Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeLecture Outline (Derivative as a Function)Monday, January 28Announcements1. Homework 2 is due Wednesday 01/30/082. There’s review session for the midterm on Wednesday evening 7-9 in room 380W.3. Midterm 1 is on Thursday 01/31/08 in Herrin 175.RecapOn Friday we revisited the tangent line and computed lots of examples of it. We also talkedabout the two seemingly different expressions for the slope of the tangent line.On Friday we also gave the definition of the derivative of a function at a point. We notedthat the derivative f0(a) is the slope of tangent line to y = f(x) at x = a.Derivative at a PointLet’s do one or two more examples with this concept before we move on to thinking aboutthe derivative as a function.(1) Sketch the graph of a function f for which f (0) = 0, f0(0) = 3, f0(1) = 0, andf(2) = −1.Let’s also connect these ideas back to the real world. Suppose we have a function y = f(x)(f(x) can be the price of producing x units, or it can be position of a particle at time x, orit can be concentration of a chemical in your blood a time x). The average rate of change ofthis function is given by:f(x2) − f(x1)x2− x1.We can do better than just the finding the average rate of change between two input values.We can find the instantaneous rate of change at x = x1by taking the limit of this expression:= limx2→x1f(x2) − f(x1)x2− x1= f0(x1).Look in your textbook, sections 2.6 and 2.7, for examples relating these ideas back toreal world problems.1The Derivative as a FunctionSo far we’ve considered the derivative of a function at a fixed point a. Today, we’re changingour perspective slightly and instead we’ll speak of the derivative as a function itself. For agiven input x, this derivative function will return for us the derivative of f at the point x.ExplicitlyDefinition: For a function f(x), the derivative of f (x), written f0(x) orddx(f(x)), isdefined to belimh→0f(x + h) − f (x)h,if it exists. The derivative f0(x) is the function which gives the slope of the line tangent tof at the point x.Note: this derivative function is very much like the limits we’ve been considering in thepast few weeks. The only difference is taht instead of having an explicit value for x in thelimit above, we leave this quantity a variable. This means that, generally speaking, thederivative f0(x) will be a function of x and not just a number.Examples: Derivative as a function1. Use the definition of the derivative to compute f0(x) where f (x) = x2.We can use geometry to make computations a bit easier.2. Using only that the derivative f0(a) is the slope of the tangent line to f at a, find thederivative of f(x) = c.We could also compute this derivative using the definition. Let’s try it out.3. Using only that the derivative f0(a) is the slope of the tangent line to f at a, find thederivative of f(x) = mx + b. Also compute thsi derivative using the definition above.4. Here is the graph of f (x) = sin(x). Sketch the graph of f0(x) on to the graph of f (x).What to know/memorizeKnow the definition of the derivative of the function f (x).Know how to graph the function f0(x) onto the graph of f
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