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Equations of Curves
• Explicit: y = f(x) Set of ordered pairs (x, y) = (x, f(x)) 2nd coordinate is given in terms of an expression
involving the 1st coordinate.
• Parametric: x = f(t), y = g(t) Set of ordered pairs (x, y) = (f(t), g(t)) 1st coordinate is given in terms of an expression
of the parameter, t 2nd coordinate is given in terms of an expression
of the parameter, t
• Implicit: f(x,y) = 0 Set of ordered pairs (x,y) such that f(x,y) = 0 f(x,y) is an expression involving x and/or y
Equations of Curves - Example
• A Circle of Radius 2
Explicit:
Parametric:
Implicit:
• Line with slope 2/3 containing the point (0, 5)
Explicit:
Parametric:
Implicit:
24y x
2cos , 2sin , 0 2x t y t t
2 2 4x y
23 5y x
3 , 2 5, x t y t t
2 3 15x y
Explicit Parametric
• If y = f(x) … Let x = g(t) … any expression of t Substitute to get y = f(x) = f(g(t)) Determine the domain for t.
• Example …
2 5y x
AKA: Parameterization
Parametric Explicit
• If x = f(t), y = g(t) … Solve x = f(t) for t, giving t = h(x) Substitute to get y = g(t) = g(h(x)) Determine the domain for x.
• Example …
2 3, 1x t y t
AKA: Eliminating the Parameter
Explicit Implicit
• If y = f(x) … Move everything to one side of the equation. (optional) Simplify. I.E. f(x) – y = 0 or y – f(x) = 0
• Example …
2 5y x
Implicit Explicit
• If f(x, y) = 0 … Solve for y ... if possible!
• Examples …
2 2 0xy x
2 2 0y x x y
cos sin 0x y y x
Implicit Differentiation
• Finding dy/dx for an implicitly defined function without explicitly solving for y. Note: The result may (will) be in terms of x & y
1. Differentiate both sides of the equation in terms of x, treating y as a function of x• i.e. use the chain rule and
2. Algebraically solve for dy/dx
d dyy
dx dx
Tangents & Normals
• Tangent Line The limit of secant lines. Slope = dy/dx
• Normal Line The line perpendicular to the tangent. Slope = –1/(dy/dx)
• Example … find the tangent and normal to the curve y2 – 2x – 4y – 1 = 0 at the point (–2, 1)
Second Derivatives Implicitly
• Find the first derivative implicitly.
• Differentiate the first derivative implicitly. The answer will be in terms of dy/dx. Substitute the 1st derivative into the 2nd derivative
to get the result in terms of x and y only.
• Higher Order Derivatives … continue likewise!
• Example: Find the 1st & 2nd derivatives of …2 2 25x y