14.2%20Prel%201-6%20Ex%201-22

14.2%20Prel%201-6%20Ex%201-22 - 466 C H A P T E R 14 C A L...

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466 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS (ET CHAPTER 13) Let us look, instead, at ( PF 1 + 2 ) 2 , and show that this is equal to K 2 . Since everything is positive, this will imply that 1 + 2 = K , as desired. ( 1 + 2 ) 2 = 2 r 2 + 4 a 2 r 2 + 2 b 2 r 2 + 2 p r 4 + 2 b 2 r 4 + b 4 r 4 = 2 r 2 + 4 a 2 r 2 + 2 b 2 r 2 + 2 ( 1 + b 2 ) r 2 = 4 r 2 ( 1 + a 2 + b 2 ) = K 2 14.2 Calculus of Vector-Valued Functions (ET Section 13.2) Preliminary Questions 1. State the three forms of the Product Rule for vector-valued functions. SOLUTION The Product Rule for scalar multiple f ( t ) of a vector-valued function r ( t ) states that: d dt f ( t ) r ( t ) = f ( t ) r 0 ( t ) + f 0 ( t ) r ( t ) The Product Rule for dot products states that: d r 1 ( t ) · r 2 ( t ) = r 1 ( t ) · r 0 2 ( t ) + r 0 1 ( t ) · r 2 ( t ) Finally, the Product Rule for cross product is d r 1 ( t ) × r 2 ( t ) = r 1 ( t ) × r 0 2 ( t ) + r 0 1 ( t ) × r 2 ( t ). In Questions 2–6, indicate whether true or false and if false, provide a correct statement. 2. The derivative of a vector-valued function is de±ned as the limit of the difference quotient, just as in the scalar-valued case. The statement is true. The derivative of a vector-valued function r ( t ) is de±ned a limit of the difference quotient: r 0 ( t ) = lim t 0 r ( t + h ) r ( t ) h in the same way as in the scalar-valued case. 3. There are two Chain Rules for vector-valued functions, one for the composite of two vector-valued functions and one for the composite of a vector-valued and scalar-valued function. This statement is false. A vector-valued function r ( t ) is a function whose domain is a set of real numbers and whose range consists of position vectors. Therefore, if r 1 ( t ) and r 2 ( t ) are vector-valued functions, the composition ( r 1 · r 2 )( t ) = r 1 ( r 2 ( t )) ” has no meaning since r 2 ( t ) is a vector and not a real number. However, for a scalar-valued function f ( t ) , the composition r ( f ( t )) has a meaning, and there is a Chain Rule for differentiability of this vector-valued function. 4. The terms “velocity vector” and “tangent vector” for a path r ( t ) mean one and the same thing. This statement is true. 5. The derivative of a vector-valued function is the slope of the tangent line, just as in the scalar case. The statement is false. The derivative of a vector-valued function is again a vector-valued function, hence it cannot be the slope of the tangent line (which is a scalar). However, the derivative, r 0 ( t 0 ) is the direction vector of the tangent line to the curve traced by r ( t ) ,at r ( t 0 ) .
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14.2%20Prel%201-6%20Ex%201-22 - 466 C H A P T E R 14 C A L...

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