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Section 2.2 Higher order derivatives
Objectives
Interpret the second derivative to understand concavity and curvature.
Use the Second Derivative Test to classify critical points when possible.
Extend derivative ideas to third and higher orders to study patterns in a function’s behavior.
Introduce second derivative \(f''\) and its relationship to concavity
If
\(f''(x)>0\text{,}\) then
\(f\) is concave up
If
\(f''(x)\lt 0\text{,}\) then
\(f\) is concave down
Get into the Leibniz notation
\(\DS{\dv[2]{y}{x}}\) ...and maybe why it’s NOT very good! The proper notation really should be
\(\DS{\dv{x}(\dv{y}{x})}\text{.}\) But the notation is still pretty standard and hence we should learn it.
Third and higher order derivatives
