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calculus1

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Unformatted text preview: SPARK CHARTS ™ C++ page 1 of 4 This downloadable PDF copyright © 2004 by SparkNotes LLC. SPARK CHARTS TM CALCULUS I SPARK CHARTS TM 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 Printed in the USA \$4.95 \$7.95 CAN SPARK CHARTS TM 50495 9781586636234 ISBN 1-58663-623-5 WHAT IS CALCULUS? 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. 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. An equation such as f ( x ) = x 2 + 1 describes how to numerically manipulate the independent variable (often x ) to get the output value f ( x ) . If y = x 2 + 1 , then y is a “function of x ,” and it is the dependent variable. A graph represents a function visually. If y = f ( x ) , then plotting many points x, f ( x ) ¡ on the plane will give a pic- ture of the function. Usually, the independent variable is represented horizontally, and the dependent variable verti- cally. A graph represents a function as long as it passes the vertical line test: for every x-value, there is at most one y- value. ¡unction Not a Function 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 ( x 1 , y 1 ) and ( x 2 , y 2 ) are two points on the line, then the slope is m = change in y change in x = y 2 − y 1 x 2 − x 1 . POLYNOMIAL FUNCTIONS A general polynomial function can be written in the form f ( x ) = a n x n + a n − 1 x n − 1 + · · · + a 1 x + a . The polynomial has degree n if a n = 0 . All polynomials are defined for all real numbers. If n is odd, then the range is all real numbers, too. If n is even, then the polynomial reaches some maximum if a n < or some minimum if a n > ....
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