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