The equation of the tangent and normal to a curve
The tangent to a curve at a point is the line which goes through that point and which has the same slope as the curve does, at that point. Similarly the normal is perpendicullar to the tangent at that same point.
Finding it is not hard provided that you keep a clear idea of the steps you need to go through. To find a straight line first you need a slope, so you need to find the slope of the curve at the point in question. The steps are:
- First differentiate your curve: f'(x) is a function which gives the gradient of the tangent at every point x on the curve y=f(x)
- substitute the x value of the point in question to get the slope of the curve at that point
- Let's say the point is (a, f(a)). So m=f'(a) is the slope of the curve and the equation of the line is y=mx+c=f'(a)x+c
- Now substitute the co-ordinates of the actual point a,f(a) to find the value of c.
If you want to find the normal rather than the tangent then you follow the same steps as above, but the slope of the line is -1/f'(a), which is perpendicular to the slope of the tangent.
Exercises:
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