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Let r(t)=〈t2,1−t,4t〉. Calculate the derivative of r(t)⋅a(t) at t=2

Assuming that a(2)=〈7,−3,7〉 and a′(2)=〈3,2,4〉

ddtr(t)⋅a(t)|t=2=______

Respuesta :

lublana

Answer:

101

Step-by-step explanation:

We are given that

r(t)=[tex]<t^2,1-t,4t>[/tex]

We have to find the derivative of r(t).a(t) at t=2

a(2)=<7,-3,7> and a'(2)=<3,2,4>

We know that

[tex]\frac{d(uv)}{dx}=u'v+v'u[/tex]

Using the formula

[tex]\frac{d(r(t)\cdot at(t))}{dt}=r'(t)\cdot a(t)+r(t)\cdot a'(t)[/tex]

[tex]\frac{d(r(t)\cdot at(t))}{dt}=<2t,-1,4>\cdot a(t)+<t^2,1-t,4t>\cdot a'(t)[/tex]

Substitute t=2

[tex]\frac{d(r(t)\cdot at(t))}{dt}_|t=2=<4,-1,4>\cdot a(2)+<4,-1,8>\cdot a'(2)[/tex]

[tex]\frac{d(r(t)\cdot at(t))}{dt}_|t=2=<4,-1,4>\cdot <7,-3,7>+<4,-1,8>\cdot <3,2,4>[/tex]

[tex]\frac{d(r(t)\cdot at(t))}{dt}_|t=2=28+3+28+12-2+32=101[/tex]

Cricetus

The derivation of the equation will be "101".

Differentiation:

Given expression is:

r(t) = 〈t², 1 - t, 4t〉

Let,

a(2) = <7, -3, 7>

a'(2) = <3, 2, 4>

As we know,

→ [tex]\frac{d(uv)}{dx}[/tex] = u'v + v'u

By using the formula, the derivation will be:

→ [tex]\frac{d(r(t).at(t))}{dt}[/tex] = r'(t).a(t) + r(t).a'(t)

                  = <2t, -1, 4>.a(t) + <t², 1 - t, 4t>.a'(t)

By substituting "t = 2", we get

                  =  <4, -1, 4>.a(2) + <4, -1, 8>. a'(2)

                  = <4, -1, 4>.<7, -3, 7> + <4, -1, 8>.<3, 2, 4>

                  = 28 + 3 + 28 + 12 - 2 + 32

                  = 101

Thus the response above is appropriate.

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