Problem 27 - Calculus of a Single Variable

Question

The height of a ball \(t\) seconds after it is thrown upward from a height of 6 feet and with an initial velocity of 48 feet per second is \(f(t)=-16 t^{2}+48 t+6\). (a) Verify that \(f(1)=f(2)\) (b) According to Rolle's Theorem, what must the velocity be at some time in the interval \((1,2) ?\) Find that time.

Step-by-Step Solution

Answer

The velocity of the ball is zero at time \( t = 1.5 \) seconds.

Step 1: Verification of \(f(1)=f(2) \)

First, we substitute \( t = 1 \) and \( t = 2 \) into the given function \[ f(t)=-16t^{2} + 48t + 6 \].\n\nWe find that \( f(1)=-16(1)^{2} + 48(1) + 6 = 38 \).\n\nNext we compute \( f(2)=-16(2)^{2} + 48(2) + 6 = 38 \).\n\nThus, \( f(1) = f(2) = 38 \).

Step 2: Applying Rolle's Theorem

Now knowing that \[ f(1) = f(2) \], according to Rolle's Theorem, there must be at least one value c in the open interval (1, 2), where the derivative of the function \( f'(c) \) equals 0.\nFirstly find the derivative \( f'(t) \) of the original function \( f(t) \). \[ f'(t) = -32t + 48 \].\nThen solve \( f'(t) = 0 \) for \( t \) in the interval (1, 2).\n\nThis gives \[ -32t + 48 = 0 \]. Solving for \( t \) gives us \( t = 1.5 \). As 1.5 is in the interval (1, 2), there is indeed a time between 1 and 2 seconds where the velocity is zero.

Key Concepts

Real-World Application of Calculus

Understanding the practical use of calculus deepens comprehension and appreciation for the subject. Let's take the example of a ball thrown upward. By applying calculus, particularly in deriving equations of motion, we can predict the ball’s height at any given time, its velocity, and acceleration. This example is not just academic; these principles are widely used in engineering projects such as constructing bridges or designing roller coasters, and in technology, such as in computer graphics and animation where objects' motions are simulated realistically. Professionals use calculus to find optimal solutions, like minimizing the materials for cost-effective structures while ensuring safety and durability. This reflects the indispensable role of calculus in problem-solving across various real-world scenarios.

Derivative of a Function

The derivative of a function at a point is the rate at which the function's value changes with respect to change in the input value. For the ball thrown upwards, the function for height over time is given by \( f(t)=-16t^2+48t+6 \). To find the velocity of the ball at any time \( t \), we calculate the derivative of this function, \( f'(t) \). So, \( f'(t) = -32t + 48 \), which gives us the velocity. At any moment, if we want to know the instantaneous change in height, we look at the derivative. Derivatives are foundational for understanding motion, economics (like cost optimization), and various changes over time.

Verification of Function Properties

Verifying a function's properties, such as continuity and differentiability, is essential to applying theorems like Rolle's. The ball’s trajectory example involves confirming that the height, represented by \( f(t) \), remains continuous and differentiable in our interval of interest. Analyzing function properties facilitates predictions and ensures that our usage of calculus theorems is valid. This scrutiny is critical in verifying the conditions under which a system behaves as predicted, which is of paramount importance in the development and testing of scientific and engineering models.

Critical Points of Functions

Critical points of functions are where the derivative is either zero or undefined. This helps identify the function's local maxima and minima – crucial for optimization problems. In the exercise, Rolle's Theorem tells us that somewhere between 1 and 2 seconds, the velocity must be zero due to the function’s height reaching a turning point. Such information is essential for finding when the ball reaches its peak and begins to descend. Recognizing these critical points is not only useful in physics but also in business for finding profit maximization or cost minimization points, and in environmental studies for evaluating species population changes over time.