Functions & Calculus Intro
Limits, average gradient, first principles
Limits, Average Gradient and First Principles
The average gradient between two points on a curve is the gradient of the secant line. As the two points get closer, the average gradient approaches the instantaneous gradient (derivative). Differentiation from first principles uses: f'(x) = lim[h→0] [f(x+h) − f(x)] / h.
Example
Average Gradient
f(x) = x². Find the average gradient between x = 1 and x = 3.
f(1) = 1, f(3) = 9
Average gradient = (9 − 1)/(3 − 1) = 8/2 = 4
This is the gradient of the secant line joining (1,1) and (3,9).
Example
Differentiation from First Principles
f(x) = x². Find f'(x) from first principles.
f'(x) = lim[h→0] [(x+h)² − x²] / h
= lim[h→0] [x² + 2xh + h² − x²] / h
= lim[h→0] [2xh + h²] / h
= lim[h→0] (2x + h)
= 2x
Note
Remember
First principles gives the general derivative formula. The derivative at a point gives the gradient of the tangent to the curve at that point. This is the foundation of calculus.
Key Vocabulary
LimitThe value a function approaches as the input approaches a certain value
Average gradientThe gradient of the line joining two points on a curve
DerivativeThe instantaneous rate of change of a function
First principlesFinding the derivative using the limit definition
Secant lineA line that intersects a curve at two points
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