Final Exam Review Topic List
Limit: Numerical method using calculator; graphical method for e-d definition; L’Hopital Rules for 0/0, ∞/∞ types; rational functions determined by their leading degree terms; Squeeze Theorem. With applications in: finding horizontal, vertical asymptotes; removable or nonremovable discontinuities; finding derivatives by definition; finding definite integrals by definition;
Continuity: Definition limx → a f(x) = f(a); Intermediate Value Theorem with application in finding solutions to equations; 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: linearity rule, product rule, quotient rule, chain rule, derivatives of elementary functions. Implicit differentiation; related rate problems. 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, local extrema, graphs of functions. Critical points, interval of increasing and decreasing, 1st derivative test. Inflection points, interval of concave up and concave down. 2nd derivative test. Absolute extrema of functions over closed and finite intervals. Graph of functions with all essential elements: intercepts, asymptotes, local extrema, inflection points, intervals of monotonicity and concavity. Mean Value Theorem: f’(c) = (f(b)-f(a))/(b-a) for some c in [a,b].
Integrals: Definition of definite integrals; meaning of definition integral in terms of signed area between the curve and the x-axis. Special Riemann sums: left, right, midpoint, trapezoid, and Simpson; calculation by hand, by calculator programs, by definition making use of formulas: 1+2+…+n=n(n+1)/2, 1^2+2^2+3^2+…+n^2=(2n+1)n(n+1)/6. Under and over estimates by left, right, midpoint, trapezoid sums in relationship to monotonicity and concavity of functions. Average value of a function; Mean Value Theorem for integrals: if f(x) is continuous and c is in (a,b). Definition of indefinite integral, antiderivative of functions. Elementary methods to find antiderivatives/indefinite integrals: power rules, reversal of derivative formulas for elementary functions, simplification/manipulation of integrants before integration, special substitution: integral of f’(x)/f(x) type integrants, linearity rule of integration, additive rule. Fundamental Theorems of Calculus: where F’(x) = f(x), and .
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; approximating definite integrals by left, right, mid, trapezoid, and Simpson rules.
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.
Partial Review Problems: All homework, quizzes, lecture example problems, sample hour exams, and sample final exams.