Question

Suppose \(s(t)\) is the position of an object moving along a line at time \(t\) as \(0 .\) Describe a process for finding the instantaneous velocity at \(t=a\)

Short answer

Answer: To find the instantaneous velocity at a specific time \(t = a\) given the position function \(s(t)\), follow these steps: 1. Understand that instantaneous velocity is given by the derivative of the position function with respect to time. 2. Find the derivative of the position function, \(s(t)\), using the limit definition of the derivative: \(v(t) = \frac{ds(t)}{dt} = \lim_{h \to 0} \frac{s(t + h) - s(t)}{h}\). 3. Evaluate the velocity function at \(t = a\): \(v(a) = \lim_{h \to 0} \frac{s(a + h) - s(a)}{h}\).

Step-by-step solution

Understanding instantaneous velocity

Instantaneous velocity at a point is the rate at which the position changes at that exact moment. Mathematically, it is given by the derivative of the position function with respect to time. So we need to find the derivative of \(s(t)\) and then evaluate it at \(t = a\).

Find the derivative of the position function

In order to find the derivative of the position function \(s(t)\), we can use the limit definition of the derivative. This is given by: \(v(t) = \frac{ds(t)}{dt} = \lim_{h \to 0} \frac{s(t + h) - s(t)}{h}\) We don't yet have the function \(s(t)\), so we can only write it in this general form.

Evaluate the velocity function at \(t = a\)

Now that we have the derivative of the position function (velocity function), we need to evaluate it at time \(t = a\). Doing this will give us the instantaneous velocity at that specific time: \(v(a) = \lim_{h \to 0} \frac{s(a + h) - s(a)}{h}\) This expression gives the instantaneous velocity at time \(t = a\). Without the specific position function \(s(t)\), we can't find the numerical value, but we now have a general process for finding the instantaneous velocity at any time given the position function.

Key concepts

Derivative of Position Function

Understanding the derivative of a position function is crucial for analyzing motion in calculus. In simpler terms, when we talk about a moving object, its position changes with time. The function that represents this changing position is known as the position function, often denoted as s(t), with t indicating time. To grasp how quickly the position is changing at any given moment, we resort to the concept of instantaneous velocity.

This is where the derivative comes into play. The derivative of the position function with respect to time, denoted as ds/dt or s'(t), tells us the rate of change of position—essentially, it is the object's velocity. Instantaneous velocity is the value of this derivative at a specific point in time. By finding this derivative, we determine how fast the object is moving and in what direction (positive or negative velocity) at exactly that particular moment.

Limit Definition of the Derivative

If we want to dive deeper into understanding derivatives, we encounter the limit definition of the derivative. This classic definition is the foundation of calculus and it explains how to find the derivative of a function at any point.

According to this definition, the derivative of a function at a point t can be found by taking the limit as h approaches zero of the average rate of change of the function over the interval from t to t + h. This is mathematically denoted as:
\[ v(t) = \lim_{h \to 0} \frac{s(t + h) - s(t)}{h} \]
The expression (s(t + h) - s(t))/h represents the average velocity over the tiny time interval h. As this interval approaches zero, the average velocity becomes the instantaneous velocity. This process reflects how the derivative provides a precise rate of change (in our case, the instantaneous velocity) at any given instant.

Evaluating the Velocity Function

Once we have established the derivative of the position function as the velocity function, the next step is to determine the object's instantaneous velocity at a particular time t = a. To do this, we evaluate the previously discussed limit definition of the derivative at that specific point.

The formula to compute the instantaneous velocity at t = a is thus:
\[ v(a) = \lim_{h \to 0} \frac{s(a + h) - s(a)}{h} \]
By substituting t with a, the calculated limit gives us the exact velocity of the object at time t = a. It's important to note that without an explicit position function s(t), we can't compute a specific numerical value for velocity. However, understanding this evaluation process is key to applying calculus concepts to real-life motion problems, enabling us to determine how objects behave at any point in time.