   Chapter 13.2, Problem 49E

Chapter
Section
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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