2014 Implicit Differentiation Calculus BC Implicit Differentiation Equation

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2014 Implicit Differentiation Calculus BC

2014 Implicit Differentiation Calculus BC

 Implicit Differentiation Equation for a line: Explicit Form <One variable given explicitly in

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.

Implicit Differentiation Equation of circle: To work explicitly; must work two equations Implicit Differentiation

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)

Implicit Differentiation Find the derivative with respect to x < Assuming - y is

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.

Implicit Differentiation (D) Product Rule (E) Chain Rule

Implicit Differentiation (D) Product Rule (E) Chain Rule

Implicit Differentiation To find implicitly. EX: Diff Both Sides of equation with respect to

Implicit Differentiation To find implicitly. EX: Diff Both Sides of equation with respect to x Solve for

EX 1: (a) Find the derivative at the point ( 5, 3 ) ,

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.

Ex 2: < Folium of Descartes >

Ex 2: < Folium of Descartes >

Why Implicit? Explicit Form: < Folium of Descartes >

Why Implicit? Explicit Form: < Folium of Descartes >

Ex 2 Graph: < Folium of Descartes > Parametric Form: Plot the Folium of

Ex 2 Graph: < Folium of Descartes > Parametric Form: Plot the Folium of Descartes on your graphing calculator and determine the portion of the folium generated when (a) t < -1 ; (b) -1 < t 0 ; (c) t > 0

2 nd Derivatives EX: Our friendly circle. Find the 2 nd Derivative. NOTICE: The

2 nd Derivatives EX: Our friendly circle. Find the 2 nd Derivative. 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.

2 nd Derivatives EX: Find the 2 nd Derivative.

2 nd Derivatives EX: Find the 2 nd Derivative.

Higher Derivatives EX: Find the Third Derivative.

Higher Derivatives EX: Find the Third Derivative.

Last update • 10/19/10 Øp. 162 11 – 29 odd

Last update • 10/19/10 Øp. 162 11 – 29 odd