Average and instantaneous rate of change, the limit definition of the derivative, tangent lines, velocity, and where a function fails to be differentiable.
From average to instantaneous
Drive \(120\) miles in \(2\) hours and your average speed is \(60\) mph. But your speedometer shows your speed right now. Calculus gets the instantaneous rate by taking averages over smaller and smaller intervals: a limit.
The derivative
Shrink the interval: replace \(b\) by \(x + h\) and let \(h \to 0\).
Where derivatives fail
A continuous function can still fail to have a derivative at:
a corner, like \(|x - 2|\) at \(x = 2\) (left and right slopes differ);
a cusp, like \(x^{2/3}\) at \(0\);
a vertical tangent, like \(\sqrt[3]{x}\) at \(0\).
And of course at any discontinuity.
Practice
Quick check
What does \(f'(a)\) represent on the graph of \(f\)?
The derivative is the limit of secant slopes: the tangent slope.
Quick check
At \(x = 0\), the function \(f(x) = |x|\) is:
The graph has a corner: the left slope is \(-1\) and the right slope is \(1\).
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