## 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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