Final_exam_practice

# Final_exam_practice - MATH 2401 Sections F4& F5...

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Unformatted text preview: MATH 2401 December 10, 2009 Sections F4 & F5 11:30-14:20 PRACTICE FINAL EXAM Name: gtID # : Note: There are eight problems in this exam, each worth 20 points. Write out your solutions neatly and explain your work. Calculators are not allowed. Problem number Points 1 2 3 4 5 6 7 8 Total: Basic formulas: Unit tangent and principal normal: T ( t ) = r ( t ) k r ( t ) k , N ( t ) = T ( t ) k T ( t ) k Arc length: L ( C ) = Z b a k r ( t ) k dt Curvature: κ = d T ds = k d T /dt k ds/dt = k v × a k k v k 3 κ = | x ( t ) y 00 ( t )- y ( t ) x 00 ( t ) | [[ x ( t )] 2 + [ y ( t )] 2 )] 3 / 2 Components of acceleration: a = a T T + a N N a T = d 2 s dt 2 = T · a = v · a k v k a N = κ ds dt 2 k T × a k = k v × a k k v k Gradient: ∇ f ( x ) = ∂f ∂x ( x ) i + ∂f ∂y ( x ) j + ∂f ∂z ( x ) k Directional derivative: f u ( x ) = ∇ f ( x ) · u Chain rule: d dt [ f ( r ( t ))] = ∇ f ( r ( t )) · r ( t ) The second-partials test: A = ∂ 2 f ∂x 2 , B = ∂ 2 f ∂y∂x , C = ∂...
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Final_exam_practice - MATH 2401 Sections F4& F5...

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