L'hopitals rule and curve sketching

Transcription

1 L'hopitals rule and curve sketching Suppose you get given some sort of crazy equation. And you want to know what it looks like on a graph, give yourself some intuition and stuffs.

2 Graph sketching: example Suppose I have some crazy function... say... xth root of x. And I want to sketch up a graph of it- nothing super complex, just a general idea... Well, the graph in question looks something like This: Okay... that's great... but how do you do that?

3 Sketching a graph: getting started 1) What are the important parts of a graph (ask class) X and Y intercepts (where y and x=0) Maxima/minima/ inflection. (When are the first and/or second derivatives equal to zero) Are their any discontinuities? Are they jump or asymptote? What happens at positive and negative infinity? Are there any other features that seem important? Could I SHIFT my graph to get something more familiar.

4 Basic example Suppose I want to graph: f (x)=x 4 10 x 2 +9 Okay- So what do we find first: Y intercept (x=0) 9 X intercepts (y=0) need to factorise (x^2-9) (x^2-1) Hence x intercepts at +-1 and +-3. Horray. Fill these in on graph

5 f (x)=x 4 10 x 2 +9 Basic example Where are maxima, minima, inflection points: F'(x)= 4x^3 20 x F''(x)= 12 x^2 20 F'(x)=0 at x=0 or +- sqrt 5 F''(x)=0 at x= +- sqrt 20/12 Okay cool, sub in and find y values for these. Mark on graph.

6 Basic example Are their discontinuities? No. none. What happens at +ve and -ve infinity? goes to inf. Any other features that seem important? In this case, it is worth noticing that the graph is symmetrical- it is in fact what we call an EVEN function Even functions are graphs where f(x)=f(-x). They have a line of symmetry down the middle Odd functions have -f(x)=f(-x). They have rotational symmetry around the origin.

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8 f ( x)=2arctan( x)+ 1 x Clicker example Find the x and y intercepts of the above: a) y intercept at y=0, no x intercept. b) x intercepts at 3 and -3, no y intercept. c) No intercepts anywhere. d) infinitely many intercepts.

9 f ( x)=2 arctan( x)+ 1 x Clicker example What are the x values for the local maxima and minima? a) b) c) d) x=0 x=±1 f ' ( x)= 2 x=± 2 x=±2 1+x 2 1 x 2

10 f ( x)=2 arctan( x)+ 1 x Clicker example What are the corresponding y values for the local maxima and minima? a) b) c) d) y= ( π 2 +1) y=±( π 2 1) E) y=±( π 2 +1) y=±( π 4 +1) y=±1

11 f ( x)=2 arctan( x)+ 1 x Clicker example What is the it as x tends to infinity? a) x f ( x)=1 b) c) d) x x x f ( x)=π f ( x)=π+1 f ( x)=2π

12 f ( x)=2 arctan( x)+ 1 x Clicker example What other features are worth noting? a) Discontinuity b) The function is even c) The function is odd d) The function is non-smooth

13 L'hopitals rule Okay, so remember at the start of the lesson when I mentioned the xth root of x. Suppose you wanted to plot that? What could you do? Taking derivatives is tricky but doable. Taking its as we approach zero, and infinity is a little harder. How do we do this?

14 The basic idea: Suppose we have a it like: x a f ( x) g ( x) Suppose it is in-determinant of the form 0/0 or inf/inf. Suppose BOTH f(x) and g(x) are NICE. (What nice means is complex, and not really in the scope of this course. As a general rule, lets just say a function is nice if we can differentiate it forever near our it point.

15 L'hoptials rule IFF the its of f and g are either both infinte or both zero, and IF they are both nice, then: x a f ( x) g ( x) = x a f ' ( x) g ' ( x) WARNING: do not mix this up with the quotient rule. EXTRA WARNING: Do not attempt to do this if your it is not 0/0 or inf/inf- it won't work then.

16 L'hopitals example x ln(x) x 0.2 Both functions are smooth near infinity (but sadly not near zero) Both functions are approach infinity- hence why we have problem. Time for L'hopitals! x ln(x) x 0.2 = x 1/ x 0.2 x 0.8= x x 0.2=0

17 L'hopitals example x π 2 tan( x) ( x π 2 ) Here we have inf times 0. This is also indeterminant, but not in a form we find useful. Damn. Luckily we can change it up a little x π/2 x π 2 cot( x) = x π/2 1 cosec 2 ( x) = x π/2 1 1 = 1

18 Clicker question x 0 + e x 1 x 3 A) x 0 + e x 1 x 3 = C) x 0 + e x 1 x 3 =0 B) x 0 + e x 1 x 3 = 1 D) L'hopitals does not apply

19 Clicker question x 0 + x ln x A) x 0 + B) x 0 + x ln x= x ln x= 1 C) x 0 + x ln x=0 D) L'hopitals does not apply

20 Clicker question x x x A) x x x= C) x x x=0 B) x x x=1 D) L'hopitals does not apply x x=e 1 x ln(x)