Maths (university) · Part 04
Applications of derivatives
Using derivatives to find peaks, best choices, rates and approximations.
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- Critical pointA value of x where f'(x) = 0 or f'(x) does not exist. Local maxima and minima can only happen at critical points.
- First derivative testCheck the sign of f' on each side of a critical point. Positive then negative means a local maximum. Negative then positive means a local minimum.
- Second derivative testAt a critical point, if f'' is positive the curve bends upward (a minimum). If f'' is negative it bends downward (a maximum).
- OptimisationTurn a real problem into a function of one variable, find its critical points, and pick the best one. Check the ends of the allowed range too.
- Mean value theoremIf f is continuous on [a, b] and differentiable inside, there is a point c between a and b where f'(c) equals the average slope.
- Related ratesWhen two or more quantities change with time and are linked by an equation, differentiate the equation with respect to time to link their rates.
- L'Hopital's ruleIf a limit gives 0/0 or infinity/infinity, differentiate the top and the bottom separately and try the limit again.
- Linear approximationNear x = a, a curve looks like its tangent line. So f(x) is about f(a) + f'(a)(x - a) for x close to a.
- Newton's methodA way to find roots by repeating x_new = x - f(x)/f'(x). Each step follows the tangent line down to the x-axis.