* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download A quick review of Mathe 114
Survey
Document related concepts
Infinitesimal wikipedia , lookup
Series (mathematics) wikipedia , lookup
Partial differential equation wikipedia , lookup
Divergent series wikipedia , lookup
Matrix calculus wikipedia , lookup
Limit of a function wikipedia , lookup
Sobolev space wikipedia , lookup
History of calculus wikipedia , lookup
Lebesgue integration wikipedia , lookup
Generalizations of the derivative wikipedia , lookup
Multiple integral wikipedia , lookup
Transcript
Calculus: Differential calculus + integral calculus 1. Establish functions for simple problems (Domain of a function - where the function is meaningful) 2. Properties of a function 1). Limit of a function when the variable approaches a finite point or +/- infinity 2). Continuity (continuous at a point, continuous on the domain) 3). Differentiation. (a). Definition of derivatives (b). Derivatives of basic functions (c). Differentiation rules (product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation) (d). Use derivatives to find maximum or minimum values (on a closed interval or on an open interval) (e). Use derivatives to find increasing/decreasing/concave up/concave down intervals (f). Graph sketching D I S A I L C D: domain I: intercepts S: symmetry (even/odd/periodic) A: asymptotes (horizontal/vertical/slant asymptotes) I: increasing/decreasing intervals L: local maximum/minimum values C: concavity (concave up/down intervals) Final sketching: (i). Locate a few special points: points on the x-/y- axes; local maximum/minimum value points (ii). Divide the domain of the function into many subintervals by the critical points, inflection points and those points where the function is not defined; (iii). Indicate if f is increasing/decreasing, concave up/down in each subinterval (iv). Draw the graph (notice the asymptotes) (g). Meaning of derivatives: rate of change, slope, related rates (h): Use derivatives to solve optimization problems 4). Integration (a). Indefinite integrals--antiderivatives (b). Definite integrals (definition of a limit of Riemann sums) (c). Integral rules, antiderivatives of special functions, Fundamental theorem of calculus (d). Substitution rule for indefinite/definite integrals (e). Use symmetry of a function to find/simplify the definite integral of the function (f). Areas, distances, net change Theorems: Squeeze Theorem, Intermediate Value Theorem, Extreme Value Theorem, Rolle's Theorem, Mean Value Theorem, First/Second Derivative Tests, Fundamental Theorem of Calculus