Engineering Calculus Notes 178

Engineering Calculus Notes 178 - Linear Combinations for...

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166 CHAPTER 2. CURVES Note the distinction between velocity , which has a direction (and hence is a vector) and speed , which has no direction (and is a scalar). For example, the point moving along the helix −→ p ( t ) = (cos 2 πt, sin 2 πt,t ) has velocity −→ v ( t ) = ˙ −→ p ( t ) = ( 2 π sin 2 πt, 2 π cos 2 πt, 1) speed d s dt = r 4 π 2 + 1 and acceleration −→ a ( t ) = ˙ −→ v ( t ) = ( 4 π 2 cos 2 πt, 4 π 2 sin 2 πt, 0) . The relation of derivatives to vector algebra is analogous to the situation for real-valued functions. Theorem 2.3.15. Suppose the vector-valued functions −→ f , −→ g : I R 3 are diFerentiable on I . Then the following are also diFerentiable:
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Unformatted text preview: Linear Combinations: for any real constants α,β ∈ R , the function α −→ f ( t ) + β −→ g ( t ) is diFerentiable on I , and d dt b α −→ f ( t ) + β −→ g ( t ) B = α V f ′ ( t ) + βVg ′ ( t ) . Products: 11 • The product with any diFerentiable real-valued function α ( t ) on I is diFerentiable on I : d dt b α ( t ) −→ f ( t ) B = α ′ ( t ) −→ f ( t ) + α ( t ) V f ′ ( t ) . 11 These are the product rules or Leibniz formulas for vector-valued functions of one variable....
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