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
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Conventions for arithmetic notation including order of operations
(precedence), proper use of parentheses
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Expression manipulation. Distibutive laws, law of signs
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Exponents and laws for exponents
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Roots, laws for roots, rational exponents, rationalizing denominators
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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
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Pythagorean theorem
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Similar triangles
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Areas of triangles, circles, and other simple plane figures
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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
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Expression manipulation. Addition, subtraction, and multiplication
of polynomials
- Rational functions and their domains, least common denominators
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Techniques for simplifying algebraic expressions
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Factoring quadratic polynomials and other simple polynomials
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Techniques for solving linear equations in one unknown
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Solving quadratic equations in one unknown, completing the square,
quadratic formula
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Solving linear equations in two or more unknowns
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Techniques for solving inequalities and both equations and
inequalities involving absolute value
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The concept of function, functional notation and substitution, domain
and range of a function
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Composition of functions
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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
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Coordinates of points in the plane
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Linear equations. Slope-intercept form especially, but also other forms
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Distance between two points
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Equations of circles, especially the unit circle
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Slopes of straight lines, parallel lines
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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/
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Angle measurement, both degrees and radians, but radians are more
important in calculus. Negative angles.
- Length of an arc of a circle
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Understanding of trig functions of angles, especially sine,
cosine, tangent, and secant. Trig functions and the unit circle
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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
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Solving right triangles
- Obtuse triangles. Law of sines, law of cosines. Solving obtuse
triangles
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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
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Exponential functions. Growth of exponential functions
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Laws for exponents. Manipulation of algebraic expressions involing
exponents, solving equations involving exponents
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Logarithms and their relation to exponential functions
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Laws for logs. Manipulation of algebraic expressions involing
logs, solving equations involving logs
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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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