The second derivative test classifies a critical point c where f′(c) = 0 by the sign of f″(c). If f″(c) > 0, the graph is concave up there, so f has a local minimum. If f″(c) < 0, it is concave down, so f has a local maximum. If f″(c) = 0, the test gives no answer.
Concavity describes which way a graph bends. Where f″(x) > 0, the graph is concave up, shaped like a cup. Where f″(x) < 0, it is concave down, shaped like a cap. An inflection point is a point on the graph where the concavity changes, which usually happens where f″ = 0 or f″ is undefined.
The absolute extrema of a function on an interval are its highest and lowest values there. By the extreme value theorem, a function that is continuous on a closed interval [a, b] always has both an absolute maximum and an absolute minimum. To find them, evaluate f at the critical numbers and at the endpoints, then compare.
The first derivative test uses the sign of f′ to find where a function increases or decreases and to classify its critical points. If f′ changes from positive to negative at a critical number c, f has a local maximum at c. If f′ changes from negative to positive, f has a local minimum.
A critical point of a function f is a point where x = c is in the domain of f and either f′(c) = 0 or f′(c) does not exist. The number c is called a critical number. Every local maximum or minimum inside an interval happens at a critical point.
The mean value theorem says that if f is continuous on [a, b] and differentiable on (a, b), then at some point c between a and b, the instantaneous rate of change equals the average rate of change: f′(c) = (f(b) − f(a))/(b − a). Geometrically, some tangent line is parallel to the secant line through the endpoints.
Indeterminate forms are limit results like 0 · ∞, ∞ − ∞, 1^∞, 0^0 and ∞^0 that don’t tell you the answer by themselves. To find these limits, rewrite the expression as a fraction of the form 0/0 or ∞/∞, then use L’Hôpital’s rule. For the three power forms, take the natural log first.
L’Hôpital’s rule finds limits of fractions that give the indeterminate forms 0/0 or ∞/∞. If lim f(x)/g(x) has one of these forms, then it equals lim f′(x)/g′(x), as long as that new limit exists. Differentiate the top and bottom separately; do not use the quotient rule.
Differentials estimate how much a function’s output changes when its input changes a little. If y = f(x) and x changes by a small amount dx, the differential of y is dy = f′(x) dx. Then the true change Δy is close to dy. Differentials are often used to estimate measurement errors.
Linear approximation estimates a function’s value with its tangent line. Near a point x = a where you know f(a) and f′(a), use the linearization L(x) = f(a) + f′(a)(x − a). For x close to a, f(x) ≈ L(x). For example, starting from the exact value √25 = 5, it gives the estimate √26 ≈ 5.1.