What is the instantaneous velocity of an object moving in accordance to # f(t)= (sin(t+pi),cos(3t-pi/4)) # at # t=(pi)/3 #?

Answer 1

The instantaneous velocity of #f(t) => for t=pi/3# is:
#f(t) = (-sqrt(3)/2, -sqrt(2)/2)#

Given: #f(t)=(sin(t+pi),cos(3t-pi/4))#
Required: The instantaneous velocity at #t=(pi)/3#
Solution Strategy: The instantaneous velocity at any time is #f(t)# itself so you need Evaluate the #f(t)=f(t)|_(t=pi/3)=f(pi/3)#
#color(green)("Evaluate at at " t=(pi)/3)# #f(t)=(sin(t+pi),cos(3t-pi/4))=(-sin(t),cos(3t-pi/4))# #f(t)=(-sin(pi/3),cos(3pi/3-pi/4))# #f'(t) = (-sqrt(3)/2, -sqrt(2)/2)#
#color(red)(Remark)#: This is deceptively easy problem. Most students are easily tricked to think that #f(t)# is not the instantaneous velocity at any time. But #f(t)# is indeed the instantaneous velocity at anytime. It is asking what it is at a given point.
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Answer 2

To find the instantaneous velocity of an object moving according to the function ( f(t) = (\sin(t+\pi), \cos(3t-\frac{\pi}{4})) ) at ( t = \frac{\pi}{3} ), differentiate the function with respect to ( t ) to get the velocity function. Then, evaluate this velocity function at ( t = \frac{\pi}{3} ) to find the instantaneous velocity.

The velocity function, ( v(t) ), is the derivative of the position function ( f(t) ): [ v(t) = \left( \frac{d}{dt} \sin(t+\pi), \frac{d}{dt} \cos(3t-\frac{\pi}{4}) \right) ]

[ v(t) = (\cos(t+\pi), -3\sin(3t-\frac{\pi}{4})) ]

Now, evaluate ( v(t) ) at ( t = \frac{\pi}{3} ): [ v\left(\frac{\pi}{3}\right) = \left( \cos\left(\frac{\pi}{3}+\pi\right), -3\sin\left(3\left(\frac{\pi}{3}\right)-\frac{\pi}{4}\right) \right) ]

[ v\left(\frac{\pi}{3}\right) = \left( \cos\left(\frac{4\pi}{3}\right), -3\sin\left(\pi-\frac{\pi}{4}\right) \right) ]

[ v\left(\frac{\pi}{3}\right) = \left( \cos\left(\frac{4\pi}{3}\right), -3\sin\left(\frac{3\pi}{4}\right) \right) ]

[ v\left(\frac{\pi}{3}\right) = \left( -\frac{1}{2}, -3\frac{\sqrt{2}}{2} \right) ]

So, the instantaneous velocity of the object at ( t = \frac{\pi}{3} ) is ( \left( -\frac{1}{2}, -\frac{3\sqrt{2}}{2} \right) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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