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MATH 1100-02/FALL 2004 – REVIEW FOR FINAL EXAM The following is an outline of all of the material that we have covered this semester. Major concepts and typical problems from the textbook are mentioned. As well as knowing how to do the given problems one should know how to do all types of problems as given on the tests and quizzes. Good luck!! 1. Chapter 9: Derivatives • Limits: Algebraic and Geometric interpretation of the limit; Right and Left hand Limits (limx→c+ f (x) and limx→c− f (x)); Limit Properties (page 621 properties I-VI); Limits of Polynomials; Limits of type 0 0 and a 0. Section 9.1 #2, 6, 8, 12, 24, 26, 30. • Continuous Functions: Definition of Continuity; How to determine if Piecewise defined functions are continuous. Section 9.2 #16, 18, 22. • Limits at Infinity: Properties (page 640); How to calculate them (example 5 page 641.) Section 9.2 #34, 38. • The Derivative: Definition of the Derivative; Interpretation of the Derivative (marginal revenue, velocity, rate of change); Average Rate of Change; Differentiability (figure 9.24 page 658.) Section 9.3 #10, 16, 30 • Rules: Constant Rule; Constant Coefficient Rule; Sum/Difference Rule; Product Rule; Quotient Rule; Chain Rule. Section 9.4 #22, 26. Section 9.5 # 10, 22, 24. Section 9.6 #12, 16, 34. Section 9.7 #22, 24, 34. • Higher Derivatives: Section 9.8 #14, 30, 34. 2. Chapter 10: Applications of the Derivative • Relative Maxima and Minima: Derivative tells when function is increasing and decreasing; When does a function have a relative max or min? (f ′ (x) = 0 of f ′ (x) is undefined); First Defrivative Test; Maxima / Minima / Points of Inflection. Section 10.1 #30, 32, 34, 48. • Concavity/Points of Inflection: 2nd Derivative tells when a function is concave up or concave down; Finding Points of Inflection (page 754); The Second Derivative Test (page 757.) Section 10.2 #4, 20, 24, 30. • Optimization: Absolute Maxima/Minima; Maximizing Revenue / Minimizing Average Cost. Section 10.3 #2, 4, 20. • Asymptotes: How to Determine Vertical and Horizontal Asymptotes; Graphing. Section 10.5 #18, 20, 26. 1 2 MATH 1100-02/FALL 2004 – REVIEW FOR FINAL EXAM 3. Chapter 11: Derivatives Continued • Derivative of the Logarithm: If y = ln(u(x)) then y ′ = u′ (x) u(x) ; Properties of the Logarithm; Change of Base. Section 11.1 #10, 16, 34, 38. • Derivative of the Exponential: If y = eu(x) then y ′ = u′ (x)eu(x) ; Change of Base. Section 11.2 #12, 14, 24, 32, 34. • Implicit Differentiation: “Pretending y = f (x);” Solving for y ′ ; Equation of Tangent Line to a Curve. Section 11.3 #6, 12, 20, 36. • Related Rates: Both variables are functions of t; Implicit Differentiation to get an equation that relates the derivatives (rates.) Section 11.5 #4, 12, 30. 4. Chapter 12: Anti-Derivatives • Power Rule for Integration: #22, 24, 32. • Exponential Functions: • Logarithmic Functions: R R R u′ (x)[u(x)]n dx = u′ (x) u(x) dx ′ u (x)e u(x)n+1 n+1 + C. Section 12.1 #26, Section 12.2 = ln |u(x)| + C. Section 12.3 #10, 14. u(x) = eu(x) + C. Section 12.3 #22, 26. • Long Division: Section 12.3 #30, 32. • Differential Equations: Separation of Variables; Antidifferentiate both sides; General and Particular Solutions. Section 12.5 #32, 36. 5. Chapter 13: Definite Integrals • Area Under a Curve/Summation Formulas: Handout: Divide interval into n pieces, Using left of right endpoints determine SR or SL , use formulas to get rid of summation symbol, take limit as n → ∞. Section 13.1 #31, 32. • Area Under a Curve/Fundamental Theorem of Calculus: Relation between the integral of a function and any antiderivative. Section 13.2 # 18, 24, 33-40, 42, 44. • Area Between Two Curves: Setting up and Integral that Represents Area; Evaluating the Integral (example 3 page 949.) Section 13.3 #9, 11. R R R • Integration by Parts: udv = uv − vdu; Pick u and dv, du = u′ dx, v = dv, plug in and continue. Section 13.6 #8, 10, 14. Rb R∞ • Improper Integrals: a f (x)dx = limb→∞ a f (x)dx. Section 13.7 #4, 6.