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AP Calculus BC

AP Calculus BC. Chapter 3. 3.1 Derivative of a Function. Definition of the Derivative:. Notation:. If f’ exists we say f has a derivative and is differentiable at x. Can have one-sided derivatives and two-sided derivatives. Alternative forms are discussed in the HW. Graph Examples.

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AP Calculus BC

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  1. AP Calculus BC Chapter 3

  2. 3.1 Derivative of a Function Definition of the Derivative: Notation: If f’ exists we say f has a derivative and is differentiable at x. Can have one-sided derivatives and two-sided derivatives. Alternative forms are discussed in the HW. Graph Examples Examples:

  3. 3.2 Differentiability Where a f’ might fail to exist 1. Corner 2. Cusp 3. Vertical Tangent 4. Discontinuity EXAMPLES: Differentiability implies local linearity. Differentiability implies continuity. Calculator: nderiv(f(x), x, a) = slope at a Graphing: EXAMPLES: on calculator Intermediate Value Theorem for Derivatives

  4. 3.3 Rules for Differentiation u and v are functions of x Derivative of a constant: Power Rule: Constant Multiple Rule: Sum/Diff. Rule: Product Rule: Quotient Rule: The first derivative tells us the slope of the tangent line.

  5. 3.3 Examples Find the first derivative of the following problems. Find the Horizontal Tangents for:

  6. 3.4 Velocity & other Rates of Change Instantaneous Rate of Change is at a specific time. (derivative) Rate of change for the Area of Circle: Evaluate when r = 5, and r =10 Displacement - Position Velocity Acceleration Avg. Velocity - Instantaneous velocity is s’ or v, plug in the value. Speed is absolute value of velocity. A particle is at rest when velocity = 0. Derivatives in Economics – “Marginal” Cost, Revenue

  7. 3.5 Derivatives of Trig. Functions On calculator, find the derivative of y = sin x - sin x cos x - csc x cot x sec x tan x EXAMPLES: Tangent and Normal lines do not change. Calculate everything the same…….

  8. 3.6 Chain Rule Chain Rule is the “Outside – Inside Rule” Example: Examples: Answers: T.A. P.T. P.T.A. Product Rule with Chain Rule

  9. 3.6 cont’d. Slopes of Parametric Curves Example: Find the Tangent and Normal Line T.L. N.L.

  10. 3.7 Implicit Differentiation Now, non-functions…….. First Derivative: Examples: Find the second derivative of: Find T.L. & N.L. at (-1,2) Power rule for Rational Exponents……

  11. 3.8 Derivatives of Inv. Trig. Triangle.. First derivative??? Examples???? For the following problems: u = f(x)

  12. 3.9 Deriv. of Exponential/Logarithmic Long Form Derivative of : For the rules: u = f(x) Examples:(Be careful of #6) Do not forget Product & Quotient Rules…….

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