   Chapter 12.2, Problem 27E

Chapter
Section
Textbook Problem

# Higher-Order DifferentiationIn Exercises 23–26, find (a) r ′ ( t ) , (b) r ″ ( t ) , and (c) r ′ ( t ) ⋅ r ″ ( t ) , and (d) r ′ ( t ) × r ″ ( t ) . r ( t ) = 〈 cos t + t sin t , sin t − t cos t , t 〉

(a)

To determine

To calculate: The derivative r'(t) for the function r(t)=cost+tsint,sinttcost,t.

Explanation

Given:

The provided function is r(t)=cost+tsint,sinttcost,t.

Formula used:

The differentiation of the function tn with respect to t is:

ntn1.

Calculation:

Consider the function r(t)=cost+tsint,sinttcost,t.

Differentiate the function with respect to t,

r(t)=ddt(r(t))=

(b)

To determine

To calculate: The second derivative r''(t) for the function r(t)=cost+tsint,sinttcost,t

(c)

To determine

To calculate: The dot product r'(t)r''(t) for the function r(t)=cost+tsint,sinttcost,t.

(d)

To determine

To calculate: The cross product r'(t)×r''(t) for the function r(t)=cost+tsint,sinttcost,t.

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