Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose a ball is thrown straight up from a height of 48 feet and given an initial upward velocity of \(8 \mathrm{ft} / \mathrm{sec} .\) Then its height at time \(t, t\) in seconds, is given by \(h(t)=-16 t^{2}+8 t+48\), for \(t \in[0,2] .\) In this problem we will look at the ball s velocity at \(t=1\). (a) At \(t=1\), is the ball heading up, or down? Explain your reasoning. (b) By calculating the average rate of change of height with respect to time, \(\frac{\Delta h}{\Delta t}\), on the intervals \([0.9,1]\) and \([1,1.1]\), give bounds for the ball s velocity at \(t=1\). (c) Improve your bounds by using the intervals \([0.99,1]\) and \([1,1.01]\). (d) Use the limit de nition of \(f^{\prime}(1)\) to nd the ball s instantaneous velocity at \(t=1\).

Step-by-Step Solution

Answer

The ball was moving downwards at \(t = 1s\). The velocity at that time was approximated to be in between the values obtained in (b) and (c). The exact velocity at \(t = 1s\) according to the limit definition of derivative, was found in part (d).

Step 1: Determine the direction of movement

For (a), one can determine whether the ball is moving up or down by comparing the height at \(t=1\) second with the height at a slightly later time. If the height at the later time is less than the height at \(t=1\), then the ball is moving downwards; otherwise, it's moving upwards. To do this, calculate \(h(1)\) and \(h(1.01)\) and compare.

Step 2: Calculate the Average Rate of Change over the given intervals

For (b), calculate the average rate of change over the intervals \([0.9,1]\) and \([1,1.1]\). This is done by applying the formula \(\frac{\Delta h}{\Delta t} = \frac{h(t2)-h(t1)}{t2 - t1}\), where \([t1,t2]\) is the interval. Calculate this for both intervals.

Step 3: Improve the bounds

For (c), repeat the process in Step 2 for the smaller intervals \([0.99,1]\) and \([1,1.01]\). Again calculate the average rates of change over these intervals.

Step 4: Apply the limit definition of derivative

For (d), use the definition of the derivative, \(f'(t) = \lim_{h-> 0}\frac{f(t+h) - f(t)}{h}\), to find the instantaneous velocity at \(t=1\). Here, \(f(t) = -16 t^{2}+8 t+48\) and \(h\) represents a smaller and smaller time interval around \(t=1\).

Key Concepts

Derivative

The concept of a derivative is central to calculus and is crucial for understanding how quantities change instantaneously. In the context of kinematics, the derivative allows us to calculate the **instantaneous velocity** of an object, which is the velocity at a specific point in time. To find the derivative of a function, we use the limit definition: \[ f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h} \] This formula measures the rate at which the function value changes as we shrink the interval \(h\) towards zero. For the given problem, the height function of the ball is: \[ h(t) = -16t^{2} + 8t + 48 \] By applying the limit definition of the derivative to this function, we can determine the ball's instantaneous velocity at any point \(t\). Specifically, solving this at \(t=1\) gives us the ball's velocity at that instant.

Average Rate of Change

The average rate of change of a function over a specific interval provides an approximation of how quickly a quantity is changing over that period. It is essentially the "velocity" over a chosen interval. Calculated as follows: \[ \frac{\Delta h}{\Delta t} = \frac{h(t_2) - h(t_1)}{t_2 - t_1} \] In our problem, this calculates the change in height (\(\Delta h\)) over the change in time (\(\Delta t\)). To find the average rate of change around \(t=1\), we examine the function on intervals like \([0.9, 1]\) and \([1, 1.1]\) to estimate how fast the ball is moving. These intervals provide bounds for velocity and help us approximate instantaneous movement until we use the derivative for precision.

Kinematics

Kinematics is the study of motion without considering the causes of this motion. It involves analyzing different quantities like velocity, acceleration, and displacement. In our scenario, kinematics focuses on the ball’s upward and downward motion. Key equations include: - **Displacement**: The overall change in position:\[ s = ut + \frac{1}{2}at^2 \] - **Velocity**: The rate at which displacement changes. The derivative of the height function gives us this. By using kinematic equations and concepts like the derivative, we can fully describe how an object moves under various conditions. The vertical motion of our ball, governed by gravity, follows these principles.

Motion Analysis

Motion analysis involves understanding how objects change position over time. It includes scratching the surface with concepts like velocity and extends through intricate analyses using calculus. This is crucial when solving physics problems involving moving objects. For the ball's motion, understanding begins with identifying direction: - At any point in time, comparing height at two successive points lets us know if the ball is ascending or descending. Through further analysis, calculating average rates for different intervals gives a clearer picture of how quickly the object transitions through space. Eventually, by pinpointing instantaneous velocity using derivatives, you clearly define the nature of motion at a precise moment, allowing accurate conclusions about the dynamic state of an object.