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SparkChartsTM—created by Harvard students for students everywhere—serve as study companions and reference tools that cover a wide range of college and graduate school subjects, including Business, Computer Programming, Medicine, Law, Foreign Language, Humanities, and Science. Titles like How to Study, Microsoft Word for Windows, Microsoft Powerpoint for Windows, and HTML give you what it takes to find success in school and beyond. Outlines and summaries cover key points, while diagrams and tables make difficult concepts easier to digest. This four-page chart includes reviews: Definition of calculus and functions Types of functions and rules Trigonometric identities Limits and continuity Taking derivatives Using derivatives
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1 Calculus 4/14/03 4:08 PM Page 1 SPARKCHARTSTM CALCULUS I SPARK CHARTS TM BACKGROUND AND FUNCTIONS Calculus is the study of “nice”—smoothly changing—functions. • Differential calculus studies how quickly functions are changing at particular points. • Integral calculus studies areas enclosed by curves. • The Fundamental Theorem of Calculus connects the two. FUNCTIONS WHAT IS A FUNCTION? A function is a rule for churning out values: for every value you plug in, there’s a unique value that comes out. • The set of all the values that can be plugged in is the domain. • The set of all the values that can be output is the range. CONSTANT FUNCTIONS—Horizontal lines A constant function y = c has only one output value. Its graph is a horizontal line at height c. LINEAR FUNCTIONS—Straight lines A linear function can be expressed in the easy-to-graph slope-intercept form y = mx + b, where b is the y -intercept (the value of f (0)) and m is the slope. The slope of a straight line measures how steep it is; if (x1 , y1 ) and (x2 , y2 ) are two points on the line, then the slope is y2 − y1 change in y . = m= x2 − x1 change in x TYPES WRITING FUNCTIONS DOWN A table is a list that keeps track of input values (such as ages of a cactus) and corresponding output values (such as the number of needles on the cactus) of a function. There may not be a universal equation that describes such a function. A rational function is a quotient of two polynomials: f (x) = p(x) q(x) . It is defined everywhere except at the roots of q(x). The zeroes of f (x) are those roots of p(x) that are not also roots of q(x). • If the degree of p(x) is greater than the degree of q(x) (deg p(x) ≥ deg q(x) ), then at points where |x| is very large, f (x) will behave like a polynomial of degree deg p(x) − deg q(x). • If deg p(x) < deg="" q(x)="" ,="" then="" when="" |x|="" is="" very="" large,="" f="" (x)="" will="" approach="" 0.="" see="" limits="" and="" continuity.="" f="" (x)="an" xn="" +="" an−1="" xn−1="" +="" ·="" ·="" ·="" +="" a1="" x="" +="" a0=""> EXPONENTIAL AND LOGARITHMIC FUNCTIONS—Very fast or very slow growth Simple exponential functions can be written in the form y = ax , where the base a is positive (and a �= 1). The function is always increasing if a > 1 and always decreasing if a < 1.="" the="" domain="" is="" all="" the="" reals;="" the="" range="" is="" the="" positive="" reals.="" exponential="" functions="" grow="" extremely="" fast—="" faster="" than="" any="" polynomial.="" the="" basic="" shape="" of="" the="" graph="" is="" always="" the="" same,="" no="" matter="" the="" value="" of=""> Logarithmic functions have the form y = loga x. The number loga b is “the power to which you raise a to get b”: loga x = y if and only if ay = x. $7.95 CAN Printed in the USA $4.95 Copyright © 2002 by SparkNotes LLC. All rights reserved. SparkCharts is a registered trademark of SparkNotes LLC. A Barnes & Noble Publication 10 9 8 7 6 5 4 3 2 REMEMBER: Logar