Exam 3 Review Topics, 2008
Basics: Concept
of optimization, maximization and minimization with constraints. Indetermined forms of limit, and L’Hopital’s
Rules. Regular partition of intervals, general Riemann
sums, special Riemann sums: Left Point Sum, Right Point Sum, Midpoint Sum. Limit
of Riemann sums. Definition of definite integral. Geometric interpretation of definite integral in terms of area.
Average value of a function. Simple
rules for definite integrals. Definition of
anti-derivatives. Definition of indefinite integral.
Fundamental Theorems of Calculus – part I and part II. Method
of substitution. Finding areas between curves. Finding volumes by slicing, and by rotation.
Techniques: Story
problems in optimization: objective functions and constraints. Formulation optimization problems. L’Hopital’ rule for 0/0 type, ∞/∞
type, 0*∞ type. Hand calculation for left point sum, right point
sum, midpoint sum. Memorize 2 special formulas: 1+2+…+n=n(n+1)/2
and 1^2+2^2+…+n^2=n(n+1)(2n+1)/6. Use definition to find definite integrals:
i.e. taking limit of Riemann sums as the partition tends to infinity. Find
anti-derivatives for all elementary functions:
power functions, exponential functions, logarithmic functions,
trigonometric functions. Find anti-derivatives (indefinite integrals) by
substitution. Elementary rules for integration: linearity, additivity,
sign reversal with end point switching. Find areas
between curves, volumes by cross-sections, and volumes by rotation.
Elementary Functions: Differentiation and integration properties of elementary functions:
power functions, exponential functions, logarithmic functions, trigonometric
functions.
Miscellaneous Techniques: Finding critical points, finding absolute extrema
on closed intervals, 1st derivative test for local extrema. Finding limits of rational functions as x → ∞.
Distance, area, volume formulas for elementary geometric objects: rectangle,
triangle, circle, ellipse, rectangular solid, cylinders.
Quadratic formula
for roots, factorization of a^2-b^2, a^3-b^3, long division, multiply and
divide to maintain and transform quantities. Basic forms and shapes of
polynomials: linear function, parabola, cubic polynomials, definitions of trigonometric
functions, exponential functions, and logarithmic functions, basic identities
of trigonometric functions including double angle and half angle formulas,
basic rules, identities, and limiting properties as x → ∞ for
exponential and logarithmic functions, special values of trigonometric
functions and exponential and logarithmic functions
Partial Review Problems: All homework, quiz, lecture example problems, and sample exams