# 3.4 - MATH 1081  Monday, February 7   ...

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Unformatted text preview: MATH 1081  Monday, February 7    Chapter 3 – Section 4    DEFINITION OF THE DERIVATIVE    Homework #4 (due 2/14):    Section 3.4 #18, 48, 56                    Section 4.1 #34, 56, 62, 72    Clicker Check­in:  Choose any letter to check in now.    Last time, we defined the average rate of change of  f ( x )   between  x = a  and  x = b  and the instantaneous rate of change of  f ( x )  at  x = a .      The average rate of change is simply the slope of the line  between two different points on the curve.    The instantaneous rate of change is the limit of the slopes of  these lines as the two points get closer and closer together.    Let’s start with  P = ( 3, f ( 3))  and Q = ( 4, f ( 4 ))  that are two  different points on the graph of a function, which we will see  on the next slide.    We will call the line between these two points a secant line.    We then move the point Q  closer and closer to the point  P .  The “limiting” line, which we call the tangent line at P, will be  THE line that passes through the graph at that single point P  and has the same direction of the curve there.    The slope of this tangent line is defined to be the limit of the  slopes of the secant lines and is often referred to as the slope of  the curve at the point P.                                 So, the instantaneous rate of change of  f ( x )  at  x = a , which on  a graph can be thought of as the slope of the tangent line to  f ( x )  at  x = a , is computed as    f (a + h) − f (a)             lim   h→ 0 h   as long as the (two‐sided) limit exists.    We will use the notation              f '(a)    to notate this very important limit.  Now, if this limit exists, then we have  x = a  being assigned  exactly one value called  f ' ( a ) .    That is, we can think of this  f '   as a function that assigns to each input value  x = a  in the  domain of the original function  f ( x ) , the output value  f ' ( a )   which equals the slope of the original curve  f ( x )  at that point  x = a .      So, for a function  f ( x )  at  x = a , the value of  f ( a )  tells us where  the point is on the curve and the value  f ' ( a )  tells us the slope  of the curve at that point.    Definition:  Derivative    The derivative of the function  f  at  x  is defined as     f ( x + h) − f ( x)         f ' ( x ) = lim h→ 0 h     provided the limit exists.      In essence, all that we have done here is to rename the specific  value  a  to the general variable  x  to emphasize its identification  as a function that can be evaluated at many different points.      The definition of derivative includes the phrase “provided the  limit exists”.  For what kinds of situations will the derivative  NOT exist at  x = a ?      • when the function is not defined at  x = a   • when the function is not continuous at  x = a   • when the function has a sharp corner at  x = a   • when the tangent line to the curve at  x = a  is vertical                From page 209 of your book,                          Example:  Find the derivative for the function.      1.   f ( x ) = 3      2.   g ( x ) = x 2       3.   h ( x ) = 4   x    Example:  Find the equation of the tangent line to the curve at    the indicated point.    1.   f ( x ) = x 2  at the point ( 2, 4 )         2.   f ( x ) = x 2  at the point ( −3, 9 )         4 3.   h ( x ) =  at the point  x = 1  x As introduced in Workshop 1, the marginal cost for the  production of  a  units is the rate of change of the cost function  C ( x )  at  x = a .      This marginal cost can be used to approximate the cost to  produce the  a + 1st unit.    That is, for a cost function  C ( x ) , the derivative  C ' ( x )  (called  the marginal cost function) measures the rate of change of cost  and can be used to approximate the cost to increase production  by one unit.    We can similarly define the marginal revenue and marginal  profit functions.  Example:  Suppose that the revenue (in dollars) generated    from the sale of  x  picnic tables is given by   x2 .    R ( x ) = 20 x − 500   Approximate the revenue from the sale of the 1001st  picnic table.    Clicker Check­Out:  Choose any letter to check out now.    Next time – Section 4.1      ...
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