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Activity: Teacher-Directed Instruction

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Activity: Teacher-Directed Instruction. 2013 Implicit Differentiation. Calculus AB. Objective . C: The swbat differentiate implicitly equations in more than one variable. L: the swbat explain to others how to find derivatives of multiple types of problems verbally and demonstratively.

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  1. Activity: Teacher-Directed Instruction

  2. 2013 Implicit Differentiation Calculus AB

  3. Objective • C: The swbat differentiate implicitly equations in more than one variable. • L: the swbat explain to others how to find derivatives of multiple types of problems verbally and demonstratively

  4. Implicit Differentiation Equation for a line: Explicit Form <One variable given explicitly in terms of the other> Implicit Form <Function implied by the equation>   Differentiate the Explicit < Explicit: , y is function of x > Differentiation taking place with respect to x. The derivative is explicit also.

  5. Implicit Differentiation Equation of circle: To work explicitly; must work two equations Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly. ( Usually non-functions) Don’t want to solve for y

  6. Implicit Differentiation Find the derivative with respect to x < Assuming - y is a differentiable function of x > Chain Rule Pretend y is some function like so becomes (A) (B) (C) Note: Use the Leibniz form. Leads to Parametric and Related Rates. = =

  7. Implicit Differentiation Find the derivative with respect to x < Assuming - y is a differentiable function of x >

  8. Implicit Differentiation (D) Product Rule

  9. Implicit Differentiation  (E) Chain Rule Product inside a chain

  10. Implicit Differentiation To find implicitly. EX: Diff Both Sides of equation with respect to x Solve for Need both x and y to find the slope.

  11. EX 1: (a) Find the derivative at the point ( 5, 3 ) , at ( -1,-3 ) (b) Find where the curve has a horizontal tangent.  (c) Find where the curve has vertical tangents.

  12. EX 1: (b) Find where the curve has a horizontal tangent. Horizontal tangent has a 0 slope

  13. EX 1:  (c) Find where the curve has vertical tangents. Vertical tangent has an undefined slope

  14. Ex 2: < Folium of Descartes >

  15. Why Implicit? Explicit Form: < Folium of Descartes >

  16. EX: Our friendly circle. Find the 2nd Derivative. 2nd Derivatives NOTICE:The second derivative is in terms of x , y , AND dy /dx. The final step will be to substitute back the value of dy / dx into the second derivative.

  17. EX: Find the 2nd Derivative. 2nd Derivatives

  18. EX: Find the Third Derivative. Higher Derivatives

  19. Last update • 10/19/10 • p. 162 1 – 29 odd