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1 Calculus 4/14/03 4:08 PM Page 1
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CALCULUS I
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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.
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