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 nicesmoothly changingfunctions. 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, theres 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 FUNCTIONSHorizontal lines A constant function y = c has only one output value. Its graph is a horizontal line at height c . LINEAR FUNCTIONSStraight 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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This note was uploaded on 11/13/2010 for the course MA 141 taught by Professor Wears during the Spring '07 term at N.C. State.

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