Velocity functions A projectile is fired vertically upward...

Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) abous the ground after $t$ seconds is given by the function $s(t)$ a. For the following functions $s(t)$. find the instantaneous velocity function $v(t) .$ (Recall that the velocity function $v$ is the derivative of the position function $s$ ) Determine the instantancous velocity of the projectile at $t=1$ and $t=2$ seconds. Evaluate $\frac{d s}{d t}$ and $\left.\frac{d s}{d t}\right|_{t=-1}$ if $s=11 t^{3}+t+1$

Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) abous the ground after $t$ seconds is given by the function $s(t)$
a. For the following functions $s(t)$. find the instantaneous velocity function $v(t) .$ (Recall that the velocity function $v$ is the derivative of the position function $s$ )
Determine the instantancous velocity of the projectile at $t=1$ and $t=2$ seconds.
Evaluate $\frac{d s}{d t}$ and $\left.\frac{d s}{d t}\right|_{t=-1}$ if $s=11 t^{3}+t+1$

Key Concepts

Function Evaluation of a Derivative

After obtaining a derivative function through differentiation, evaluating it at specific points determines the rate of change at those particular values. This process is essential for finding instantaneous values such as velocity at a given time, and it involves substituting the specific value into the derivative function to compute the result.

Instantaneous Velocity

Instantaneous velocity is the rate at which an object’s position changes at a specific moment in time. In calculus, it is defined as the derivative of the position function with respect to time, which yields the velocity function. This concept is crucial in analyzing the motion of objects where velocity may vary continuously over time.

Derivative

A derivative is a fundamental concept in calculus that describes the instantaneous rate of change of a function with respect to a variable. It is obtained through the process of differentiation and provides key information about the behavior of the function such as increasing or decreasing trends, instantaneous slopes, and rates of change in various contexts.

Power Rule

The power rule is a basic technique for differentiation which states that the derivative of a term in the form of t^n is n*t^(n-1), where n is any real number. This rule simplifies the process of finding the derivative for polynomial functions and is fundamental to solving many calculus problems involving rates of change.

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