Properties of the Derivative Lecture 9.

Transcription

1 Properties of the Derivative Lecture 9. Recall that the average rate of change of a function y = f(x) over the interval from a to a + h, with h 0, is the slope of the line between y x f(a + h) f(a) =, h (a, f(a)) an (a + h, f(a + h)), an the the instant rate of change or erivative of f(x) at a is the limit Untitle-2 f (a) = f f(a + h) f(a), x x=a h the slope of the line tangent to y = f(x) at (a, f(a)). slope = f (a) (a + h, f(a + h)) (a, f(a)) x = h y

2 More generally, given a function y = f(x), the erivative of f is the function y = f (x) whose omain is the set of all x at which f (x) exists an is then given by the rule f (x) = the erivative of f at the point x. For many functions f we can with relative ease can calculate approximately their erivatives at particular points using a calculator or other technical ais. How to o this, an more, is iscusse at some length in the text for the TI-83 an Excel. We urge you to spen some time seeing how this works an becoming familiar with it. We will bypass that here in class, but you shoul not take that as a reason to skip that valuable knowlege! Now, though, let s calculate the actual erivatives of some simple functions, check out some stanar rules of ifferentiation, an then put most of our effort into learning some of the significance of erivatives. 2

3 Example 1. First, a constant function f(x) = a. This is a linear function with a horizontal line for a graph, so its slope, an hence its erivative is zero!! That is, using some of our new notation x a = 0. Example 2. A linear function f(x) = mx + b has for a graph a line with slope m. So it s erivative is simply the constant function y = m. Let s just check that using the appropriate limit of a ifference quotient: f f(x + h) f(x) (x) h m(x + h) + b (mx + b) h as we saw from the geometry! mx + mh mx h m = m 3

4 Example 3. Now let s try the quaratic function f(x) = ax 2. Calculating the ifference quotient for this function an then evaluating its limit, we fin that for all x f (x) f(x + h) f(x) h a(x + h) 2 ax 2 h ax 2 + 2ahx + ah 2 ax 2 h 2ahx + ah 2 h 2ax + ah = 2ax. So we have neat formula for the erivative x (ax2 ) = 2ax. 4

5 A Couple Rules of Differentiation. Later we shall assemble several rules that will greatly help us calculate the erivatives of a great many important functions incluing all of the ones we ve stuie in this course! But for now here are a couple of anies: Aition. If f an g are ifferentiable at x, then so is their sum an (f(x) + g(x)) = x x f(x) + x g(x). Constant Multiple. If f is ifferentiable at x an if a is a constant, then af is ifferentiable at x, an x (af)(x) = a x f(x). Example 4. What we have so far makes it easy to fin the erivative of any quaratic. Let s try it with f(x) = 3x 2 5x + 7: x f(x) = x (3x2 5x + 7) = x (3x2 ) + x ( 5x) + x (7) = 3 x (x2 ) 5 x (x) + 0 = 3 2x 5 = 6x 5. 5

6 Next, let s consier some of the behavior of the graphs of functions. So suppose we have a function f. Then we say that the function f is increasing on the interval a x b if for all a x 1 < x 2 b f(x 1 ) < f(x 2 ). the function f is ecreasing on the interval a x b if for all a x 1 < x 2 b f(x 1 ) > f(x 2 ). The function f has a horizontal tangent at some point a if the erivative f (a) = 0. Let s check out these concepts in a simple example: 6

7 Example 5. Fin the intervals on which the function is Positive (f(x) > 0), Negative (f(x) < 0), Zero (f(x) = 0). Untitle y = f(x) Positive: Negative: Zero Next fin all intervals on which the function y = f(x) is/has Increasing: Decreasing: Horizontal Tangent: 7

8 If the function f is ifferentiable, then this behavior can be etecte from the erivative! For example, to say that the function is increasing on an interval a x b, simply means that for each pair a x 1 < x 2 b on that interval the average rate of change from x 1 to x 2 is positive, an thus, the erivative at any point on the interval is non-negative! A similar argument shows that the function is ecreasing on the interval if an only if its erivative is non-positive on that interval! Let s summarize all of this in the table: f(x) f (x) increasing f (x) > 0 ecreasing f (x) < 0 horizontal tangent f (x) = 0 Okay. With that information in han, let s try a few examples: 8

9 Untitle-1 Example 6. Sketch the graph of the erivative of the function: y = f(x) Example 7. Sketch the graph of the erivative of the function: Untitle-1 1 y = f(x) 9

10 Untitle-1 Example 8. Sketch the graph of the erivative of the function: a b c y = f(x) x-intercepts of f (x): f (x) { positive: negative: Untitle-1 Example 9. Sketch the graph of the erivative of the function: y = f(x) x-intercepts of f (x): f (x) { positive: negative: 10

11 Example 10. Sketch the graph of the function y = f(x) whose Untitle-1 erivative is: y = f (x) Example 11. Sketch the graph of the function y = f(x) whose erivative is: Untitle-1 1 y = f (x) 11

12 Untitle-1 Example 12. Sketch the graph of the function y = f(x) whose erivative is: a b c y = f (x) Horizontal tangents of f(x): f(x) { increasing: ecreasing: Untitle-1 Example 13. Sketch the graph of the function y = f(x) whose erivative is: y = f (x) Horizontal tangents of f(x): f(x) { increasing: ecreasing: 12