at that point.

Finding the Gradient of a Tangent This leads us to the question of how to find the gradient of a tangent. Whilst a tangent is a straight line, we can only find the coordinates of one point on the tangent – where the line touches the curve.

This leads us to the question of how to find the gradient of a tangent.

Whilst a tangent is a straight line, we can only find the coordinates of one point on the tangent – where the line touches the curve.

Finding the Gradient of a Secant To find the gradient of a tangent, we will consider a similar problem – finding the gradient of a secant. A secant is a line that crosses the curve in two places. Now we can find two points – and hence the gradient of the secant.

To find the gradient of a tangent, we will consider a similar problem – finding the gradient of a secant.

A secant is a line that crosses the curve in two places.

Now we can find two points – and hence the gradient of the secant.

Finding the Gradient of a Secant Let’s call the first x coordinate x . The y coordinate that corresponds is f(x) . Let’s make the second point some distance, h , away. The x coordinate of the second point is x + h . The y coordinate that corresponds is f(x + h) . x x + h f (x) f (x + h)

Let’s call the first x coordinate x .

The y coordinate that corresponds is f(x) .

Let’s make the second point some distance, h , away.

The x coordinate of the second point is x + h .

The y coordinate that corresponds is f(x + h) .

Finding the Gradient of a Secant Gradient = rise run x x + h f (x) f (x + h) rise run

Gradient = rise

run

Finding the Gradient of a Secant The gradient of the secant is NOT the same as the gradient of the curve, which remember is what we would like. If we make the distance h smaller, the gradient will be closer to the gradient of the tangent.

The gradient of the secant is NOT the same as the gradient of the curve, which remember is what we would like.

If we make the distance h smaller, the gradient will be closer to the gradient of the tangent.

Finding the Gradient of a Secant Make the distance h shorter. The gradient of the secant is now closer to the gradient of the tangent. Make the distance shorter again. As h -> 0 , the secant becomes a tangent and the gradient of the secant becomes the gradient of the tangent. x x + h f (x) f (x + h) x + h 2 f (x + h 2 ) x + h 3 f (x + h 3 )

Make the distance h shorter.

The gradient of the secant is now closer to the gradient of the tangent.

Make the distance shorter again.

As h -> 0 , the secant becomes a tangent and the gradient of the secant becomes the gradient of the tangent.

Finding the Gradient of a Tangent We can now find the gradient of the tangent by finding: Remember if we can find the gradient of the tangent to the curve at a point, we can find the gradient of the curve at that point.

We can now find the gradient of the tangent by finding:

Remember if we can find the gradient of the tangent to the curve at a point, we can find the gradient of the curve at that point.

Terminology and Notation The process of finding the gradient of the tangent to a curve is differentiation . The gradient function is called the derivative . These are all different notations for the derivative

The process of finding the gradient of the tangent to a curve is differentiation .

The gradient function is called the derivative .