An object moves along the x-axis so that its position at any time t≥0 is given by x(t)=cos(2t2−t+5) . Find the velocity of the object as a function of t.
To find the velocity of the object, we need to take the derivative of the position function with respect to time.
We have x(t) = cos(2t^2 - t + 5)
Taking the derivative, we use the chain rule:
x'(t) = -sin(2t^2 - t + 5) * (4t - 1)
Therefore, the velocity of the object as a function of t is given by:
v(t) = -sin(2t^2 - t + 5) * (4t - 1)
To find the velocity of the object, we need to take the derivative of the position function with respect to time.
Given that the position function is x(t) = cos(2t^2 − t + 5), we can find the derivative by applying the chain rule of differentiation.
Let's go step by step:
1. Start by differentiating the outer function (cosine):
The derivative of cos(u) is −sin(u).
So, the derivative of x(t) will have the outer function −sin(2t^2 − t + 5).
2. Now, differentiate the inner function (2t^2 − t + 5) with respect to t.
The derivative of 2t^2 is 4t.
The derivative of −t is −1.
The derivative of 5 is 0.
3. Combine the derivatives obtained in steps 1 and 2:
The derivative of the inner function 2t^2 − t + 5 with respect to t is 4t - 1.
4. Multiply the derivative of the outer function (−sin(2t^2 − t + 5)) with the derivative of the inner function (4t - 1):
The velocity function, v(t), is given by v(t) = (−sin(2t^2 − t + 5))(4t - 1).
So, the velocity of the object as a function of time is v(t) = (−sin(2t^2 − t + 5))(4t - 1).
To find the velocity of the object, we need to take the derivative of the position function x(t) with respect to time (t).
Using the chain rule, we can find the derivative:
dx(t)/dt = d/dt[cos(2t^2 - t + 5)]
Applying the chain rule, we differentiate the function inside the cosine function:
dx(t)/dt = -sin(2t^2 - t + 5) * d/dt[2t^2 - t + 5]
To differentiate 2t^2 - t + 5, we apply the power rule:
d/dt[2t^2 - t + 5] = 4t - 1
Now we substitute this result back into the derivative formula:
dx(t)/dt = -sin(2t^2 - t + 5) * (4t - 1)
Therefore, the velocity of the object as a function of t is:
v(t) = dx(t)/dt = -sin(2t^2 - t + 5) * (4t - 1)