Section 2 6 Derivatives and Rates of Change If an object travels in a straight line at a constant velocity v and f t is the distance from a reference point at time t then f t f a is the distance traveled from time a to time t We know f t f a v t a or distance rate time If the velocity was not constant we talked about the average velocity over a time interval and the instantaneous velocity at a See section 2 1 If we think of a graph of a continuous function in terms of distance and time then the xaxis is the time variable the value of f x is the distance The average velocity is the average rate of change AVROC of the function and is the slope of the secant line connecting P a f a and Q x f x Slope of Secant f x f a x a f a h f a x a h where h The instantaneous velocity is the instantaneous rate of change ROC and is the limit of the slopes of the secant lines as x approaches but is not equal to a if this limit exists This is the slope of the tangent line to the graph at a and is called f a read f prime at a Slope of Tangent f a lim x a f x f a x a lim f a h f a h 0 h if this limit exists Equation of Tangent line The tangent line at a f a passes through a f a and has slope f a so the equation using pt slope is y f a f a x a or y f a x a f a Examples 1 f x mx b is linear Then any secant line is the line itself and has the same slope m lim m m h 0 The derivative of any line is the slope of the line the derivative of a constant is 0 f x x 2 a 3 Find f 3 First write the secant slope for 3 h and 3 and simplify it Then let h approach 0 2 2 secant slope tangent slope 3 h 9 2 9 6h h 9 h h f 3 lim 6 h 6 6h h h 2 h 6 h 6 h h h 0 The rate of change of f x at 3 is 6 The height is increasing 6 times as fast as x is increasing The equation of the tangent line is y f 3 x 3 f 3 6 x 3 9 I prefer to leave it in this form but webassign will probably want it in y mx b form y 6x 9 Remember that means the change in y m x for a line and y f a x for a curve For the example above if x 0 1 y 0 6 This means that f 3 1 f 3 is about 0 6 so the square of 3 1 is about 9 6 It is actually 9 61 9 6 is the y value on the tangent line when x is 3 1 Exercise Let x 0 1 y 0 6 and compare the value of the tangent line at x 2 9 to f 2 9 Linear property of the derivative If A and B are numbers and f a and g a both exist then Af Bg a Af a Bg a Example Using examples above if h x 4 x 2 5 x 10 h 3 4 6 5 0 29 Just as the sign of the slope of a line tells whether the line is rising or falling as x moves to the right the derivative tells whether or not the graph is rising or falling at a as x moves to the right of a We know that f x x 2 is rising to the right of 0 and falling to the left of 0 f 3 6 is positive because f is increasing as x moves to the right of 3 Find f 3 and you will see it is negative as f is decreasing as x moves to the right of 3
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