MTH 122 Calculus II Essex County College Division of Mathematics and Physics 1 Lecture Notes #11 Sakai Web Project Material
|
|
- Phillip Parsons
- 5 years ago
- Views:
Transcription
1 MTH Calculus II Essex County College Division of Mathematics and Physics Lecture Notes # Sakai Web Project Material Introduction Figure : Graph of y sin ( x y ) = x cos (x + y) with red tangent line. With few exceptions, we have only dealt with relationships between two variables some functional, others not functional. In those rare cases where we had more than two variable relationships we invariably found some way to reduce it to two variables. By now you should have a good idea how to graph simple relationships between two variables, and also use what you ve learned in pre-calculus and calculus to refine these basic graphs. When confronted with difficult two variable problems, you should at least be able to use a computer to construct a graph. So if you re asked to graph y = sin x, I hope you don t say, give me a calculator! However, if you re asked to graph y sin ( x y ) = x cos (x + y), and find the line tangent to this curve where x and y are positive proper fractions, and x = y, you would use technology to do so. Yes, in this case I d certainly want to use a computer to help visualize the problem. I don t think anyone could do the graph by hand, but some of you may be able to find the tangent line requested without graphing the relationship. In any case This document was prepared by Ron Bannon (ron.bannon@mathography.org) using L A TEX ε. Last revised October 0, 00. Found this by implicit differentiation and I used the point where x = y = π/4. y π 4 = + π ( π x π ) 4
2 I would still like visual confirmation that I m right and that s where a computer can help. The graph of y sin ( x y ) = x cos (x + y) and the requested tangent line, for the impatient among us, is given in Figure (page ). Parameters Now we are going to intentionally introduce a third variable, often called a parameter into our two variable problems. Yes, as before we may decide to eliminate this third variable in an attempt to land on familiar ground of two variable relationships. However, no matter how frustrating this becomes, you should always know that technology can deal with our limited capacity to resolve abstractions that are not yet familiar to us. For example, there was a time when jumping from linear relationships to more complex quadratics relationships proved insurmountable in your academic careers, so you may have resorted to using a graphing utility. So please consider having a graphic calculator 3 nearby for those times your brain is too stretched to move on. As an example, suppose we have a simple linear equation, y = x + 3, where we now define x as a function of t as follows: x = f (t). Certainly, if x is a function of t, we can also state that y is a functions of t as follows: For this particular example we might say that y = g (t). x = f (t) = t, and y = g (t) = 4t + 3. Yes, you probably think this is pointless, and in fact taking a simple problem and making it impossible. For example, I think it s really easy to graph y = x + 3 (Figure ), but not so easy to graph (f (t), g (t)) (Figure 3, page 3). Let s do both by using a computer to see what happens. 3 This may mean your computer!
3 First, the traditional way Now the unfamiliar way. Figure : Graph of y = x + 3, software image on right Figure 3: Graph of (f (t), g (t)), software image on the right. Problem: Now let s try to eliminate the parameter in this particular example. Work: 3
4 3 Graphing a Parametric Equation You should carefully look into using software to graph parametric equations, however, you should still be able to reason through simple examples. Here s one x = f (t) = t + t, and y = g (t) = t, where 3 t 3. Here, I would like to suggest that you construct a simple table (Table, page 4) and then connect the dots in order. You ll see that if you follow t from 3 forward that the graph moves in a certain direction. For this example, the initial point is (f ( 3), g ( 3)) = (5, 6), and the termial point is (f (3), g (3)) = (, 6). Table : x = t + t and y = t point t x y No No. 6 4 No. 3 No No. 5 3 No No Figure 4: x = t + t and y = t in red, for 3 t 3. 4
5 Problem: Now s your chance to do some real work.. Verify Table (page 4) is correct.. Plot the points, in order, directly on the given graph (Figure 4, page 4). Notice the direction that these points follow. 3. Differentiate both f (t) and g (t) and try to make sense out of these derivatives and the way these points are moving. 4. Try to eliminate the parameter to get a relationship between x and y You should also see that the graph of the relationship between x and y may differ, as in this example. 4 This relationship between x and y describes a set of points. The parametric equations have a great advantage though if we consider t to be time, we can view these points as location in time and can easily follow these points to see how they re moving. 5
6 3. Easily Tricked? Technology will certainly prove helpful, but using your head should be your main goal. Learning this material will require you to not only use your head, but also to explore technology to see why your head may not always produce the correct answer. For example, suppose you re asked to graph f (t) = cos t g (t) = sin t, where t R. And you decide to eliminate the paramter t. Work: Okay, give it a shot. Now graph out the relationship between x and y. Now look at the actual graph (Figure 5) of the parametric equations. Notice anything strange? Now, looking back over the both f and g, you should note that x and 0 y. Tricky, I know. 4 Using Software Here s an interesting problem, graph x = cos (3t) and y = cos (t + cos (3t)). I really don t think you can do this 5 using a table and plotting points, so I want to recommend that you either use a calculator 6 or a computer 7. Since we have a site license for Mathematica, here s the Mathematica code, ParametricPlot[{Cos[3 t], Cos[t + Cos[3 t]]}, {t, 0, Pi}], that you should try-out today. Make sure your graph looks like the one in Figure 6. You should also play around with the range for t in this example. 5 At least I can t. I have literally no patients for this type of problem. 6 Yes, you may have to read your manual. 7 I m use an application called Grapher, which is free on Mac OS X. 6
7 - 0 Figure 5: Graph of (f (t), g (t)) Figure 6: x = cos (3t) and y = cos (t + cos (3t)) 7
8 5 Finding Derivatives Given that both x and y are functions of t, I think you ll agree that it is pretty easy to find f and g. 8 Actually we already did this in a prior example and tried to make sense out of them, especially as they related to the way the graph moves (has direction). Now, for another example, suppose x = f (t) = t 3 and y = g (t) = t t, you can differentiate both x and y with respect to t quite easily. Here goes... f (t) = dx dt = 3t, g (t) = dy dt = t. More difficult to find would be find dy/dx. Let s start by finding a relationship between x and y. That will put us on familiar ground. Work: Okay, that was easy and you should have found that y = x /3 x /3. So we have and if we like, we can even find dy dx = 3x /3 3x /3, d y dx = 4 9x 5/3 + 9x 4/3. Normally, we don t need derivatives beyond these. You should realize that this largely depends on your ability to find a relationship between x and y, and that, my friend, may be very difficult to do! So let s look at another way to do these 8 We re differentiating with respect to t, so f (t) = dx dt and g (t) = dy dt. 8
9 derivatives that does not depend on finding a relationship between x and y. The first derivative: x = f (t) () y = g (t) () F (x) = y relationship between x and y (3) F (x) = g (t) substitute (4) F (f (t)) = g (t) substitute (5) F (f (t)) f (t) = g (t) differentiate (6) F (x) f (t) = g (t) substitute (7) This (Equation 8) is often written as F (x) = g (t) /f (t) voilà (8) d y dx = dy dx = dy dt dx dt. (9) Taking this very same equation (Equation 9) we can find the second derivative. 9 ( ) d dy = d y dx dx dx second derivative (0) ( ) d dy dt dx dx dt So in our example above we have the first derivative dy dx = 3x /3 3x /3 = g (t) f (t) = t 3t = 3x /3 3x /3 And the second derivative ( ) d t d y dx = dt 3t 3t = t 4 9t 5 = 4 9x 5/3 + 9x 4/3 second derivative () just do the substitution just do the substitution Let s summarize. There s no need to find a relationship between x and y to find the find the derivative dy/dx. Just use this formula: dy dx = g (t) f (t). It should also be easier to analyze the derivative when it s a function of t. For this particular case we have dy dx = t 3t, 9 The trick here is to replace y in Equation 9 with dy/dx. 9
10 and we should note that we ll get a horizontal tangent at t =, that is, at the point (f () g ()) = (, ). You ll also get a vertical tangent at (f (0) g (0)) = (0, 0). Furthermore, our graph will be increasing for t <, and decreasing for t >. Here s the graph (Figure 7, page 0) Figure 7: Graph of x = f (t) = t 3 and y = g (t) = t t. The second derivative is a bit-more-tricky! d y dx = d dt ( ) dy dx dx dt However, I want to stress that the second derivative we found is pretty easy to analyze. We can see that we get a sign change at t = 0 and t =. Simple sign analysis will show that our curve will be concave down for 0 < t < and concave. That is, between the points (0, 0) and (8, 0) we should observe a concave down structure. 6 You Try!. Graph the parametric curve given by x = t 5 4t 3, and y = t. 0 0 Yes, you can use a calculator. 0
11 . Visually determine the point on the parametric curve graph in () where the tangent does not exist. 3. Determine the points on the parametric curve graph in () where the tangent is vertical. 4. Determine the single point on the parametric curve graph in () that has two different t values. What are these values. Also determine the the slopes of the line tangent at these two t values. 5. Graph the parametric curve given by x = t, and y = t 7 + t Using the second derivative, find where our parametric curve will be concave up or concave down. Your answer should be given in t. Yes, you can use a calculator.
12 7 Answers to You Try!. Graph the parametric curve given by x = t 5 4t 3, and y = t Figure 8: Graph of x = t 5 4t 3, and y = t.. Visually determine the point on the parametric curve graph in () where the tangent does not exist. At the sharp corner, (0, 0). If you take a look at the first derivative dy dx = t 5t 4 t, you ll see that this occurs when t = 0. The derivative is also undefined at t = ± /5, but that s not where the sharp corner is. 3. Determine the points on the parametric curve graph in () where the tangent is vertical. From the work above, we can see that it occurs when t = ± /5, this gives: ( ( ) 5/ ( ) ) ( 3/ 4, ( ) 5/ ( ) ) 3/ + 4, Determine the single point on the parametric curve graph in () that has two different t values. What are these values. Also determine the the slopes of the line tangent at these two t values. By inspection this appears to happen when y = 4, and that s the same as saying t = ±. The slope of the tangent line at this point is ±/8.
13 5. Graph the parametric curve given by x = t, and y = t 7 + t Figure 9: Graph of x = t, and y = t 7 + t Using the second derivative, find where our parametric curve will be concave up or concave down. Your answer should be given in t. d y dx = 35t3 + 5t 4 Only one real solution to the second derivative being zero, and that occurs when t = 0. Simple sign analysis gives us concave up for t > 0, and concave down for t < 0. Now, look at the graph. Not easy, I know... 8 Area As always, I want to stress that you graph even if it s just a rough sketch the area you re trying to determine before you set-up an integral. It may not be easy, but technology is available that helps, and it s incumbent upon you to learn your tools. For example if you re asked to find the area between the x axis and the curve defined by y = x ; or the area 3 defined between y = e x and the positive x-axis... you should always try to visualize it first... the answer may be staring at you. Let s move on to parametric equations. Typically, in the past, we had some integral that will look like this b a F (x) dx. () Now, we are going to have parametric equations of the form x = f (t) and g (t) = y, where Answer: It s just half of a unit circle, so the area is π/. No, I did not do the integration. 3 You won t be able to do this one, but it s well known (related to the normal function) to mathematical statisticians and the answer is π/. Again, no need to do the integration, but I did need to know a very famous integral related to this. 3
14 α t β. Let s make a simple substitution into the Integral () above and we ll get b a F (x) dx = b a y dx = β α g (t) f (t) dt. (3) Example: Determine the area under (between the graph and the x-axis) the parametric curve given by the following parametric equations. x = (t sin t) y = ( cos t) 4π t 4π. To do this problem you need to graph (Figure 0, page 4) it first. Notice that you have four Figure 0: Graph of x = (t sin t), y = ( cos t), 4π t 4π, not properly scaled! equal sections, so I ll just look at one section for this example I ll use 0 t π and multiply its area by four. 4 b a 4π F (x) dx = 4 = 4 9 Arc Length 0 π 0 y dx a = (0 sin 0) and b = (π sin π) ( cos t) ( cos t) dt dx = ( cos t) dt = 48π Yes, you too can do this integration. Well, you should know by now that I am also going to approach arc length from a visual perspective too. For example, suppose I were to ask you to determine the length of arc for the parametric equations g (t) = y = sin t and f (t) = x = cos t for 0 t π? Yes, graph this one and you ll see that it s fairly easy. 4 Now let s do the calculus! If you eliminate the parameter you ll get y = x where x. You can now use the arc length formula that you learn prior. That is, if the derivatives are continuous on the interval 4 Half the unit cirlce, so its length is π. 4
15 of interest, then the length of the curve is Or if you prefer, L = L = b a d c f (a) f (b) + + ( ) dy dx. dx ( ) dx dy. dy Let s now instead try to develop an analogous model for the parametric form. Same as we did in the past, but this time we ll use parameter t. b ( ) dy L = + dx a dx f (b) ( g = + ) (t) f f (t) dt (t) = f (a) (f (t)) + (g (t)) dt Giving it a try on our example. L = = = = = π 0 π 0 π 0 π + ( ) dy dx dx (f (t)) + (g (t)) dt ( sin t) + (cos t) dt dt You re going to get confused, so please draw a picture before proceeding forward on arc length problems. There s also some (many) details I am leaving off here and you may want to read the book for a more complete picture. 0 Surface Area You may recall that we also did surface area and found that b [ ] dy S = π r (x) + dx. Here y is a function of x. dx a This is easily transformed to the parametric form as follows S = π f (b) f (a) g (t) (f (t)) + (g (t)) dt. 5
16 A simple example using parametric equations follows. Example: Show that the surface area of a unit sphere is 4π. Let s use f (t) = sin t and g (t) = cos t on π/ t π/. 5 Here s the integral. π/ S = π = π = 4π π/ π/ π/ cos t (cos t) + ( sin) dt cos t dt Now why don t you try doing this with y = x for x. 6 You try... You should be able to do the following problems. Yes, it s a good idea to use a graphing aid. 7. Graph f (t) = x = e t cos 5t, g (t) = y = e t sin 5t, 0 t π.. Find an equation of the tangent(s) to the curve x = cos t + cos t, y = sin t + sin t at the point (, ). Then graph the curve and the tangent(s). 3. Find the length of the loop of the curve x = 3t t 3, y = 3t. 4. Find the exact area of the surface obtained by rotating the curve x = 3t t 3, y = 3t about the x-axis for 0 t. 5. Find the area enclosed by the x-axis and the curve x = + e t, y = t t. 5 Yes, you need to graph this! 6 It is really just as easy. 7 Mathematica, or whatever you prefer. I prefer Grapher, but you need to make your own decisions. 6
17 Answers to You try.... Graph f (t) = x = e t cos 5t, g (t) = y = e t sin 5t, 0 t π Figure : Graph of (f (t), g (t)) for 0 t π. This would be pure hell to do by hand.. Find an equation of the tangent(s) to the curve x = cos t + cos t, y = sin t + sin t at the point (, ). Then graph the curve and the tangent(s). You ll need to figure out where this point occurs on the graph. Yes, I suggest you graph it before doing anything else, including the point and the tangent. By inspection, this point occurs at t = π/. 8 To find the slope at this point you will need the derivative evaluated at this point. Here s the derivative, dy cos t + cos t = dx sin t sin t. And here s the value of the derivative at this point. dy =. dx So the line tangent is t= π y = (x + ) y = x You ll need to use your head here. Just try to reason on this one, mainly because it is easier than thuggishly doing the math. 7
18 0 Figure : Graph of (x = cos t + cos t, y = sin t + sin t) for 0 t π, and the line tangent to this curve at the point (, ). 3. Find the length of the loop of the curve x = 3t t 3, y = 3t. You should, of course, figure out what the loop is and the best way to do this is to visualize it. The way to do this is by graphing various ranges for t, and also trying to figure out the range by looking at the equations. Just factoring the expression for x will give you a good idea... the loop occurs for 3 t 3. Again, use a graphing device. Here s the graph Figure 3: Graph of ( x = 3t t 3, y = 3t ) for 3 t 3, where the loop occurs between ( 3, 3 ). 8
19 Now let s compute the length of this loop. = = = 3 (36t ) + (9 8t + 9t 4 ) dt 3t + 3 dt 4. Find the exact area of the surface obtained by rotating the curve x = 3t t 3, y = 3t about the x-axis for 0 t. We just did this graph, but now t is being restricted to 0 t. Here s the graph And here s the integration. Figure 4: Graph of ( x = 3t t 3, y = 3t ) for 0 t. = π = 48π 5 0 3t (36t ) + (9 8t + 9t 4 ) dt 5. Find the area enclosed by the x-axis and the curve x = + e t, y = t t. I think you re getting the point about graph. And the area between this curve and the x-axis is found by integration. A = 0 ( t t ) e t dt = 3 e Okay, I m tired and I used Mathematica! 9
20 3 Figure 5: Graph of ( x = + e t, y = t t ) for 0 t. Assignment:. You should read 0. and 0., and then be able to do the following problems. 0. Curves Defined by Parametric Equations:, 5, 9, 4, 6,, 4, 5, 8, 9, 33, 37, 40, 4, Calculus with Parametric Curves:, 4, 6, 7, 0,, 4, 5, 8,, 6, 3, 36, 37, 39, 4, 45, 5, 65, 74. WebAssign problems, similar to the ones above, will be posted and you need to get started right away!. The second unit exam will be on section sections 7.6, 7.7, 7.8, 8., 8., 0., 0.. The date will be discussed in class. Calculators, including cell phones and computers, are not allowed. One sheet of notes is allowed, but it can not contain any worked problems! 9 9 I will collect your note sheet, along with the exam. If I see any worked problems you will be assigned a zero on the exam. 0
1 Tangents and Secants
MTH 11 Web Based Material Essex County College Division of Mathematics and Physics Worksheet #, Last Update July 15, 010 1 1 Tangents and Secants The idea of a it is central to calculus and an intuitive
More informationUpdated: August 24, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University
Updated: August 24, 216 Calculus III Section 1.2 Math 232 Calculus III Brian Veitch Fall 215 Northern Illinois University 1.2 Calculus with Parametric Curves Definition 1: First Derivative of a Parametric
More informationHere are some of the more basic curves that we ll need to know how to do as well as limits on the parameter if they are required.
1 of 10 23/07/2016 05:15 Paul's Online Math Notes Calculus III (Notes) / Line Integrals / Line Integrals - Part I Problems] [Notes] [Practice Problems] [Assignment Calculus III - Notes Line Integrals Part
More informationFirst of all, we need to know what it means for a parameterize curve to be differentiable. FACT:
CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of
More informationExam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA:
MA 114 Exam 3 Spring 217 Exam 3 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test.
More informationCALCULUS II. Parametric Equations and Polar Coordinates. Paul Dawkins
CALCULUS II Parametric Equations and Polar Coordinates Paul Dawkins Table of Contents Preface... ii Parametric Equations and Polar Coordinates... 3 Introduction... 3 Parametric Equations and Curves...
More informationIn this chapter, we will investigate what have become the standard applications of the integral:
Chapter 8 Overview: Applications of Integrals Calculus, like most mathematical fields, began with trying to solve everyday problems. The theory and operations were formalized later. As early as 70 BC,
More information10.1 Curves Defined by Parametric Equations
10.1 Curves Defined by Parametric Equations Ex: Consider the unit circle from Trigonometry. What is the equation of that circle? There are 2 ways to describe it: x 2 + y 2 = 1 and x = cos θ y = sin θ When
More informationTangents of Parametric Curves
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 32 Notes These notes correspond to Section 92 in the text Tangents of Parametric Curves When a curve is described by an equation of the form y = f(x),
More informationLecture 15. Lecturer: Prof. Sergei Fedotov Calculus and Vectors. Length of a Curve and Parametric Equations
Lecture 15 Lecturer: Prof. Sergei Fedotov 10131 - Calculus and Vectors Length of a Curve and Parametric Equations Sergei Fedotov (University of Manchester) MATH10131 2011 1 / 5 Lecture 15 1 Length of a
More informationEuler s Method for Approximating Solution Curves
Euler s Method for Approximating Solution Curves As you may have begun to suspect at this point, time constraints will allow us to learn only a few of the many known methods for solving differential equations.
More informationMAT1B01: Curves defined by parametric equations
MAT1B01: Curves defined by parametric equations Dr Craig 24 October 2016 My details: acraig@uj.ac.za Consulting hours: Thursday 11h20 12h55 Friday 11h30 13h00 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
More informationMATH 1020 WORKSHEET 10.1 Parametric Equations
MATH WORKSHEET. Parametric Equations If f and g are continuous functions on an interval I, then the equations x ft) and y gt) are called parametric equations. The parametric equations along with the graph
More informationMTH 120 Fall 2007 Essex County College Division of Mathematics Handout Version 6 1 October 3, 2007
MTH 10 Fall 007 Essex County College Division of Mathematics Handout Version 6 1 October, 007 1 Inverse Functions This section is a simple review of inverses as presented in MTH-119. Definition: A function
More information5/27/12. Objectives. Plane Curves and Parametric Equations. Sketch the graph of a curve given by a set of parametric equations.
Objectives Sketch the graph of a curve given by a set of parametric equations. Eliminate the parameter in a set of parametric equations. Find a set of parametric equations to represent a curve. Understand
More informationIncreasing/Decreasing Behavior
Derivatives and the Shapes of Graphs In this section, we will specifically discuss the information that f (x) and f (x) give us about the graph of f(x); it turns out understanding the first and second
More informationTo graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6. ) is graphed below:
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
More informationStudy Guide for Test 2
Study Guide for Test Math 6: Calculus October, 7. Overview Non-graphing calculators will be allowed. You will need to know the following:. Set Pieces 9 4.. Trigonometric Substitutions (Section 7.).. Partial
More informationPlane Curve [Parametric Equation]
Plane Curve [Parametric Equation] Bander Almutairi King Saud University December 1, 2015 Bander Almutairi (King Saud University) Plane Curve [Parametric Equation] December 1, 2015 1 / 8 1 Parametric Equation
More informationHot X: Algebra Exposed
Hot X: Algebra Exposed Solution Guide for Chapter 11 Here are the solutions for the Doing the Math exercises in Hot X: Algebra Exposed! DTM from p.149 2. Since m = 2, our equation will look like this:
More informationDirection Fields; Euler s Method
Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this
More information: = Curves Defined by Parametric Equations. Ex: Consider the unit circle from Trigonometry. What is the equation of that circle?
10.1 Curves Defined by Parametric Equations Ex: Consider the unit circle from Trigonometry. What is the equation of that circle? of 8* * # 2+-12=1 There are 2 ways to describe it: x 2 + y 2 = 1 x = cos!
More informationNAME: Section # SSN: X X X X
Math 155 FINAL EXAM A May 5, 2003 NAME: Section # SSN: X X X X Question Grade 1 5 (out of 25) 6 10 (out of 25) 11 (out of 20) 12 (out of 20) 13 (out of 10) 14 (out of 10) 15 (out of 16) 16 (out of 24)
More informationMath 206 First Midterm October 5, 2012
Math 206 First Midterm October 5, 2012 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 8 pages including this cover AND IS DOUBLE SIDED.
More informationMAT01B1: Curves defined by parametric equations
MAT01B1: Curves defined by parametric equations Dr Craig 10 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
More informationHot X: Algebra Exposed
Hot X: Algebra Exposed Solution Guide for Chapter 10 Here are the solutions for the Doing the Math exercises in Hot X: Algebra Exposed! DTM from p.137-138 2. To see if the point is on the line, let s plug
More informationAP Calculus. Slide 1 / 213 Slide 2 / 213. Slide 3 / 213. Slide 4 / 213. Slide 4 (Answer) / 213 Slide 5 / 213. Derivatives. Derivatives Exploration
Slide 1 / 213 Slide 2 / 213 AP Calculus Derivatives 2015-11-03 www.njctl.org Slide 3 / 213 Table of Contents Slide 4 / 213 Rate of Change Slope of a Curve (Instantaneous ROC) Derivative Rules: Power, Constant,
More informationIncreasing/Decreasing Behavior
Derivatives and the Shapes of Graphs In this section, we will specifically discuss the information that f (x) and f (x) give us about the graph of f(x); it turns out understanding the first and second
More informationPolar Coordinates. 2, π and ( )
Polar Coordinates Up to this point we ve dealt exclusively with the Cartesian (or Rectangular, or x-y) coordinate system. However, as we will see, this is not always the easiest coordinate system to work
More informationMath 231E, Lecture 34. Polar Coordinates and Polar Parametric Equations
Math 231E, Lecture 34. Polar Coordinates and Polar Parametric Equations 1 Definition of polar coordinates Let us first recall the definition of Cartesian coordinates: to each point in the plane we can
More informationUniversity of California, Berkeley
University of California, Berkeley FINAL EXAMINATION, Fall 2012 DURATION: 3 hours Department of Mathematics MATH 53 Multivariable Calculus Examiner: Sean Fitzpatrick Total: 100 points Family Name: Given
More informationMAT 003 Brian Killough s Instructor Notes Saint Leo University
MAT 003 Brian Killough s Instructor Notes Saint Leo University Success in online courses requires self-motivation and discipline. It is anticipated that students will read the textbook and complete sample
More informationCalculus II (Math 122) Final Exam, 11 December 2013
Name ID number Sections B Calculus II (Math 122) Final Exam, 11 December 2013 This is a closed book exam. Notes and calculators are not allowed. A table of trigonometric identities is attached. To receive
More informationGoals: Course Unit: Describing Moving Objects Different Ways of Representing Functions Vector-valued Functions, or Parametric Curves
Block #1: Vector-Valued Functions Goals: Course Unit: Describing Moving Objects Different Ways of Representing Functions Vector-valued Functions, or Parametric Curves 1 The Calculus of Moving Objects Problem.
More informationCCNY Math Review Chapter 2: Functions
CCN Math Review Chapter : Functions Section.1: Functions.1.1: How functions are used.1.: Methods for defining functions.1.3: The graph of a function.1.: Domain and range.1.5: Relations, functions, and
More informationPolar Coordinates
Polar Coordinates 7-7-2 Polar coordinates are an alternative to rectangular coordinates for referring to points in the plane. A point in the plane has polar coordinates r,θ). r is roughly) the distance
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More informationBackground for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
More informationLab 2B Parametrizing Surfaces Math 2374 University of Minnesota Questions to:
Lab_B.nb Lab B Parametrizing Surfaces Math 37 University of Minnesota http://www.math.umn.edu/math37 Questions to: rogness@math.umn.edu Introduction As in last week s lab, there is no calculus in this
More information2.9 Linear Approximations and Differentials
2.9 Linear Approximations and Differentials 2.9.1 Linear Approximation Consider the following graph, Recall that this is the tangent line at x = a. We had the following definition, f (a) = lim x a f(x)
More informationMA 114 Worksheet #17: Average value of a function
Spring 2019 MA 114 Worksheet 17 Thursday, 7 March 2019 MA 114 Worksheet #17: Average value of a function 1. Write down the equation for the average value of an integrable function f(x) on [a, b]. 2. Find
More informationENGI Parametric & Polar Curves Page 2-01
ENGI 3425 2. Parametric & Polar Curves Page 2-01 2. Parametric and Polar Curves Contents: 2.1 Parametric Vector Functions 2.2 Parametric Curve Sketching 2.3 Polar Coordinates r f 2.4 Polar Curve Sketching
More informationHSC Mathematics - Extension 1. Workshop E2
HSC Mathematics - Extension Workshop E Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong Moss
More informationDerivatives and Graphs of Functions
Derivatives and Graphs of Functions September 8, 2014 2.2 Second Derivatives, Concavity, and Graphs In the previous section, we discussed how our derivatives can be used to obtain useful information about
More informationBasics of Computational Geometry
Basics of Computational Geometry Nadeem Mohsin October 12, 2013 1 Contents This handout covers the basic concepts of computational geometry. Rather than exhaustively covering all the algorithms, it deals
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More informationDifferentiation. The Derivative and the Tangent Line Problem 10/9/2014. Copyright Cengage Learning. All rights reserved.
Differentiation Copyright Cengage Learning. All rights reserved. The Derivative and the Tangent Line Problem Copyright Cengage Learning. All rights reserved. 1 Objectives Find the slope of the tangent
More informationAlgebra 2 Semester 1 (#2221)
Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the 2016-2017 Course Guides for the following course: Algebra 2 Semester
More information3.1 Maxima/Minima Values
3.1 Maxima/Minima Values Ex 1: Find all critical points for the curve given by f (x)=x 5 25 3 x3 +20x 1 on the interval [-3, 2]. Identify the min and max values. We're guaranteed max and min points if
More informationSAMLab Tip Sheet #1 Translating Mathematical Formulas Into Excel s Language
Translating Mathematical Formulas Into Excel s Language Introduction Microsoft Excel is a very powerful calculator; you can use it to compute a wide variety of mathematical expressions. Before exploring
More informationMath 126C: Week 3 Review
Math 126C: Week 3 Review Note: These are in no way meant to be comprehensive reviews; they re meant to highlight the main topics and formulas for the week. Doing homework and extra problems is always the
More informationMA 113 Calculus I Fall 2015 Exam 2 Tuesday, 20 October Multiple Choice Answers. Question
MA 113 Calculus I Fall 2015 Exam 2 Tuesday, 20 October 2015 Name: Section: Last digits of student ID #: This exam has ten multiple choice questions (five points each) and five free response questions (ten
More informationR f da (where da denotes the differential of area dxdy (or dydx)
Math 28H Topics for the second exam (Technically, everything covered on the first exam, plus) Constrained Optimization: Lagrange Multipliers Most optimization problems that arise naturally are not unconstrained;
More information10 Polar Coordinates, Parametric Equations
Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates
More information15. PARAMETRIZED CURVES AND GEOMETRY
15. PARAMETRIZED CURVES AND GEOMETRY Parametric or parametrized curves are based on introducing a parameter which increases as we imagine travelling along the curve. Any graph can be recast as a parametrized
More information12 Polar Coordinates, Parametric Equations
54 Chapter Polar Coordinates, Parametric Equations Polar Coordinates, Parametric Equations Just as we describe curves in the plane using equations involving x and y, so can we describe curves using equations
More informationMath 113 Exam 1 Practice
Math Exam Practice January 6, 00 Exam will cover sections 6.-6.5 and 7.-7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second
More information3 Vectors and the Geometry of Space
3 Vectors and the Geometry of Space Up until this point in your career, you ve likely only done math in 2 dimensions. It s gotten you far in your problem solving abilities and you should be proud of all
More information: = Curves Defined by Parametric Equations. Ex: Consider the unit circle from Trigonometry. What is the equation of that circle?
10.1 Curves Defined by Parametric Equations Ex: Consider the unit circle from Trigonometry. What is the equation of that circle? of 8* * # 2+-121 There are 2 ways to describe it: x 2 + y 2 1 x cos! : 9
More informationTopic 5.1: Line Elements and Scalar Line Integrals. Textbook: Section 16.2
Topic 5.1: Line Elements and Scalar Line Integrals Textbook: Section 16.2 Warm-Up: Derivatives of Vector Functions Suppose r(t) = x(t) î + y(t) ĵ + z(t) ˆk parameterizes a curve C. The vector: is: r (t)
More informationObjectives. Materials
Activity 13 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the
More information4 Visualization and. Approximation
4 Visualization and Approximation b A slope field for the differential equation y tan(x + y) tan(x) tan(y). It is not always possible to write down an explicit formula for the solution to a differential
More informationExam 2 Preparation Math 2080 (Spring 2011) Exam 2: Thursday, May 12.
Multivariable Calculus Exam 2 Preparation Math 28 (Spring 2) Exam 2: Thursday, May 2. Friday May, is a day off! Instructions: () There are points on the exam and an extra credit problem worth an additional
More informationy= sin( x) y= cos( x)
. The graphs of sin(x) and cos(x). Now I am going to define the two basic trig functions: sin(x) and cos(x). Study the diagram at the right. The circle has radius. The arm OP starts at the positive horizontal
More informationWHICH GRAPH REPRESENTS THE FOLLOWING PIECEWISE DEFINED FUNCTION FILE
22 March, 2018 WHICH GRAPH REPRESENTS THE FOLLOWING PIECEWISE DEFINED FUNCTION FILE Document Filetype: PDF 404.36 KB 0 WHICH GRAPH REPRESENTS THE FOLLOWING PIECEWISE DEFINED FUNCTION FILE Which of the
More informationGradient Descent - Problem of Hiking Down a Mountain
Gradient Descent - Problem of Hiking Down a Mountain Udacity Have you ever climbed a mountain? I am sure you had to hike down at some point? Hiking down is a great exercise and it is going to help us understand
More informationDynamics and Vibrations Mupad tutorial
Dynamics and Vibrations Mupad tutorial School of Engineering Brown University ENGN40 will be using Matlab Live Scripts instead of Mupad. You can find information about Live Scripts in the ENGN40 MATLAB
More informationDifferentiation. J. Gerlach November 2010
Differentiation J. Gerlach November 200 D and diff The limit definition of the derivative is covered in the Limit tutorial. Here we look for direct ways to calculate derivatives. Maple has two commands
More informationThe diagram above shows a sketch of the curve C with parametric equations
1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate
More informationMEI GeoGebra Tasks for AS Pure
Task 1: Coordinate Geometry Intersection of a line and a curve 1. Add a quadratic curve, e.g. y = x 2 4x + 1 2. Add a line, e.g. y = x 3 3. Use the Intersect tool to find the points of intersection of
More information1 Finding Trigonometric Derivatives
MTH 121 Fall 2008 Essex County College Division of Matematics Hanout Version 8 1 October 2, 2008 1 Fining Trigonometric Derivatives 1.1 Te Derivative as a Function Te efinition of te erivative as a function
More informationParametric Curves and Polar Coordinates
Parametric Curves and Polar Coordinates Math 251, Fall 2017 Juergen Gerlach Radford University Parametric Curves We will investigate several aspects of parametric curves in the plane. The curve given by
More informationParametric Curves and Polar Coordinates
Parametric Curves and Polar Coordinates Math 251, Fall 2017 Juergen Gerlach Radford University Parametric Curves We will investigate several aspects of parametric curves in the plane. The curve given by
More informationSolution Guide for Chapter 20
Solution Guide for Chapter 0 Here are the solutions for the Doing the Math exercises in Girls Get Curves! DTM from p. 351-35. In the diagram SLICE, LC and IE are altitudes of the triangle!sci. L I If SI
More informationThe first thing we ll need is some numbers. I m going to use the set of times and drug concentration levels in a patient s bloodstream given below.
Graphing in Excel featuring Excel 2007 1 A spreadsheet can be a powerful tool for analyzing and graphing data, but it works completely differently from the graphing calculator that you re used to. If you
More informationf xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is
1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,
More informationTo graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
More informationf sin the slope of the tangent line is given by f sin f cos cos sin , but it s also given by 2. So solve the DE with initial condition: sin cos
Math 414 Activity 1 (Due by end of class August 1) 1 Four bugs are placed at the four corners of a square with side length a The bugs crawl counterclockwise at the same speed and each bug crawls directly
More information2. Periodic functions have a repeating pattern called a cycle. Some examples from real-life that have repeating patterns might include:
GRADE 2 APPLIED SINUSOIDAL FUNCTIONS CLASS NOTES Introduction. To date we have studied several functions : Function linear General Equation y = mx + b Graph; Diagram Usage; Occurence quadratic y =ax 2
More information10.2 Calculus with Parametric Curves
CHAPTER 1. PARAMETRIC AND POLAR 91 1.2 Calculus with Parametric Curves Example 1. Return to the parametric equations in Example 2 from the previous section: x t + sin() y t + cos() (a) Find the Cartesian
More informationIn math, the rate of change is called the slope and is often described by the ratio rise
Chapter 3 Equations of Lines Sec. Slope The idea of slope is used quite often in our lives, however outside of school, it goes by different names. People involved in home construction might talk about
More information9.1 Parametric Curves
Math 172 Chapter 9A notes Page 1 of 20 9.1 Parametric Curves So far we have discussed equations in the form. Sometimes and are given as functions of a parameter. Example. Projectile Motion Sketch and axes,
More information1) Complete problems 1-65 on pages You are encouraged to use the space provided.
Dear Accelerated Pre-Calculus Student (017-018), I am excited to have you enrolled in our class for next year! We will learn a lot of material and do so in a fairly short amount of time. This class will
More informationMAT137 Calculus! Lecture 12
MAT137 Calculus! Lecture 12 Today we will study more curve sketching (4.6-4.8) and we will make a review Test 2 will be next Monday, June 26. You can check the course website for further information Next
More informationIntroduction to creating and working with graphs
Introduction to creating and working with graphs In the introduction discussed in week one, we used EXCEL to create a T chart of values for a function. Today, we are going to begin by recreating the T
More informationGraphing by. Points. The. Plotting Points. Line by the Plotting Points Method. So let s try this (-2, -4) (0, 2) (2, 8) many points do I.
Section 5.5 Graphing the Equation of a Line Graphing by Plotting Points Suppose I asked you to graph the equation y = x +, i.e. to draw a picture of the line that the equation represents. plotting points
More informationMath Lab 6: Powerful Fun with Power Series Representations of Functions Due noon Thu. Jan. 11 in class *note new due time, location for winter quarter
Matter & Motion Winter 2017 18 Name: Math Lab 6: Powerful Fun with Power Series Representations of Functions Due noon Thu. Jan. 11 in class *note new due time, location for winter quarter Goals: 1. Practice
More informationChapter 10 Homework: Parametric Equations and Polar Coordinates
Chapter 1 Homework: Parametric Equations and Polar Coordinates Name Homework 1.2 1. Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the
More informationMath (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines
Math 18.02 (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines February 12 Reading Material: From Simmons: 17.1 and 17.2. Last time: Square Systems. Word problem. How many solutions?
More informationCalculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes
1 of 11 1) Give f(g(1)), given that Calculators ARE NOT Permitted On This Portion Of The Exam 28 Questions - 55 Minutes 2) Find the slope of the tangent line to the graph of f at x = 4, given that 3) Determine
More information1 MATH 253 LECTURE NOTES for FRIDAY SEPT. 23,1988: edited March 26, 2013.
1 MATH 253 LECTURE NOTES for FRIDAY SEPT. 23,1988: edited March 26, 2013. TANGENTS Suppose that Apple Computers notices that every time they raise (or lower) the price of a $5,000 Mac II by $100, the number
More informationSection 4.3: Derivatives and the Shapes of Curves
1 Section 4.: Derivatives and the Shapes of Curves Practice HW from Stewart Textbook (not to hand in) p. 86 # 1,, 7, 9, 11, 19, 1,, 5 odd The Mean Value Theorem If f is a continuous function on the closed
More informationGenerating Functions
6.04/8.06J Mathematics for Computer Science Srini Devadas and Eric Lehman April 7, 005 Lecture Notes Generating Functions Generating functions are one of the most surprising, useful, and clever inventions
More informationExam 1 Review. MATH Intuitive Calculus Fall Name:. Show your reasoning. Use standard notation correctly.
MATH 11012 Intuitive Calculus Fall 2012 Name:. Exam 1 Review Show your reasoning. Use standard notation correctly. 1. Consider the function f depicted below. y 1 1 x (a) Find each of the following (or
More informationMath 124 Final Examination Autumn Turn off all cell phones, pagers, radios, mp3 players, and other similar devices.
Math 124 Final Examination Autumn 2016 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name Turn off all cell phones, pagers, radios, mp3 players, and other similar devices. This
More informationMAT01B1: Surface Area of Solids of Revolution
MAT01B1: Surface Area of Solids of Revolution Dr Craig 02 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday 11h20 12h55 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
More information2.2 Volumes of Solids of Revolution
2.2 Volumes of Solids of Revolution We know how to find volumes of well-established solids such as a cylinder or rectangular box. What happens when the volume can t be found quite as easily nice or when
More informationLab 4. Recall, from last lab the commands Table[], ListPlot[], DiscretePlot[], and Limit[]. Go ahead and review them now you'll be using them soon.
Calc II Page 1 Lab 4 Wednesday, February 19, 2014 5:01 PM Recall, from last lab the commands Table[], ListPlot[], DiscretePlot[], and Limit[]. Go ahead and review them now you'll be using them soon. Use
More informationArea rectangles & parallelograms
Area rectangles & parallelograms Rectangles One way to describe the size of a room is by naming its dimensions. So a room that measures 12 ft. by 10 ft. could be described by saying its a 12 by 10 foot
More informationMEI GeoGebra Tasks for A2 Core
Task 1: Functions The Modulus Function 1. Plot the graph of y = x : use y = x or y = abs(x) 2. Plot the graph of y = ax+b : use y = ax + b or y = abs(ax+b) If prompted click Create Sliders. What combination
More information