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Unformatted text preview: Calculus Refresher, version 2008.4 c 19972008, Paul Garrett, [email protected] http://www.math.umn.edu/˜garrett/ 1 Contents (1) Introduction (2) Inequalities (3) Domain of functions (4) Lines (and other items in Analytic Geometry) (5) Elementary limits (6) Limits with cancellation (7) Limits at infinity (8) Limits of exponential functions at infinity (9) The idea of the derivative of a function (10) Derivatives of polynomials (11) More general power functions (12) Quotient rule (13) Product Rule (14) Chain rule (15) Tangent and Normal Lines (16) Critical points, monotone increase and decrease (17) Minimization and Maximization (18) Local minima and maxima (First Derivative Test) (19) An algebra trick (20) Linear approximations: approximation by differentials (21) Implicit differentiation (22) Related rates (23) Intermediate Value Theorem, location of roots (24) Newton’s method (25) Derivatives of transcendental functions (26) L’Hospital’s rule (27) Exponential growth and decay: a differential equation (28) The second and higher derivatives (29) Inflection points, concavity upward and downward (30) Another differential equation: projectile motion (31) Graphing rational functions, asymptotes (32) Basic integration formulas (33) The simplest substitutions (34) Substitutions (35) Area and definite integrals (36) Lengths of Curves (37) Numerical integration 2 (38) Averages and Weighted Averages (39) Centers of Mass (Centroids) (40) Volumes by Cross Sections (41) Solids of Revolution (42) Surfaces of Revolution (43) Integration by parts (44) Partial Fractions (45) Trigonometric Integrals (46) Trigonometric Substitutions (47) Historical and theoretical comments: Mean Value Theorem (48) Taylor polynomials: formulas (49) Classic examples of Taylor polynomials (50) Computational tricks regarding Taylor polynomials (51) Prototypes: More serious questions about Taylor polynomials (52) Determining Tolerance/Error (53) How large an interval with given tolerance? (54) Achieving desired tolerance on desired interval (55) Integrating Taylor polynomials: first example (56) Integrating the error term: example 3 1. Introduction The usual trouble that people have with ‘calculus’ (not counting general math phobias) is with algebra , not to mention arithmetic and other more elementary things. Calculus itself just involves two new processes, differentiation and integration , and applications of these new things to solution of problems that would have been impossible otherwise. Some things which were very important when calculators and computers didn’t exist are not so important now. Some things are just as important. Some things are more important. Some things are important but with a different emphasis....
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 Fall '11
 GUICHARD
 Calculus, Geometry, Inequalities, Limits, lim

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