   Chapter 12.2, Problem 63E

Chapter
Section
Textbook Problem

# Differentiation State the definition of the derivative of a vector-valued function. Describe how to find the derivative of a vector-valued function and give its geometric interpretation.

To determine

The definition of the derivative of vector-valued function and explain how to find the derivative of the vector-valued function and give its geometric interpretation.

Explanation

Let r(t)=f(t)i+g(t)j be the vector-valued function.

The derivative for the function r(t) is determined by the limit,

r(t)=limΔt0r(t+Δt)r(t)Δt

For all t wherever the above limit exists.

The above vector-valued function r(t) is differentiable at t=t0 if r(t) is defined at t0. A vector-valued function r(t)=f(t)i+g(t)j+h(t)k is differentiable if the components f, g and h are differentiable functions of t and then, the derivative r(t) of the function r(t) is determined by,

r(t)=f(t)i+g(t)j+h(t)k

If r(t) is differentiable at t=t0 and r(t)0, then r(t) is a vector tangent to the curve of r(t) at the point t=t0 and points in the direction of increasing values of t

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