DOCSLIB.ORG

Parametric Curves a Curve C Consisting of Points (X,Y) Where X and Y Are Both Functions of a Third Variable, Say T (Called a Parameter)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Parametric Curves a Curve C Consisting of Points (X,Y) Where X and Y Are Both Functions of a Third Variable, Say T (Called a Parameter) Parametric Curves A curve C consisting of points (x,y) where x and y are both functions of a third variable, say t (called a parameter). Thus, we have each point (x,y) determined by the parametric equations x= f( t) and y= gt( ) . Note: Often, but not always, t represents time and these equations are called equations of motion and the curve C is the trajectory of the moving particle. Examples: (a) Sketch the curve defined by the parametric equations xt=−+2 yt=52 − We will eliminate the parameter t and sketch. (b) Sketch the curve defined by the parametric equations ( xy,) =( 1,1) +− t( 1,1) This is a compact form of writing the equations separately. Separate this into two individual equations. Eliminate the parameter and sketch the line. Find the coordinates of the point of intersection of the above two curves (c) A spider and a fly crawl so that their positions at time t (in seconds) are: Spider: ( xy,) =−+−( 2,5) t( 1, 2) Fly:( xy ,) =( 1,1) +− t( 1,1) Question: Does the spider meet the fly? Explain As you should have noticed, parametric equations tend to contain more information. Eliminating the parameter in parametric equations containing trig functions. Note: We often use trig identities, especially cos22tt+= sin 1 Examples: Eliminate the parameter t if: (a) xt= cos (b) xt = cos+= 2( c) x 2sin t y=cos( 2 t) yt =sin+= 3 y3cos t 6 Slopes of Tangent lines of Parametric Curves Method: dy dx 1. Find and dt dt dy dy dt 2. = dx dx dt dy Example: Find if x=−+2 ty = 52 − t dx Solution: dy dy dx dy −2 =−= 2 and 1. Hence, ===−dt 2 dt dt dx dx 1 dt Example: Find an equation of the tangent line to the parametric curve xt=−=+322 yt at the point where t = 1. dy dy dx dy 1 Solution: =1 and =−==2t . Hence, dt dt dt dxdx −2 t dt 11 Slope of tangent at t = 1 is m = = − and (x,y) = (2, 3) t −22 1 Therefore, the equation of the tangent line is: yx−=−32( −) 2 In order to draw sketches of these parametric curves we also need to be able to find the second derivative to discuss concavity. Recall from Calculus 1500: To find the second derivative we differentiated the first derivative. That is exactly what we do with parametric equations. Just be careful! 7 Example: Find the first and second derivatives of the parametric curve xt=−=+322 yt Solution: As above, the first derivative is dy dy dx dy 1 =1 and =−==2t . Hence, dt dt dt dxdx −2 t dt 1 That is, y′ = − and the second derivative is 2t dy( ′) dy2 which most people write as . In other words, you find the dx dx2 11 parametric derivative using x=−3 t21 NewY ==−=− y′ t − 22t 1 −2 2 t dy d( NewY ) 1 So = =2 = dx23 dx−−24 t t Be very careful not to confuse this with the derivative of a quotient. It has nothing to do with quotients! dy d2 y 11 Example: Find and at t=2 if x = ty23 = t dx dx2 23 dy dy 22d dy t d y dx 1 Solution: =dt = =t and = = dxdx t dx2 dx t dt dy d2 y 1 When t = 2 = 2 and = dx dx2 2 We are now in a position to follow curve sketching steps, very similar to the curve sketching steps you followed in math 1500. 8 Other than plotting some interesting points we will follow these steps: 1. Find the x and y intercepts. 2. Determine the “starting” and “ending” locations of the curve by finding the limits of x and y as t→± ∞. 3. Find all horizontal and vertical tangent lines. 4. Use a first derivative chart to find the direction the curve is following in strategic intervals. 5. Sketch the curve. Using the above information. 6. Check the concavity of your curve using the values of the second derivative in strategic intervals. Example: Follow the above steps in order to sketch the curve determined by the parametric equations xttytt=−=−3233 3 Solution: Intercepts xy= 00= 03=tt32−=03tt3− 0=−t2 ( t 3) 0=−+tt( 33)( t ) tt= 0,3 =0, 3, − 3 ∴−yxintercepts are: 0,18 −intercepts are: 0,3 3 −9, − 3 3 − 9 Beginning and Ending Positions limx= lim t32 − 3 t = lim tt 2( − 3) = −∞ ( ∴ starts far to the left ) tt→−∞ →−∞ ( ) t→−∞ 32 2 limx= lim( t − 3 t) = lim tt( − 3) =∞∴ ( ends far to the right ) tt→∞ →∞ t→∞ limy= lim t32 − 3 t = lim tt − 3 = ±∞ tt→±∞ →±∞ ( ) t→±∞ ( ) ( ∴starts far down and ends far up ) Coordinates of points of horizontal and vertical tangent lines. ( These are our horizontal and vertical “min/max” positions. ) dy dx dy Be very careful – we will be using and NOT !!! dt dt dx 9 dy dx =33t22−=36tt− dt dt 03=−+(t 1)( t 1) 03=tt( − 2) dx dy tt=−≠1, 1 and 0 =0,2 and ≠0 dt dt ∴∴The curve has horizontal tangents The curve has vertical tangents at (−−2, 2) and ( −4, 2) at (0,0) and (− 4,2) The Chart We will use the critical values of t to determine the directions of the curve t −1 01 2 dx ++−−+ dt dy +−−++ dt x →→←←→ y ↑↓↓↑↑ curve Use the above information, especially the curve directions in the chart to sketch the curve. dy d2 y Check the concavity [Note: Now we will be using and .] dx dx2 10 dy dy33 t 2 − Note: It is tempting but not advisable =dt = 2 dxdx 36 t− t to reduce this fraction. dt 2 33t − 22 d 63tt( −−− 6 t) ( 6 t 63)( t − 3) 36tt2 − ← [This is the Quotient Rule] 2 2 dy2 (36tt− ) = dt = dx22dx 36t− t dt 18ttttt3− 36 232 − 18 + 18 +− 18 18 = 3 (36tt2 − ) 2 −18tt2 +− 18 18 −18(tt −+ 1) = 33= (36tt2 − ) 27tt3 ( − 2) Therefore, the numbers which appear on the number line are t = 0 and 2. t 02 dy2 − +− dx2 curve ∩ ∪∩ Practice: Follow the steps outlined above to analyze and sketch the curve described by the parametric equations: 1. xt=−=−323 t y 39 t 2. xt=−=−2344 yt t 3. xt=−=−23 t yt3 t 4. xt=+=+222 t yt t 5. xt= 22 yt= − 2 t 6. xt=22 +21 t − yt = +− t 2 7. xttytt=−=−33332 8. xt= 32 yt= − 2 t t 9. xt=−=−3232 t yt t 10. xe=−=t t y4 e2 11 .
Load more

Recommended publications
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • 2.3 Geometry of Curves: Arclength, Curvature, Torsion Overview
    8 Chapter 2 draft 2.3 Geometry of curves: arclength, curvature, torsion Overview: The geometry of curves in space is described independently of how the curve is parameterized. The key notion of curvature measures how rapidly the curve is bending in space. In 3-D, an additional quantity, tor- sion, describes how much the curve is wobbling out of a plane. Alternative methods of calculation for curvature and torsion are developed. 2.3.1 Idea of curved vs. straight as independent of parametrization The goal of this subsection is to create some measurement of curvedness that is a property of the road, not the driver. A first question: which of the two circles is “more curved”? Figure 1: The two circles have different “curviness” You most likely decided that the one with the smaller radius is more curved. How can we quantify that? How should we relate more general curves to circles? What could play the role of the radius? How curved is a line? 2.3.2 Circular model: tied to radius of circle If a smaller radius leads to a more curved circle, it follows that the measurement of curvature should increase as the radius of a circle decreases. Curvature has a long history including work in ancient Greece, but the curvature of a general curve in the plane was solved by Newton, building on earlier work by Oresme, Huyghens, and others (see Lodder[TBA REF] and references therein). Newton defined the curvature for a circle as the reciprocal of the radius. Newton then went deeper and developed the general case in terms of the radius of the best circular approx- imation, made instantaneously at each point.
    [Show full text]
  • CALCULUS Author Sushma Sarin, Freelance Author Units (1-14)
    ALAGAPPA UNIVERSITY [Accredited with ‘A+’ Grade by NAAC (CGPA:3.64) in the Third Cycle and Graded as Category–I University by MHRD-UGC] (A State University Established by the Government of Tamil Nadu) KARAIKUDI – 630 003 Directorate of Distance Education B.Sc. (Mathematics) I - Semester 113 14 CALCULUS Author Sushma Sarin, Freelance Author Units (1-14) "The copyright shall be vested with Alagappa University" All rights reserved. No part of this publication which is material protected by this copyright notice may be reproduced or transmitted or utilized or stored in any form or by any means now known or hereinafter invented, electronic, digital or mechanical, including photocopying, scanning, recording or by any information storage or retrieval system, without prior written permission from the Alagappa University, Karaikudi, Tamil Nadu. Information contained in this book has been published by VIKAS® Publishing House Pvt. Ltd. and has been obtained by its Authors from sources believed to be reliable and are correct to the best of their knowledge. However, the Alagappa University, Publisher and its Authors shall in no event be liable for any errors, omissions or damages arising out of use of this information and specifically disclaim any implied warranties or merchantability or fitness for any particular use. Vikas® is the registered trademark of Vikas® Publishing House Pvt. Ltd. VIKAS® PUBLISHING HOUSE PVT. LTD. E-28, Sector-8, Noida - 201301 (UP) Phone: 0120-4078900 • Fax: 0120-4078999 Regd. Office: 7361, Ravindra Mansion, Ram Nagar, New Delhi 110 055 • Website: www.vikaspublishing.com • Email: [email protected] Work Order No. AU/DDE/DE1-238/Preparation and Printing of Course Materials/2018 Dated 30.08.2018 Copies - 500 SYLLABI-BOOK MAPPING TABLE Calculus Syllabi Mapping in Book BLOCK I: DIFFERENTIATION, POLAR CO-ORDINATES AND ASYMPTOTES UNIT-I: Differentiation – Introduction, Parametric differentiation, Unit 1: Differentiation Logarithmic differentiation, differentiation of implicit functions.
    [Show full text]
  • Advanced Placement Calculus Formula and Theorem Review
    Malden Catholic High School Advanced Placement* Calculus Formula and Theorem Review Tom MacWilliam, '09 [email protected] May 3, 2009 *This review is not affiliated with the College Board or AP program. 1 Contents Table of Contents 5 1 Prerequisites for Calculus 6 1.1 Increments . 6 1.2 Slope . 6 1.3 Point-Slope Equation . 6 1.4 Slope-Intercept Equation . 6 1.5 General Linear Equation . 6 1.6 Properties of Logarithms . 6 1.7 Change of Base Formula . 6 2 Limits and Continuity 7 2.1 Properties of Limits . 7 2.2 Sandwich Theorem . 7 3 Derivatives 7 3.1 Derivative . 7 3.2 Derivative at a Point . 7 3.3 Derivative of a Constant Function . 7 3.4 Power Rule . 7 3.5 Constant Multiple Rule . 8 3.6 Sum and Difference Rule . 8 3.7 Product Rule . 8 3.8 Quotient Rule . 8 3.9 Velocity and Acceleration . 8 3.10 Derivatives of Trigonometric Functions . 8 3.11 Chain Rule . 8 3.12 Parametric Derivative . 8 3.13 Derivatives of Inverse Trigonometric Functions . 9 3.14 Function-Cofunction Inverse Identities . 9 3.15 Calculator Conversion Identities . 9 3.16 Derivative of ax .................................. 9 3.17 Derivative of ln x ................................. 9 3.18 Derivative of loga x ................................ 9 4 Applications of Derivatives 10 4.1 Absolute Extreme Values . 10 4.2 Extreme Value Theorem . 10 4.3 Candidates for Local Extrema . 10 4.4 Mean Value Theorem . 10 4.5 Increasing and Decreasing Functions . 10 2 4.6 First Derivative Test for Local Extrema . 10 4.7 Concavity Test .
    [Show full text]