Suppose velocity (in miles per hour) is given by v(t)=3 t,...

Suppose velocity (in miles per hour) is given by $v(t)=3 t$, where $t$ is measured in hours. We are interested in the distance traveled from $t=0$ to $t=k$, where $k$ is a constant. (a) By solving the differential equation $d s / d t=3 t$ and using the initial condition $s(0)=s_{0}$, nd the distance function $s(t) .$ Using $s(t)$, nd i. $s(0)$. ii. $s(k)$. iii. the distance traveled between $t=0$ and $t=k$. (b) Find the area under the graph of $v(t)$ from $t=0$ to $t=k$. Verify that your answers to part (a) iii and (b) are the same.

Suppose velocity (in miles per hour) is given by $v(t)=3 t$, where $t$ is measured in hours. We are interested in the distance traveled from $t=0$ to $t=k$, where $k$ is a constant.
(a) By solving the differential equation $d s / d t=3 t$ and using the initial condition $s(0)=s_{0}$, nd the distance function $s(t) .$ Using $s(t)$, nd
i. $s(0)$.
ii. $s(k)$.
iii. the distance traveled between $t=0$ and $t=k$.
(b) Find the area under the graph of $v(t)$ from $t=0$ to $t=k$. Verify that your answers to part (a) iii and (b) are the same.

Key Concepts

Differential Equations

This concept involves equations that relate a function to its derivatives. In the problem, the differential equation ds/dt = 3t is solved to find the position function s(t). The process requires integrating the velocity function, which represents how the position changes over time.

Initial Conditions

An initial condition provides a starting value that is used to determine the constant of integration after solving a differential equation. The condition s(0) = s? is essential to obtain a specific solution for the position function that meets the problem's setup.

Definite Integrals

Definite integrals are used to calculate the accumulation of a quantity over an interval. When integrating the velocity function from t = 0 to t = k, the definite integral directly gives the total distance traveled between these two points in time.

Area Under a Curve

This concept is linked to finding the definite integral of a function, representing the accumulated value of the function over an interval. In the context of the question, the area under the graph of the velocity function from t = 0 to t = k corresponds to the distance traveled, showing the equivalence between integration and physical accumulation.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges differentiation and integration by stating that integration can be reversed by differentiation. It explains why the definite integral of a derivative (such as the velocity function) over an interval equals the net change in the original function (the position function) over that interval.

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