Mathematics background needed for calculus
Clark University
You need to know a fair amount of mathematics before embarking on
a study of calculus.
Listed below are topics in mathematics that are used in calculus.
Some are essential for the development of the subject. They're marked
with the symbol
. Others are used
incidently in applications of calculus. Most of them we assume
that you know and we won't review them at all, but we'll remind you a
bit about a few of them as we use them. Most of the topics are used in the
first semester of calculus, but a few aren't used until later.
- Topics from arithmetic. We assume you know these:
- Kinds of numbers. Fractions and decimals. We'll refer to integers
(whole numbers, either positive, negative, or zero), rational and
irrational numbers. The number line
Conventions for arithmetic notation including order of operations
(precedence), proper use of parentheses
Expression manipulation. Distibutive laws, law of signs
Exponents and laws for exponents
Roots, laws for roots, rational exponents, rationalizing denominators
Absolute value, order (less than, etc.), and their properties
- Factorials (e.g., 5! is the product of the integers from 1 through 5)
- Topics from geometry
Pythagorean theorem
Similar triangles
Areas of triangles, circles, and other simple plane figures
Perimeters of simple plane figures, circumference of circles
- Volumes of spheres, cones, cylinders, pyramids
- Surface areas of spheres and other simple solid figures
- Topics from algebra. We use algebra constantly. You've got to know
algebra well. Topics:
Translating word problems into algebra
Expression manipulation. Addition, subtraction, and multiplication
of polynomials
- Rational functions and their domains, least common denominators
Techniques for simplifying algebraic expressions
Factoring quadratic polynomials and other simple polynomials
Techniques for solving linear equations in one unknown
Solving quadratic equations in one unknown, completing the square,
quadratic formula
Solving linear equations in two or more unknowns
Techniques for solving inequalities and both equations and
inequalities involving absolute value
The concept of function, functional notation and substitution, domain
and range of a function
Composition of functions
Uniform motion in a straight line. When objects move with constant velocity, the relation
among distance, time, and velocity
- Notation and concepts from set theory. We only use a bit of the notation from
set theory and only the most basic concepts
- Sets, membership in sets, subsets, unions, intersections, empty set
- Open and closed intervals and their notations
- Topics from analytic geometry. Mainly the basics, straight lines,
circles, a little on quadratic functions
Coordinates of points in the plane
Linear equations. Slope-intercept form especially, but also other forms
Distance between two points
Equations of circles, especially the unit circle
Slopes of straight lines, parallel lines
Graphs of functions. Vertical line test
- Symmetries of functions, even and odd functions.
Transformation of functions
- Graph of a quadratic function is a parabola
- Graph of y = 1/x is a rectangular hyperbola
- Topics from trigonometry. For a review of trigonometry, see
"Dave's Short Course in Trig" at
http://www.clarku.edu/~djoyce/trig/
Angle measurement, both degrees and radians, but radians are more
important in calculus. Negative angles.
- Length of an arc of a circle
Understanding of trig functions of angles, especially sine,
cosine, tangent, and secant. Trig functions and the unit circle
Right triangles, trig functions sine, cosine, and tangent of
acute angles. Values of these trig functions for standard angles
of 0, π/6, π/4, π/3, π/2
Solving right triangles
- Obtuse triangles. Law of sines, law of cosines. Solving obtuse
triangles
Basic trig identities. Pythagorean identities, trig functions in terms
of sines and cosines
- Other trig identities. Double angle formulas
for sine and cosine, addition formulas for sine and cosine
- Exponential functions and logarithms. Although these are topics
in algebra, they deserve to be separated for emphasis
Exponential functions. Growth of exponential functions
Laws for exponents. Manipulation of algebraic expressions involing
exponents, solving equations involving exponents
Logarithms and their relation to exponential functions
Laws for logs. Manipulation of algebraic expressions involing
logs, solving equations involving logs
An understanding of mathematical proof. We'll develop more in
calculus. You should be able to follow proofs like the ones you've already
seen in geometry, algebra, and your other mathematics courses
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