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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find f'(2), where f(t) = u(t) · v(t), u(2) = ⟨1, 2, −1⟩, u'(2) = ⟨3, 0, 4⟩, and v(t) = ⟨t, t2, t3⟩.

To determine

To find: The value of f(2) .

Explanation

Given:

f(t)=u(t)v(t) , u(2)=1,2,1 , u(2)=3,0,4 , and v(t)=t,t2,t3 .

Formula used:

Write the expression to find the derivative of dot product of the two vectors u(t) and v(t) .

ddt[u(t)v(t)]=u(t)v(t)+u(t)v(t) (1)

The derivative term f(2) is the derivative of the real-valued function f(t)=u(t)v(t) at t=2 .

Write the real-valued function as follows.

f(t)=u(t)v(t)

Differentiate on both sides of the expression with respect to t .

ddt[f(t)]=ddt[u(t)v(t)]

Rewrite the expression as follows.

f(t)=ddt[u(t)v(t)]

Substitute 2 for t in the expression,

f(2)=ddt[u(2)v(2)] (2)

Use the formula in equation (1) and rewrite the expression in equation (2) as follows.

f(2)=u(2)v(2)+u(2)v(2) (3)

Calculation of vector v(2) :

Substitute 2 for t in the expression v(t)=t,t2,t3 ,

v(2)=2,22,23=2,4,8

Calculation of vector v(t) :

The vector v(t) is the derivative of the vector function v(t)=t,t2,t3 .

Differentiate each component of the vector function v(t)=t,t2,t3 to obtain the vector v(t) as follows.

ddt[v(t)]=ddtt,t2,t3v(t)=ddt(t),ddt(t2),ddt(t3)

Use the following formula and compute the expression.

ddttn=ntn1

Compute the expression v(t)=ddt(t),ddt(t2),ddt(t3) as follows

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Sect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.1 P-41ESect-13.1 P-42ESect-13.1 P-43ESect-13.1 P-44ESect-13.1 P-45ESect-13.1 P-46ESect-13.1 P-49ESect-13.1 P-50ESect-13.1 P-53ESect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 P-33ESect-13.2 P-34ESect-13.2 P-35ESect-13.2 P-36ESect-13.2 P-37ESect-13.2 P-38ESect-13.2 P-39ESect-13.2 P-40ESect-13.2 P-41ESect-13.2 P-42ESect-13.2 P-43ESect-13.2 P-44ESect-13.2 P-45ESect-13.2 P-46ESect-13.2 P-47ESect-13.2 P-48ESect-13.2 P-49ESect-13.2 P-50ESect-13.2 P-51ESect-13.2 P-52ESect-13.2 P-53ESect-13.2 P-54ESect-13.2 P-55ESect-13.2 P-56ESect-13.2 P-57ESect-13.2 P-58ESect-13.3 P-1ESect-13.3 P-2ESect-13.3 P-3ESect-13.3 P-4ESect-13.3 P-5ESect-13.3 P-6ESect-13.3 P-7ESect-13.3 P-8ESect-13.3 P-9ESect-13.3 P-10ESect-13.3 P-11ESect-13.3 P-12ESect-13.3 P-13ESect-13.3 P-14ESect-13.3 P-15ESect-13.3 P-16ESect-13.3 P-17ESect-13.3 P-18ESect-13.3 P-19ESect-13.3 P-20ESect-13.3 P-21ESect-13.3 P-22ESect-13.3 P-23ESect-13.3 P-24ESect-13.3 P-25ESect-13.3 P-26ESect-13.3 P-27ESect-13.3 P-28ESect-13.3 P-29ESect-13.3 P-30ESect-13.3 P-31ESect-13.3 P-32ESect-13.3 P-33ESect-13.3 P-38ESect-13.3 P-39ESect-13.3 P-42ESect-13.3 P-43ESect-13.3 P-44ESect-13.3 P-45ESect-13.3 P-46ESect-13.3 P-47ESect-13.3 P-48ESect-13.3 P-49ESect-13.3 P-50ESect-13.3 P-53ESect-13.3 P-55ESect-13.3 P-56ESect-13.3 P-58ESect-13.3 P-59ESect-13.3 P-60ESect-13.3 P-62ESect-13.3 P-63ESect-13.3 P-64ESect-13.3 P-65ESect-13.3 P-66ESect-13.3 P-67ESect-13.4 P-1ESect-13.4 P-3ESect-13.4 P-4ESect-13.4 P-5ESect-13.4 P-6ESect-13.4 P-7ESect-13.4 P-8ESect-13.4 P-9ESect-13.4 P-10ESect-13.4 P-11ESect-13.4 P-12ESect-13.4 P-13ESect-13.4 P-14ESect-13.4 P-15ESect-13.4 P-16ESect-13.4 P-19ESect-13.4 P-20ESect-13.4 P-21ESect-13.4 P-22ESect-13.4 P-23ESect-13.4 P-24ESect-13.4 P-25ESect-13.4 P-26ESect-13.4 P-27ESect-13.4 P-28ESect-13.4 P-29ESect-13.4 P-30ESect-13.4 P-31ESect-13.4 P-32ESect-13.4 P-34ESect-13.4 P-35ESect-13.4 P-36ESect-13.4 P-37ESect-13.4 P-38ESect-13.4 P-39ESect-13.4 P-40ESect-13.4 P-41ESect-13.4 P-42ESect-13.4 P-44ESect-13.4 P-45ESect-13.4 P-46ECh-13 P-1RCCCh-13 P-2RCCCh-13 P-3RCCCh-13 P-4RCCCh-13 P-5RCCCh-13 P-6RCCCh-13 P-7RCCCh-13 P-8RCCCh-13 P-9RCCCh-13 P-1RQCh-13 P-2RQCh-13 P-3RQCh-13 P-4RQCh-13 P-5RQCh-13 P-6RQCh-13 P-7RQCh-13 P-8RQCh-13 P-9RQCh-13 P-10RQCh-13 P-11RQCh-13 P-12RQCh-13 P-13RQCh-13 P-14RQCh-13 P-1RECh-13 P-2RECh-13 P-3RECh-13 P-4RECh-13 P-5RECh-13 P-6RECh-13 P-7RECh-13 P-8RECh-13 P-9RECh-13 P-10RECh-13 P-11RECh-13 P-12RECh-13 P-13RECh-13 P-14RECh-13 P-15RECh-13 P-16RECh-13 P-17RECh-13 P-18RECh-13 P-19RECh-13 P-20RECh-13 P-21RECh-13 P-22RECh-13 P-23RECh-13 P-1PCh-13 P-2PCh-13 P-3PCh-13 P-4PCh-13 P-5PCh-13 P-6PCh-13 P-7PCh-13 P-8PCh-13 P-9P

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