Final Exam Review Topic List

Limit: Finding limit using graphs (for piecewise continuous functions); finding limit numerically using calculator; L’Hopital Rules for 0/0, ∞/∞ types; rational functions determined by their leading degree terms. With applications in: finding horizontal asymptotes, vertical asymptotes using side limits; removable or nonremovable discontinuities; finding derivatives by definition.

Continuity: Definition lim_{x → a} f(x) = f(a); property of: existence of absolute extrema in closed and finite intervals.

Derivatives: Definition, f’(x) = lim _{h → 0} (f(x+h) – f(x)) / h; find derivatives by definition; estimate derivatives numerically, graphically; find derivatives algebraically. Geometric meaning of derivatives: slope of tangent line/slope of function; average rate of change; instantaneous rate of change. Rules/properties of: linearity rule, product rule, quotient rule, chain rule; derivatives of elementary functions. Derivatives related to the Fundamental Theorem of Calculus. Implicit differentiation; related rate problems; story problem of related rate. Practical meaning of derivatives in terms of rates of change; geometrical interpretation of derivatives in terms of the slope of tangent line. Equations of tangent lines. With applications in: finding linear approximation of functions, differential of functions; approximation of function values by linearization. Critical points, maximum and minimum points; interval of increasing and decreasing, 1^st derivative test. Inflection points, interval of concave up and concave down; 2^nd derivative test. Absolute extrema of functions over closed and finite intervals. Applied/story problems of optimization. Graph of functions with all essential elements: intercepts, asymptotes, local extrema, inflection points, intervals of monotonicity, intervals of concavity. Graphical relationship between function and its first derivative and second derivative functions. Mean Value Theorem: f’(c) = (f(b)-f(a))/(b-a) for some c in [a,b].

Integrals: Definition of definite integrals; meaning of definite integral in terms of signed area between the curve and the x-axis. Special Riemann sums: left endpoint, right endpoint, and midpoint rule/approximation; calculation by hand. Average value of a function and the Mean Value Theorem for integrals: if f(x) is continuous and c is in (a,b). Definition of antiderivative/indefinite integral. Elementary methods to find antiderivatives/indefinite integrals: power rules, reversal of derivative formulas for elementary functions, simplification/manipulation of integrants before integration, method of substitution, linearity rule of integration, additive rule. Fundamental Theorems of Calculus: where F’(x) = f(x), and ; area of region between curves; volume of solid by cross-section method; volume of rotational symmetrical solid by washer-method and cylindrical shell method; length of curves.

Elementary Functions: 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; Derivatives of power functions, exponential functions, logarithmic functions, trigonometric functions; antiderivatives of power functions, exponential functions, trigonometric functions.

Calculator Skills: Sketch graphs, tracing intersection points and roots, finding numerical limits.

Miscellaneous Techniques: quadratic formula for roots, factorization of a^2-b^2, a^3-b^3, long division, multiply and divide to maintain and transform quantities. Distance and speed. Area, volume formulas for elementary geometric objects: rectangle, triangle, circle, ellipse, rectangular solid, cylinders.

Critical Skills: Proficiency in college algebra; proper use of parentheses.

Partial Review Problems: All homework, quizzes, lecture example problems, sample hour exams, and sample final exams.

Study Tips: Repeat homework/quiz/test problems until you can do them without any help, not by memorization but by reasoning. Before you can do this don’t try any sample final exam.