1. Equation of a tangent.
Instantaneous rate of change is given by the gradient of the tangent to the given point on a curve.
Draw a tangent, pick up two points, estimate the gradient of the tangent.
In Calculus we can determine the exact equation of the tangent to a curve at a given point.

2. Gradient & Intercept Review
Definition
The general equation of a straight line can be written as:
y = mx + c
The value of m tells us the gradient of the line.
The value of c tells us where the line crosses the y-axis.
Explain that the equation of a line can always be arranged to be in the form y = mx + c.
It is often useful to have the equation of a line in this form because it tells us the gradient of the line and where it cuts the x-axis. These two facts alone can enable us to draw the line without have to set up a table of values.
Ask pupils what they can deduce about two graphs that have the same value for m. Establish that if they have the same value for m, they will have the same gradient and will therefore be parallel.
This is called the y-intercept and it has the coordinate (0, c).
For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

3. Equation of tangent using calculus
Problem: Find the equation of the tangent to the curve y=x2+2 at a point where x=1.
To find the gradient, m, follow these steps:
Find dy/dx
Find the gradient at x=1
Write the equation as y=mx+c, we know the gradient now.
Find the y-value of the point.
Substitute the point into the equation to find c.

4. Problem: Find the equation of the tangent to the curve y=x2+2 at a point where x=1.
Therefore m=2 and y=2x+c
When x=1, y=12+2=3, so (1,3) lies on the parabola.
3=2(1)+c, c=1
Equation of the tangent is y=2x+1

5. Check with your GDC: Find the gradient at x=1 Draw the graph
Draw a tangent at this point

6. Your turn
Question 1: Find the equation of the tangent to the curve y=2-x3 at a point x=-2.