,
the derivative 
is approximately equal to the gradient of the secant.
Activities
Numerical methods - small changes - Taylor polynomials
Derivative - gradient of tangent - approximation to gradient of secant
For a small change ,
the derivative
is approximately equal to the gradient of the secant.
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Small changes in the value of a function can be estimated by approximating the function by the tangent.
Example
If the radius of a sphere is increased by 1.5%, what is the percentage increase in its volume?
We need to compare the change in volume with the actual volume.
Therefore the volume increases by approximately 4.5%.
Approximating a function by the equation of the tangent
Consider the equation of the tangent drawn at the point where x=x0.
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Close to the point where the tangent is drawn, the function can be approximated by the tangent.
(Substituting y=0 in the equation of the tangent and rearranging for x gives the formula for Newton's method.)
Taylor polynomials
Taylor polynomials are a way of approximating a function close to a particular point.
The linear approximation (tangent) is the 1st order Taylor polynomial.
A quadratic approximation (parabola) is given by the 2nd order Taylor polynomial.
A cubic approximation is given by the 3rd order Taylor polynomial.
As the order is increased, a better approximation is achieved:
Taylor polynomials - examples and more details
Taylor
polynomial calculators
Taylor
polynomials - interactive examples
