MATH 12002 - CALCULUS I § 2.1: Derivatives and Rates of Change Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 1
Introduction Our main goal in this section is to define and determine the relationships between several quantities: Velocity (average and instantaneous) Slope (of secant lines and tangent lines) Rate of Change (average and instantaneous) Derivative of a Function We will see that velocity is actually a special case of a rate of change, so we will concentrate on the last three concepts. D.L. White (Kent State University) 2 / 1
Velocity Recall that as an introduction to limits, we considered average velocity on an interval and defined instantaneous velocity at a specific time as a limit of average velocities on smaller and smaller time intervals. More precisely, suppose an object is moving along a straight line so that its position at time t is given by s ( t ). The average velocity on the interval t = a to t = a + h is given by v avg = ∆ s ∆ t = s ( a + h ) − s ( a ) = s ( a + h ) − s ( a ) . ( a + h ) − a h The instantaneous velocity at time t = a is then s ( a + h ) − s ( a ) v ( a ) = lim h → 0 v avg = lim . h h → 0 The instantaneous velocity v ( a ) is a measure of how fast the position is changing with respect to time at the time t = a . In other words, it is the “rate of change” of position with respect to time. D.L. White (Kent State University) 3 / 1
Rates of Change More generally, we make the following definition: Definition Let y = f ( x ) be a function and let a be a number in the domain of f . The average rate of change of y with respect to x on the interval x = a to x = a + h is R avg = ∆ y ∆ x = f ( a + h ) − f ( a ) . h The instantaneous rate of change of y with respect to x at the point x = a is defined to be f ( a + h ) − f ( a ) R inst = lim h → 0 R avg = lim h . h → 0 D.L. White (Kent State University) 4 / 1
Rates of Change Notes: Average or instantaneous velocity is the average or instantaneous rate of change of position with respect to time. The units of the rate of change of y with respect to x are y -units per x -unit. Notice that the slope of the line passing through the points P ( a , f ( a )) and Q ( a + h , f ( a + h )) on the graph of f is m = ∆ y ∆ x = f ( a + h ) − f ( a ) = f ( a + h ) − f ( a ) = R avg . ( a + h ) − a h Thus average rate of change on an interval is the slope of the line passing through the points on the graph at the endpoints of the interval. D.L. White (Kent State University) 5 / 1
Tangent Lines Let y = f ( x ) be a function and let a be a number in the domain of f . A line passing through two points on the graph is called a secant line . For example, the following figure illustrates the secant line through the points P ( a , f ( a )) and Q ( a + h , f ( a + h )) on the graph of f : § 2.1 Figure 6 Notice that the “rise” (∆ y ) is f ( a + h ) − f ( a ) and the “run” (∆ x ) is h , and so the slope of the secant line is f ( a + h ) − f ( a ) , which is the h average rate of change of y with respect to x on the interval [ a , a + h ]. D.L. White (Kent State University) 6 / 1
Tangent Lines Now instantaneous rate of change of y with respect to x is found by taking the limit as h → 0 of the average rate of change on [ a , a + h ]. This is also the limit as h → 0 of the slope of the secant line through P ( a , f ( a )) and Q ( a + h , f ( a + h )). What is the graphical meaning of this limit? As h → 0, the point Q moves along the graph toward the point P , as shown in the following figure: § 2.1 Figure 5 In the limit, we think of Q moving all the way to P and the secant line becomes a tangent line at the point P ( a , f ( a )) (line t in the figure). D.L. White (Kent State University) 7 / 1
Tangent Lines Definition Let y = f ( x ) be a function and let a be a number in the domain of f . The tangent line to the graph of f at x = a is the line passing through the point ( a , f ( a )) with slope f ( a + h ) − f ( a ) m = lim h . h → 0 Notes: f ( a + h ) − f ( a ) The tangent line exists only if lim exists. h h → 0 By the Point-Slope formula, the equation of the tangent line is y − f ( a ) = m ( x − a ) . This says that the instantaneous rate of change of y with respect to x at x = a is equal to the slope of the tangent line to the graph of f at x = a . D.L. White (Kent State University) 8 / 1
Tangent Lines Slope of Secant Line Through Average Rate f ( a + h ) − f ( a ) ( a , f ( a )) = = of Change on h and [ a , a + h ] ( a + h , f ( a + h )) h → 0 h → 0 h → 0 ❄ ❄ ❄ Instantaneous Slope of f ( a + h ) − f ( a ) Rate of = lim = Tangent Line h Change h → 0 at x = a at x = a D.L. White (Kent State University) 9 / 1
The Derivative f ( a + h ) − f ( a ) The expression lim has come up in enough different situations h h → 0 (instantaneous velocity, instantaneous rate of change, slope of a tangent line) that we want to give it its own name. Definition Let y = f ( x ) be a function and let a be a number in the domain of f . The derivative of f evaluated at x = a is f ( a + h ) − f ( a ) f ′ ( a ) = lim , h h → 0 if this limit exists. We can now add f ′ ( a ) to our chart of related quantities: D.L. White (Kent State University) 10 / 1
The Derivative Slope of Secant Line Through Average Rate f ( a + h ) − f ( a ) ( a , f ( a )) = = of Change on h and [ a , a + h ] ( a + h , f ( a + h )) h → 0 h → 0 h → 0 ❄ ❄ ❄ Instantaneous Slope of Rate of f ( a + h ) − f ( a ) = f ′ ( a ) = lim = Tangent Line h Change h → 0 at x = a at x = a D.L. White (Kent State University) 11 / 1
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