Fill in the blanks to complete each of the following theoremstatements:The derivative of a vector function r(t) is given by r'(t) =limh→0__________________.The Chain Rule for Vector-Valued Functions: Let t = f (τ) bea differentiable real-valued function of τ, and let r(t) bea differentiable vector function with either two or threecomponents such that f (τ ) is in the domain of r for everyvalue of τ on some interval I. Then dr/dτ =_______________ .Let C be the graph of a twice-differentiable vector functionr(t) defined on an interval I and with unit tangent vector T(t). Then the curvature κ of C at a point on the curve is given by κ = ||__________|| / ||__________|| and κ = ||_____x______||/||_________||3Let y = f (x) be a twice-differentiable function. Then thecurvature of the graph of f is given by κ =| | / ( )3/2Let C be the graph of a vector function r(t) = ⟨x(t), y(t)⟩ inthe xy-plane, where x(t) and y(t) are twice-differentiablefunctions of t such that x'(t) and y'(t) are not simultaneouslyzero. Then the curvature κ of C at a point on the curve is given by κ =| | / ( )3/2.

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Fill in the blanks to complete each of the following theorem
statements:

The derivative of a vector function r(t) is given by r'(t) =
lim_{h→0__________________}.
The Chain Rule for Vector-Valued Functions: Let t = f (τ) be
a differentiable real-valued function of τ, and let r(t) be
a differentiable vector function with either two or three
components such that f (τ ) is in the domain of r for every
value of τ on some interval I. Then dr/dτ =_______________ .
Let C be the graph of a twice-differentiable vector function
r(t) defined on an interval I and with unit tangent vector T(t). Then the curvature κ of C at a point on the curve is given by κ = ||__________|| / ||__________|| and κ = ||_____x______||/||_________||³
Let y = f (x) be a twice-differentiable function. Then the
curvature of the graph of f is given by κ =| | / ( )^3/2
Let C be the graph of a vector function r(t) = ⟨x(t), y(t)⟩ in
the xy-plane, where x(t) and y(t) are twice-differentiable
functions of t such that x'(t) and y'(t) are not simultaneously
zero. Then the curvature κ of C at a point on the curve is given by κ =| | / ( )^3/2.

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