# 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 #?

The instantaneous velocity of

By signing up, you agree to our Terms of Service and Privacy Policy

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) ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- What is the equation of the line normal to #f(x)=(5-x)^2 # at #x=2#?
- How do you find the slope of a tangent line to the graph of the function #F(t)= 3- (8/3t)# at point: (2/4, -7/3)?
- Does the limit #lim_(x->3) (f(x)-f(3))/(x-3)# always exist?
- How do you find the equation of a line tangent to #y=-5x^2+4x-3# at x=1?
- How do you find the average value of the function #f(x) = 12/(sqrt(2x-1))# on the interval of [1,5]?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7