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; finding
derivatives by definition.
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 rate 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, 1st
derivative test. Inflection points, interval of concave up
and concave down; 2nd derivative test. Absolute
extrema of functions over closed and finite
intervals. Applied/story problems of optimization.
Graph of functions with all essential elements: intercepts, local extrema, inflection points, intervals of increasing and
decreasing, 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 rules/formulas
for numerical 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 . Find specific values of the antiderivative
of a function which is given graphically.
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.