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AP Calculus Chapter 3 Section 2

AP Calculus Chapter 3 Section 2. Differentiability. How f´(a) Might Fail to Exist. A function will not have a derivative at a point P(a, f(a)) where the slopes of the secant lines fail to approach a limit as x  a. How f´(a) Might Fail to Exist.

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AP Calculus Chapter 3 Section 2

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  1. AP CalculusChapter 3 Section 2 Differentiability

  2. How f´(a) Might Fail to Exist • A function will not have a derivative at a point P(a, f(a)) where the slopes of the secant linesfail to approach a limit as x  a.

  3. How f´(a) Might Fail to Exist • Figures 3.11 – 3.14 on page 109 illustrate four different instances where this occurs. • 1) a corner: f(x) = |x| • 2) a cusp: f(x) = x2/3 • 3) a vertical tangent: f(x) = ³√x • 4) a discontinuity: • Removable • Jump • Infinite

  4. How f´(a) Might Fail to Exist • Find where a function is not differentiable • Find all points in the domain of f(x) = |x - 2| +3 where f is not differentiable. Solution: Think graphically.

  5. How f´(a) Might Fail to Exist • Most of the functions we encounter in calculus are differentiable wherever they are defined. • Which means that they will not have corners, cusps, vertical tangent lines, or points of discontinuity within their domains. • Their graphs will be unbroken and smooth, with a well-defined slope at each point.

  6. How f´(a) Might Fail to Exist • Polynomials are differentiable, as are rational functions, trigonometric functions, exponential functions, and logarithmic functions. • Composites of differentiable functions are differentiable, as well as, sums, products, integer powers, and quotients of differentiable functions, where defined.

  7. Differentiability Implies Local Linearity • A good way to think of differentiable functions is that they are locally linear. • That is, a function that is differentiable at a closely resembles its own tangent line very close to a. • Differentiable curves will “straighten out” when we zoom in on them at a point of differentiability. See Exploration 1 on page 110.

  8. Derivatives on a Calculator • Many graphing calculators can approximate derivatives numerically with good accuracy at most points of their domains. • A graphing calculator will use the symmetric difference quotient to approximate the derivative at a.

  9. Derivatives on a Calculator • The book uses NDER(f(x), a) to represent finding the derivative of f at a. • For a TI-83 plus or higher, click on MATH, then scroll down to select nDeriv(expression, variable, value) • See examples 2 and 3 on pages 111 & 112. • Do exploration 2 on page 112. • Alert: the calculator can return a false derivative value at a non-differentiable point

  10. Differentiability Implies Continuity • Theorem: If f has a derivative at x = a, then f is continuous at x = a. • Proof: Show that See page 113 for answer.

  11. Intermediate Value Theorem for Derivatives • Theorem: If a and b are any 2 points in an interval on which f is differentiable, then f´ takes on every value between f´(a) and f´(b). • The question of when a function is a derivative of some function is one of the central questions in all of calculus. The answer found by Newton and Leibniz would revolutionize mathematics.

  12. Homework • Pages 114 – 115, Exercises 3 – 39 (mult. of 3)

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