Problem 37 - A Graphical Approach to Precalculus with Limits

Question

The position in feet of a car along a straight racetrack after \(t\) seconds is approximated by \(s(t)\) Find the car's velocity in feet per second after 3 seconds. $$s(t)=9 t^{2}$$

Step-by-Step Solution

Answer

The velocity is 54 feet per second after 3 seconds.

Step 1: Understand the Relationship

The function given for the car's position is \(s(t) = 9t^2\). This is a function of time \(t\), and to find the velocity, we need the rate of change of the position function with respect to time, which requires calculus - specifically, differentiation.

Step 2: Differentiate the Position Function

To find the velocity function, differentiate the position function \(s(t)\) with respect to \(t\). Use the power rule for differentiation: if \(f(t) = at^n\), then \(f'(t) = nat^{n-1}\). Applying this to \(s(t) = 9t^2\):\[ v(t) = \frac{d}{dt}(9t^2) = 18t \]Thus, the velocity function is \(v(t) = 18t\).

Step 3: Evaluate the Velocity Function at 3 Seconds

Substitute \(t = 3\) into the velocity function to find the car's velocity after 3 seconds:\[ v(3) = 18 \times 3 = 54 \]Hence, the velocity of the car after 3 seconds is 54 feet per second.

Key Concepts

Understanding Calculus in Motion

Calculus is a branch of mathematics that helps us understand how things change. It's widely used in various fields such as physics, engineering, economics, and more. One of the most important concepts in calculus is the derivative. Derivation is all about finding the rate at which something changes.

For example, in our exercise, we're looking at a car's movement over time. The position of the car is given as a function of time, which is represented by the equation \(s(t) = 9t^2\). To find out how fast the car is moving, or its velocity, we need to use calculus to find out how the position changes with time. This involves calculating the derivative of the position function, which gives us a new function - the velocity function.

The Differentiation Process

Differentiation is the process of finding the derivative of a function. It helps us understand how a function's output changes as its input changes. In simpler terms, it tells us the slope of the function at any given point.

In our exercise, the specific type of differentiation used is called the 'power rule'. This rule is a handy shortcut that tells us how to find the derivative of functions in the form of \(at^n\), where \(a\) and \(n\) are constants.

• The formula for the power rule is: if \(f(t) = at^n\), then the derivative \(f'(t) = nat^{n-1}\).
• This means we multiply the exponent by the coefficient and then decrease the exponent by one.

Applying this rule to the car's position function \(s(t) = 9t^2\), we get its derivative, which is the velocity function \(v(t) = 18t\). This tells us how fast the car's position is changing at any time \(t\).

Using the Power Rule for Velocity

The power rule is an essential tool for finding derivatives quickly and easily. By using this rule, we can find how fast an object is moving without having to do complex long calculations.

Let's look at how the power rule helped us determine the car's velocity. For the function \(s(t) = 9t^2\), the power rule gave us:

• Derivative \(f(t) = 18t\), which is a linear function representing velocity.

At any time \(t\), we can find the velocity by simply plugging in the value of \(t\). For example, after 3 seconds, we substitute \(t = 3\) into the velocity function \(v(t) = 18t\) to get \(v(3) = 54\).
Thus, the car's velocity at 3 seconds is 54 feet per second, found using a straightforward application of the power rule.