A particle is traveling along a straight line.

Its position at time t is given by

s(t)=7t^(2)+7.

Find the velocity at time t=7.

To find the velocity at a specific time, we need to calculate the derivative of the position function with respect to time (t).

Given that the position function is s(t) = 7t^2 + 7, we can find the derivative, which represents the velocity.

To calculate the derivative, we can use the power rule of differentiation. According to the power rule, the derivative of t^n is n * t^(n-1).

Applying the power rule to the position function s(t) = 7t^2 + 7, we get:

s'(t) = d/dt (7t^2 + 7)
= 2 * 7t^(2-1) + 0
= 14t

Now, we can substitute t = 7 into the derivative expression:

v(7) = 14 * 7
= 98

Therefore, the velocity at time t = 7 is 98 units per time (e.g., meters per second).

To find the velocity at time t=7, we need to differentiate the position function with respect to time. The derivative of the position function gives us the velocity function.

Given that the position function is s(t) = 7t^2 + 7, we can find the velocity function v(t) by taking the derivative of s(t) with respect to t.

Step 1: Differentiate s(t) with respect to t
Using the power rule of differentiation, we differentiate each term separately:
d/dt(7t^2) = 2*7t^(2-1) = 14t
d/dt(7) = 0 (since a constant's derivative is zero)

Thus, the velocity function v(t) is given by v(t) = 14t.

Step 2: Evaluate the velocity at t=7
Substituting t=7 into the velocity function, we get:
v(7) = 14 * 7 = 98

Therefore, the velocity of the particle at t=7 is 98.

ds/dt = 14 t

if t = 7
14 * 7 = 98