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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Using Two MethodsIn Exercises 37 and 38, find (a) d d t [ r ( t ) u ( t ) ] and (b) d d t [ r ( t ) × u ( t ) ] in two different ways.

(i) Find the product first, then differentiate.

(ii) Apply the properties of Theorem 12.2.

r ( t ) = t i + 2 t 2 j + t 3 k , u ( t ) = t 4 k

(a)

To determine

To calculate: The expression ddt[r(t)u(t)] for the functions r(t)=ti+2t2j+t3k and u(t)=t4k.

Explanation

Given:

The functions r(t)=ti+2t2j+t3k and u(t)=t4k.

Formula used:

Let r and u be differentiable functions vector-valued functions of t, let w be a differentiable real valued functions of t, and let c be a scalar, then

1. ddt[cr(t)]=cr'(t)

2. ddt[r(t)×u(t)]=r(t)×u'(t)+r'(t)×u(t)

3. ddt[r(t)±u(t)]=r'(t)±u'(t)

4. ddt[r(t)u(t)]=r(t)u'(t)+r'(t)u(t)

5. ddt[r(t)u(t)]=r(t)u'(t)+r'(t)u(t)

If v1=x1i+y1j+z1k and v2=x2i+y2j+z2k are two vectors.

Then, dot product is given as:

v1v2=(x1i+y1j+z1k)(x2i+y2j+z2k)=x1x2+y1y2+z1z2

Calculation:

(i)

Consider the following functions:

r(t)=ti+2t2j+t3k and u(t)=t4k

Evaluate the dot product for r(t) and  u(t) and obtain the following result:

r(t)u(t)=(ti+2t2j+t3k)(t4k)=t(0)+2t2(0)+t7=t7

Differentiate the dot product obtained with respect to t as follows:

ddt(r(t)u(t))=ddt(t7)=7t6

Hence, the value of ddt[r(t)u(t)] is 7t6

(b)

To determine

To calculate: The expression ddt[r(t)×u(t)] for the functions r(t)=ti+2t2j+t3k and u(t)=t4k.

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Chapter 12 Solutions

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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