If a rock is thrown upward on the planet Mars with a velocity of 14 m/s, its height in meters t seconds later is given by y = 14t − 1.86t^2.
Estimate the instantaneous velocity when t = 1.
10.28
To estimate the instantaneous velocity when t = 1, we need to find the derivative of the height equation with respect to time (t).
The derivative of y = 14t - 1.86t^2 can be found by applying the power rule for differentiation.
The power rule states that if we have a function of the form y = ax^n, its derivative is given by dy/dx = nax^(n-1).
Applying the power rule to our equation, we find:
dy/dt = d/dt (14t - 1.86t^2)
Taking the derivative of each term separately, we get:
dy/dt = d/dt (14t) - d/dt (1.86t^2)
The derivative of 14t is simply 14, as the derivative with respect to t of any constant multiplied by t is just the constant.
The derivative of 1.86t^2 can be found using the power rule once again:
d/dt (1.86t^2) = 1.86 * 2t^(2-1) = 3.72t
Therefore, we have:
dy/dt = 14 - 3.72t
To estimate the instantaneous velocity when t = 1, we substitute t = 1 into the derivative equation:
dy/dt = 14 - 3.72(1) = 14 - 3.72 = 10.28 m/s
Therefore, the estimated instantaneous velocity when t = 1 is 10.28 m/s.
To estimate the instantaneous velocity when t = 1, we need to find the derivative of the height equation y = 14t - 1.86t^2 with respect to t. The derivative will give us the rate of change of y with respect to t, which represents the velocity at any given time.
To find the derivative, we can differentiate each term of the equation using the power rule:
dy/dt = d(14t)/dt - d(1.86t^2)/dt
The derivative of 14t with respect to t is simply 14.
To differentiate 1.86t^2, we use the power rule. The power rule states that d(x^n)/dx = nx^(n-1), where n is a constant and x is a variable. In this case, n = 2, so we get:
d(1.86t^2)/dt = 2 * 1.86 * t^(2-1) = 3.72t
Putting it all together:
dy/dt = 14 - 3.72t
Now, to estimate the instantaneous velocity when t = 1, substitute t = 1 into the derivative equation:
v = dy/dt = 14 - 3.72(1)
v = 14 - 3.72
v ≈ 10.28 m/s
Therefore, the estimated instantaneous velocity when t = 1 is approximately 10.28 m/s.
dy/dt = velocity up = 14 - 3.72 t
when t = 1
dy/dt = 14 - 3.72 = 10.28