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