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