 | Term | Definition |
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Chain Rule: One Independent Variable |
dw/dt = (∂w/∂x )(dx/dt) + (∂w/∂y )(dy/dt) where w=f(x,y), x = g(t), y= h(t) |
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Chain Rule: Two Independent Variables |
∂w/∂s = (∂w/∂x )(∂x/∂s) + (∂w/∂y )(∂y/∂s) and ∂w/∂t = (∂w/∂x )(∂x/∂t) + (∂w/∂y )(∂y/∂t) where w=f(x,y), x = g(s,t), y= h(s,t) |
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Chain Rule: Implicit Differentiation |
∂z/∂x = - (∂F(x,y,z)/∂x)/(∂F(x,y,z)/∂z) and ∂z/∂y = - (∂F(x,y,z)/∂y)/(∂F(x,y,z)/∂z) |
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Directional Derivative |
cosθ ∂f/∂x + sinθ ∂f/∂y = ∇f(x,y) ⋅ <u> where u is a unit vector |
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direction of maximum increase |
maximum value of ||∇f(x,y)|| |
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direction of minimum increase |
minimum value of -||∇f(x,y)|| |
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gradient normal to level curves |
∇f(x₀,y₀) is normal to the level curve through (x₀,y₀) |
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equation of a tangent plane |
∇F(x₀,y₀,z₀) ⋅<x-x₀, y-y₀, z-z₀> |
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angle of inclination of a plane |
cos θ = |<n> ⋅<k>|/(||<n>|| * ||<k||) = |<n>⋅<k>|/||<n>|| |
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second partials test |
d = ∂²f(a,b)/∂x²∂²f(a,b)/∂y² - [∂²f(a,b)/∂x∂y]² . if d>0 and ∂²f(a,b)/∂x²>0 then relative min. if d>0 and ∂²f(a,b)/∂x²>0 then relative max. if d<0 then saddle point. d = 0 test inconclusive |
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least squares regression line |
a =( n∑xy - ∑x∑y)/(n∑x² -(∑x)²) and b = (1/n) (∑y-a∑x) for f(x) = ax + b |
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sum of the squared errors |
S = ∑[f(x) - y] |
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Langrange's theorem |
Let f and g have continuous first partial derivatives such that f has an exteremum at a point (x₀,y₀) on the smooth constraint curve g(x,y) = c. If ∇g(x₀,y₀) ≠0 then there is a real number such that ∇f(x₀,y₀) = λ∇g(x₀,y₀). |
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Method of Langrange Multipliers |
1. Simultaneously solve the equations ∇f(x,y) = λ∇g(x,y) and g(x,y) = c by solving the following system of equations. ∂f/∂x = λ∂gf/∂x , ∂f/∂y = λ∂gf/∂y, g(x,y) = c 2. Evaluate each solution point obtained in the first step. The largest value yields the maximum of f subject to the constraint g(x,y) = c, and the smallest value yields the minimum of s subject to the constraint g(x,y) = c |
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