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4 Pages

### 0901

Course: M 0901, Fall 2009
School: University of Texas
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Word Count: 325

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Locating Contents 2 a Vertex 3 The Peak for a Square-Root 4 Graphing with Intercepts Locating a Vertex The graphing trix I did in class show a lot of things about about a function, but theres a lot they dont show. Thats what these examples are about. The best way to think about it is to look at the example I did, the quadratic y = x(x 1). This has roots at x = 0; x = 1 and we know the vertex is half way in...

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Locating Contents 2 a Vertex 3 The Peak for a Square-Root 4 Graphing with Intercepts Locating a Vertex The graphing trix I did in class show a lot of things about about a function, but theres a lot they dont show. Thats what these examples are about. The best way to think about it is to look at the example I did, the quadratic y = x(x 1). This has roots at x = 0; x = 1 and we know the vertex is half way in between. But where? Check out the picture below: 2 1 y 0 -2 x 0 2 Theres three possible graphs, with three vertices, all at dierent locations. Which is x(x 1)? I did this in class: since you know the vertex is half-way in between, you calculate x = 1 and plug that in to get y = 1 ( 1 1) = 1 . 2 22 4 Voila! the vertex is at 1 , 1 and you check, thats the red graph. 2 4 The Peak for a Square-Root Same deal with square the root function: y = 1 x2 . I know it has domain [1, 1] and it loks parabol-ish, and that it goes to the axis at a 90o angle . . . I know all this, but which of the graphs below is right? 2 1 y 0 -2 x 0 2 Vertex to the rescue the vertex is still at x = 0, so all I got to do is plug in, to locate the height: y = 1 02 = 1 and its red-graph again. Graphing with Intercepts Last example: y = 1 |1 x|. I know it has a vertex at (1, 1) and its been turned up-side down. That means ...

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University of Texas - M - 1113
Contents2 3 4 5 6 Lecture topics 11 - 13 - 98 Worked Example: Solving an Equation with ln Worked Example: Log-differentiation and a faster way A way of computing e (oh goddess, restrain me) FinishLecture topics 11 - 13 - 981) Finished the graph
University of Texas - M - 1124
Contents2 3 4 5 6 7 8 Lecture 11 - 24 - 98: First Partial Fractions Example Solving For the A, B, C Solving for C is harder Integrating &amp; getting the final answer A Warning, and Elizabeth A Second Example &amp; a picture Solving The Equations &amp; Doing Th
University of Texas - M - 1006
More about implictly dened functions I said that equations like 2x 3y = 1 dene y as a function of x implicitly because even though it isnt written out y = so and of x, you could go ahead and solve: 3y = 2x + 1 and y = 2 x 1 . 3 3 The nice thing abo
University of Texas - M - 1012
Contents2 3 4 5 Lecture topics 10 - 12 - 98 Worked example Page 1 Worked example Page 2 Worked example and GraphLecture topics 10 - 12 - 981) Worked an example: f (x) = x2 (x 1)3 . Dierentiate and simplify, nd all critical points, and classify
University of Texas - M - 0925
Contents2 Lecture topics 09 - 25 - 98 3 Why is there a pattern in the derivatives of the co-trig functions? 4 A worked example of quotiet-chain ruleLecture topics 09 - 25 - 981) Dierentiated and simplied y = (x + 2)2 (x 1)2 and located the turn
University of Texas - M - 1014
Contents2 3 4 5 6 Lecture topics 10 - 14 - 98 Worked example Page 1 Worked example Page 2 Worked example and Graph Why the inflection point happensLecture topics 10 - 14 - 981) Did a max/min problem on a closed interval: Let f (x) = x +sin x, 2
University of Texas - M - 1002
Contents2 3 4 5 Lecture topics 10 - 02 - 98 Mt. St. Helens: Explosion Mt. St. Helens: Before and After Mt. St. Helens: Circle of DestructionLecture topics 10 - 02 - 98Mt. St. Helens: ExplosionMt. St. Helens: Before and AfterMt. St. Helens: C
University of Texas - M - 0918
Extra example of differentiate and simplify1 Example Let f (x) = x2 + x2 a) Dierentiate b) Simplify c)Find all c where f (c) = 0.f (x) =x2 +1 x2= x2+1 x2= (2x) +2 x3dierentiated2x4 2 2x4 2 2(x4 1) = 3+ 3= = x x x3 x3simplied
University of Texas - M - 0924
What does the 1/g rule really mean?The1 grule says1 g=g g2But whats it mean?What you can take, right away, from a derivative, are two things: i) Where a derivative is zero matches where the function has a turning point: it either bott
University of Texas - M - 0928
Contents2 3 4 5 Lecture topics 09 - 28 - 98 Whats it mean when the derivative does not exist? Part 1 Whats it mean when the derivative does not exist? Part 2 Whats it mean when the derivative does not exist? Part 3Lecture topics 09 - 28 - 98 2)
University of Texas - M - 0930
Contents2 Lecture topics 09 - 30 - 98 3 Another example of implicit differentiationLecture topics 09 - 30 - 981) Did basic dierentiation rules using the d d d 2 3 3 dx (x sin x); dx (x + 1) ; dx (sin(x ). 2) Found 3) Found 4) Found 5) Found 7) F
University of Texas - M - 1005
Contents2 3 4 5 6 Lecture topics 10 - 05 - 98 Worked Word Problem Page 1 Worked Word Problem Page 2 Worked Word Problem Page 3 Worked Word Problem Page 4Lecture topics 10 - 05 - 981) Did variant word problem: Mt. St. Helens erupts, sending a clo
University of Texas - M - 1007
Contents2 Lecture topics 10 - 07 - 98 3 Worked worked out derivative from classLecture topics 10 - 07 - 981) Warned that Oct 21 was the last day for an easy drop. 2) Stated Rolles Theorem, then the Mean Value theorem. 3) Remarked that the MVT al
University of Texas - M - 1009
Contents2 3 4 5 6 7 Lecture topics 10 - 09 - 98 The Second Derivative Test Why I Dont Like The Second Derivative Test Why I Still Dont Like . . . Let The Computer Do The Work Maybe the Second Derivative Test Isnt That BadLecture topics 10 - 09 - 9
University of Texas - M - 1028
Contents2 3 4 5 Lecture topics 10 - 28 - 98 Even Functions Integrating Even Functions Odd FunctionsLecture topics 10 - 28 - 981) Used formula for true area:b|f (x)| dx =a{f &gt;0}f (x) dx f (x) dx{f &lt;0}and computed 0| sin x| dx2)
University of Texas - M - 1104
Contents2 3 4 5 Lecture topics 11 - 04 - 98 Slices of the hemisphere I Return of Slices of the hemisphere II Revenge of Slices of the hemisphere IIILecture topics 11 - 04 - 981) I worked the problem: Find the area between the curves x = y 2 , x
University of Texas - M - 1106
Contents2 3 4 5 Lecture topics 11 - 06 - 98 Rotating a curve about the x - axis, I Rotating a curve about the x - axis, II Rotating a curve about the x - axis, IIILecture topics 11 - 06 - 981) Computed the volume of a half-sphere, by slicing par
University of Texas - M - 0902
Contents2 A cosine - sine trick 3 Finding sin 30A cosine - sine trickThe pic below shows two angles, = 30o and = 600 . Ok; when I take cos 30o , cos is on the x axis, so thats the length of the lavender line on the x-axis. When I take sin 60o
Wisconsin Milwaukee - LIB - 1768
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Cal Poly - PROJ - 339
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Richmond - M - 212
Math 212 Kerckhove Fall 2005 First Set of HW Assignments Week 1: Hand-in HW due 9/2 5.5 / 22,31,42,46,62 5.6 / 14,16,26,28,34a,38,41 Daily HW 8/31 9/2 5.5 / 7,11,15,17,19,29,33,41,49,59 5.6 / 7,9,11,13,15,19,21,25,27,33,37,395.7 / 16,18,26,28,305
Cal Poly - PROJ - 339
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Math 212 Exam 2 Spring 00 KerckhoveName Pledge1. Two farmers, husband and wife, wish to purchase a eld in which to grow yams. The eld is rectangular in shape and lies in a valley between a riverbed and a hillside. The Farmers estimate that land
Richmond - M - 212
Math 212 | Final Exam KerckhoveName: Pledge:1. Consider the system of di erential equations given bydx dt dy dt= ,2x = 2x , 2yx0 = 10 y 0 = 3a. Find the function xt. b. Use your answer to part a and an integrating factor to nd the function
Richmond - M - 212
Math 212 | Final Exam Spring '95 | Kerckhove 1. Evaluate the following integrals.Name PledgeaZt cost dt dZ0 1bZt cost dt2cZ0sin x dx21 dx 1+x2eZ1 y , 2y , 3 dy2. Here's a plot of sinx on the interval 0 x :5.2
Richmond - M - 212
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Richmond - M - 212
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Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
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Richmond - M - 212
m212_commentary_on_review.nb1Commentary on M212 Ch 5 Review The Computational Problems (10,15,16,19,23,24)d I used #10 to talk about the fundamental theorem, T 0 f HxL x = f HT L and about the relationship between the graph of d F HT L and th
Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
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Richmond - M - 212
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Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
Richmond - M - 212
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