Lesson 11: Implicit Differentiation (handout)
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Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
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Lesson 11: Implicit Differentiation (handout)
- 1. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Sec on 1.3 Notes The Limit of a Func on V63.0121.001: Calculus I Professor Ma hew Leingang New York University January 31, 2011 Announcements First wri en HW due Wednesday February 2 . . Get-to-know-you survey and photo deadline is February 11 . Announcements Notes First wri en HW due Wednesday February 2 Get-to-know-you survey and photo deadline is February 11 . . Guidelines for written homework Notes Papers should be neat and legible. (Use scratch paper.) Label with name, lecture number (001), recita on number, date, assignment number, book sec ons. Explain your work and your reasoning in your own words. Use complete English sentences. . . . 1 .
- 2. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Rubric Notes Points Descrip on of Work 3 Work is completely accurate and essen ally perfect. Work is thoroughly developed, neat, and easy to read. Complete sentences are used. 2 Work is good, but incompletely developed, hard to read, unexplained, or jumbled. Answers which are not explained, even if correct, will generally receive 2 points. Work contains “right idea” but is flawed. 1 Work is sketchy. There is some correct work, but most of work is incorrect. 0 Work minimal or non-existent. Solu on is completely incorrect. . . Written homework: Don’t Notes . . Written homework: Do Notes . . . 2 .
- 3. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Written homework: Do Notes Written Explanations . . Written homework: Do Notes Graphs . . Objectives Notes Understand and state the informal defini on of a limit. Observe limits on a graph. Guess limits by algebraic manipula on. Guess limits by numerical informa on. . . . 3 .
- 4. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Notes Limit . . . Zeno’s Paradox Notes That which is in locomo on must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10) . . Outline Notes Heuris cs Errors and tolerances Examples Precise Defini on of a Limit . . . 4 .
- 5. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Heuristic Definition of a Limit Notes Defini on We write lim f(x) = L x→a and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. . . Outline Notes Heuris cs Errors and tolerances Examples Precise Defini on of a Limit . . The error-tolerance game Notes A game between two players (Dana and Emerson) to decide if a limit lim f(x) exists. x→a Step 1 Dana proposes L to be the limit. Step 2 Emerson challenges with an “error” level around L. Step 3 Dana chooses a “tolerance” level around a so that points x within that tolerance of a (not coun ng a itself) are taken to values y within the error level of L. If Dana cannot, Emerson wins and the limit cannot be L. Step 4 If Dana’s move is a good one, Emerson can challenge again or give up. If Emerson gives up, Dana wins and the limit is L. . . . 5 .
- 6. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 The error-tolerance game Notes L . a To be legit, the part of the graph inside the blue (ver cal) strip must also be inside the green (horizontal) strip. Even if Emerson shrinks the error, Dana can s ll move. . . Outline Notes Heuris cs Errors and tolerances Examples Precise Defini on of a Limit . . Playing the E-T Game Notes Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on Dana claims the limit is zero. If Emerson challenges with an error level of 0.01, Dana needs to guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round. . . . 6 .
- 7. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Playing the E-T Game Notes Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on If Emerson re-challenges with an error level of 0.0001 = 10−4 , what should Dana’s tolerance be? A tolerance of 0.01 works because |x| < 10−2 =⇒ x2 < 10−4 . . . Playing the E-T Game Notes Example Describe how the the Error-Tolerance game would be played to determine lim x2 . x→0 Solu on Dana has a shortcut: By se ng tolerance equal to the square root of the error, Dana can win every round. Once Emerson realizes this, Emerson must give up. . . Graphical version of E-T game Notes with x2 y No ma er how small an error Emerson picks, Dana can find a fi ng tolerance band. . x . . . 7 .
- 8. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 A piecewise-defined function Notes Example |x| Find lim if it exists. x→0 x Solu on The func on can also be wri en as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? . . The E-T game with a piecewise Notes function |x| Find lim if it exists. x→0 x y 1 . x −1 . . One-sided limits Notes Defini on We write lim f(x) = L x→a+− and say “the limit of f(x), as x approaches a from the right, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a and greater than a. . . . 8 .
- 9. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 One-sided limits Notes Defini on We write lim f(x) = L x→a+− and say “the limit of f(x), as x approaches a from the le , equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a and less than a. . . Another Example Notes Example 1 Find lim+ if it exists. x→0 x Solu on . . The error-tolerance game with 1/x Notes y 1 Find lim+ if it exists. L? x→0 x . x 0 . . . 9 .
- 10. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Weird, wild stuff Notes Example (π ) Find lim sin if it exists. x→0 x Solu on . . Function values Notes x π/x sin(π/x) 1 π π/2 1/2 2π 1/k kπ 2 π/2 2/5 5π/2 π . 0 2/9 9π/2 2/13 13π/2 2/3 3π/2 2/7 7π/2 3π/2 2/11 11π/2 . . What could go wrong? Notes Summary of Limit Pathologies How could a func on fail to have a limit? Some possibili es: le - and right- hand limits exist but are not equal The func on is unbounded near a Oscilla on with increasingly high frequency near a . . . 10 .
- 11. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 Meet the Mathematician Notes Augustin Louis Cauchy French, 1789–1857 Royalist and Catholic made contribu ons in geometry, calculus, complex analysis, number theory created the defini on of limit we use today but didn’t understand it . . Outline Notes Heuris cs Errors and tolerances Examples Precise Defini on of a Limit . . Precise Definition of a Limit Notes Let f be a func on defined on an some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write lim f(x) = L, x→a if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. . . . 11 .
- 12. . V63.0121.001: Calculus I . Sec on 1.3:. Limits January 31, 2011 The error-tolerance game = ε, δ Notes L+ε L L−ε . a−δ a a+δ . . Summary Notes Many perspectives on limits Graphical: L is the value the func on “wants to go to” near a Heuris cal: f(x) can be made arbitrarily close to L by taking x sufficiently close to a. Informal: the error/tolerance game Precise: if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ, then |f(x) − L| < ε. Algebraic: next me . . Notes . . . 12 .