1. Informal and formal (i.e., δ-ε ) definitions of limit. Be able to do a δ-ε proof for a linear function.
2. Definition of continuity at a point. Be able to look at a graph and tell if a function
is continuous at a. Be able to examine a function and determine if it is continuous at a point, e.g.
let
(Think one-sided limits.)3. Use of limits to find asymptotes (both vertical and horizontal).
4. Definition of derivative in its two forms:
and 
, and be able to evaluate these limits to find the derivative. 5. Interpretations of the derivative as both slope of tangent line and rate of change of a function.
6. If f is differentiable at
, then f is continuous at 
.7. Derivative formulas: power, sum, product, quotient, difference, chain rule. Be comfortable with both prime and Leibniz notation.
8. Derivatives of trig, exponential, logarithmic, inverse trig functions.
9. Linear approximation and differentials; know how to approximate using them.
10. Implicit differentiation and related rates.
11. Higher order (second, third, etc.) derivatives and their notations.
12. Finding critical numbers and absolute extrema of a continuous function on a closed interval.
13. Mean Value Theorem: know the statement and what it means.
14. Finding intervals of increasing/decreasing, concave up/concave down.
15. Finding relative extrema via First or Second Deriviative Tests, finding inflection points.
16. L'Hopital's Rule; be able to find limits for
indeterminate forms.17. Sketch graph of a function from information obtained from analysis of first/second derivatives, asymptotes.
18. Optimization problems.
19. Velocity and acceleration.
20. Rectangle approximations for area and net area/Riemann sums.
21. Definition of definite integral
![∫_a^bf(x) x = lim_ (n ∞) Underoverscript[∑, i = 1, arg3] f(x_i^*) Δ x](HTMLFiles/index_7.gif){kind=small}
. .22. Fundamental Theorem of Calculus (parts I and II), e.g.,
![d/(d x) ∫_a^x (1 + t^2)^(1/2) t = (1 + x^2)^(1/2), ∫_1^3x^2x = [1/3x^3 ] _1^3 = 7/3](HTMLFiles/index_8.gif){kind=small}
.23. Antiderivatives and indefinite integrals.
24. Integrals of power functions, trig functions, exponential functions .
25. Use of integrals to find areas between curves.
26. Use of integrals to find volumes using the washer/disk and shell methods.
27. Know and use arc length formula.
28. Integration by parts.
29. Integrals of products of powers of trig functions, e.g.
.30. Integration via trigonometric substitution, e.g.
.31. Integration via partial fraction decomposition.
32. Improper integrals (both kinds).
33. Infinite sequences; be able to determine convergence/divergence.
34. Definitions of infinite series, sequence of partial sums and convergence/divergence of series.
35. Geometric series, p-series: what they are, which ones converge, which ones diverge.
36. Apply various tests to determine convergence/divergence: integral test, comparison tests, alternating series test, ratio and root tests.
37. Absolute and conditional convergence; Absolute convergence implies convergence.
38. Interval of convergence for a power series.
39. Taylor/Maclaurin polynomials, and Taylor/Maclaurin series for
.40. Integration and differentiation of power series representations.
41. Remainder estimates for series (e.g., integral test, alternating series, Taylor).
42. Standard form for hyperbolas, ellipses, and parabolas.
43. Parametric equations: graphing, first and second derivatives, arc length.
44. Polar coordinates: graphing, conversion to and from Cartesian coordinates, integration in polar coordinates.
45. Vectors, dot product, projections of vectors, cross product.
46. Equations of lines and planes in 3-space.
47. Vector-valued functions; their derivatives and integrals.
48. Partial differentiation, interpretation of partial derivatives, tangent planes.
49. Chain rule for multivariable functions
50. Directional derivatives and the gradient vector.
51. Finding extreme values of multivariable functions, Lagrange multipliers.
52. Double and triple integrals in cartesian coordinates.
53. Vector fields, conservative vector field.
54. Line integral of f along C, Fundamental Theorem of line integrals, independence of path.
55. Green's Theorem