Download Final Review - Mathematical and Statistical Sciences

A List of Concepts, Theorems and Methods
Mathematics 113/114, Section D2
Below are listed most of the concepts, important theorems and methods covered in the Math
113/114 class. You should be familiar with things on this list. In the case of theorems, make sure
you know the content of all of them, and know when and how to apply them. The methods listed
should be familiar: know what type of situations each is useful in (so that when presented with a
question, you can quickly decide what to do.)
The material in the appendices should mostly be review. You should be comfortable enough
with the material there to understand everything we covered from Chapters I-V. In addition, I
recommend perusal of the textbook reference pages. Be familiar with trigonometric functions, their
elementary properties and identities, their values at the most important angles. The derivatives and
antiderivatives of the most common functions (trigonometric, exponential, inverse trigonometric,
logarithmic) should be readily at hand.
Finally, it is good to have an example in mind for the concepts and definitions you learn. It makes
learning a lot easier, and having a few functions in mind can be a big help with multiple choice
questions.
Concepts
Chapter I: Functions and Models Function, domain, codomain, range, independent and dependent variable, graph of function, Vertical Line Test, piecewise defined fns, even, odd, periodic, increasing, decreasing, linear function, slope, intercept, polynomial, coefficient, degree,
power function, rational fn, algebraic fns, trigonometric and inverse trigonometric functions
and their properties, exponential and logarithmic functions, transformations of fns (vertical
and horizontal shift and stretch, reflections), pointwise addition and multiplication, composition, Laws of Exponents, the number e, one-to-one (injective) fns, Horizontal Line test,
inverse fn, Laws of Logarithms.
Chapter II: Limits and Derivatives Limit of a function (at a number or at infinity), infinite
limit, limit from one side, vertical and horizontal asymptotes, Limit Laws (+, −, ·, ÷, power,
root), Direct Substitution Property, Squeeze Theorem; continuous functions, their behavior
and examples, The Intermediate Value Theorem; differentiable functions, derivative, tangent
line and its equation, differentiability and continuity.
Chapter III: Differentiation Rules Methods to compute derivatives (+, −, power rule), derivatives of polynomials, exponential functions, trigonometric functions. Product Rule, Quotient
Rule, Chain Rule, derivatives of logarithmic functions, implicit differentiation (finding tangents to curves at a given point, or the point where the tangent is horizontal), derivative of an
inverse function, logarithmic differentiation, related rates, linear approximation, linearization
of a function at a point, differentials, rates of change in sciences.
Chapter IV: Applications of Differentiation Local or absolute maximum or minimum values, The Extreme Value Theorem, Fermat’s Theorem, critical number, The Closed Interval
Method, Rolle’s Theorem, The Mean Value Theorem and corollaries, I/D Test, First Derivative Test, concavity, Concavity Test, Second Derivative Test, Guidelines of Curve Sketching,
methods for solving Optimization Problems, First Derivative Test for Absolute Extreme Values, Newton’s Method, Antiderivatives, theorem about most general antiderivative.
CONTINUED ON THE FOLLOWING PAGE
Chapter V: Integrals Antiderivatives and rectilinear motion, a few examples of antiderivatives in
physics (position function, velocity, acceleration), area under a curve, Riemann-sum, sample
points, definite integral, integrable functions, examples, Midpoint Rule, integrals as (difference
of) areas, integral over adjacent intervals, elementary properties of definite integral: addition,
constant multiple, comparison properties, The Fundamental Theorem of Calculus (Part I and
II), indefinite integrals, the Net Change Theorem, distance traveled, Substitution Rule for
indefinite and definite integrals, integrals of symmetric functions.
Appendix A: Numbers, Inequalities and Absolute Values Integers, rational and irrational
numbers, decimal representation, real numbers. The number line (real line), inequalities,
set, element, union, intersection, set notation. Intervals: open intervals, closed intervals
(notation with various brackets), infinite intervals. Inequalities, and solving them, presenting
the solution in interval notation. Absolute value and its properties, triangle inequality.
Appendix B: Coordinate Geometry and Lines The coordinate plane (Cartesian Coordinate
system), coordinate axes, origin, quadrants, x-coordinate, y-coordinate, distance formula.
Equations of lines: slope, point-slope form, slope-intercept form. Parallel and perpendicular
lines, relationship of their slopes.
Appendix C: Graphs of Second-Degree Equations Equation of a circle, parabolas, ellipses,
hyperbolas.
Appendix D: Trigonometry Angles measured in degrees or radians, conversion between the
two. Trigonometric functions (sin x, cos x, tan x, cot x, sec x, csc x), computing sin x and
cos x for an angle in standard position using the unit circle around the origin, other trigonometric functions from sin and cos . Special angles. Trigonometric identities, e.g. Pythagoras’
Theorem. Graphs of trigonometric functions (range, symmetries, periodicity).
Appendix E: Sigma notation what the Sigma notation means, properties with respect to addition, subtraction, constant multiple.
Theorems
Vertical and Horizontal Line Test, Laws of Exponents, Laws of Logarithms, Limit Laws, Direct
Substitution Property for Rational Functions, Squeeze Theorem, The Intermediate Value Theorem,
Power Function Rule, Product Rule, Quotient Rule, Chain Rule, The Extreme Value Theorem,
Fermat’s Theorem, Rolle’s Theorem, The Mean Value Theorem, I/D Test, First Derivative Test,
Concavity Test, Second Derivative Test, First Derivative Test for Absolute Extreme Values, The
Fundamental Theorem of Calculus (Part I and II), Substitution Rule for indefinite and definite
integrals.
Methods
Finding the domain of a function by solving inequalities. Transforming the graph of a function
by shifts, stretches and reflections; determining if a function is one-to-one, determining the inverse
of a one-to-one function. Computing limits: Limit Laws, simplification for rational functions,
multiplying by the conjugate to get rid of a root, simplifying fraction by dominant term, application
sin x
of the fact lim
= 1). Finding parameters for which a piecewise defined function is continuous.
x→0 x
Finding horizontal and vertical asymptotes of a graph. (Examining left- and right hand limits at
a point of discontinuity.) Proving existence of a solution with the Intermediate Value Theorem.
Computing derivatives: recognizing when the product rule, quotient rule and chain rule should be
used, correctly applying them (even multiple times for the same function). Computing derivatives
from the definition. Finding parameters for which a piecewise defined function is differentiable.
Implicit Differentiation, finding horizontal tangents of a graph, Logarithmic Differentiation, Method
for Solving Related Rates Problems. Finding local and absolute extrema of a function by various
methods; Closed Interval Method. Methods for Solving Optimization Problems, Process of Curve
Sketching (and methods to answer all questions along the way), manipulating formulas with the
sigma notation, Computing Integrals, finding the derivative of a function defined as an integral
function (FTC).