Problem 80
Question
The position of a moving body is given by \(p(t)=\) \((2.718281828)^{t} .\) Calculate the average velocity of the body over a time interval of the form \([1,1+\Delta t]\) for a sequence of small values of \(\Delta t .\) Display your results in the form of a table. Formulate a guess \(m\) for the instantaneous velocity of the body at time \(t=1 .\) Plot \(p .\) In the same coordinate plane, add the graph of the straight line that passes through (1,2.718281828) and that has slope \(m .\) Does the resulting figure support your conjectured value? Explain.
Step-by-Step Solution
Verified Answer
The instantaneous velocity at \( t = 1 \) is 2.718281828.
1Step 1: Calculate Position at Endpoints
To find the average velocity, we first calculate the position at the endpoints of the interval. At time \( t = 1 \), the position is \( p(1) = e^1 = 2.718281828 \). At time \( t = 1 + \Delta t \), the position is \( p(1 + \Delta t) = e^{1 + \Delta t} = e \cdot e^{\Delta t} \approx 2.718281828 \cdot e^{\Delta t} \).
2Step 2: Calculate Average Velocity
The formula for average velocity over the interval \([1, 1 + \Delta t]\) is given by \( \frac{p(1 + \Delta t) - p(1)}{\Delta t} = \frac{2.718281828 \cdot e^{\Delta t} - 2.718281828}{\Delta t} \).
3Step 3: Substitute Values for \(\Delta t\)
We calculate the average velocity for small values of \(\Delta t\).1. \( \Delta t = 0.1 \) \( \Rightarrow \text{Average Velocity} = \frac{2.718281828 (e^{0.1} - 1)}{0.1} = 3.00416349 \)2. \( \Delta t = 0.01 \) \( \Rightarrow \text{Average Velocity} = \frac{2.718281828 (e^{0.01} - 1)}{0.01} = 2.744812447 \)3. \( \Delta t = 0.001 \) \( \Rightarrow \text{Average Velocity} = \frac{2.718281828 (e^{0.001} - 1)}{0.001} = 2.721847673 \)
4Step 4: Conjecture Instantaneous Velocity
As \( \Delta t \) approaches zero, the average velocity approaches the derivative at \( t = 1 \). We conjecture that the instantaneous velocity at \( t = 1 \) is \( m = 2.718281828 \), the derivative of \( e^t \) evaluated at \( t = 1 \).
5Step 5: Plot Graphs
Plot the graph of \( p(t) = e^t \). At \( t = 1 \), the position is approximately 2.718281828. Draw a tangent line through \((1, 2.718281828)\) with slope \( m = 2.718281828 \).
6Step 6: Analyze and Conclude
Observe that the tangent line closely matches the slope of the curve at \( t = 1 \), which supports our conjecture that the instantaneous velocity of the body at \( t = 1 \) is indeed \( m = 2.718281828 \).
Key Concepts
Instantaneous VelocityPosition FunctionDerivativeTangent Line
Instantaneous Velocity
Imagine you are observing a car drive past. At a specific moment, you want to know how fast the car is going. This speed at an exact point in time is called "instantaneous velocity." It is different from average velocity, which measures speed over an interval of time. Instantaneous velocity provides insight into the motion of an object at a precise moment.
You can picture it as looking at the speedometer at that moment. In calculus, instantaneous velocity is determined using the concept of a limit, which mathematically estimates the speed as the time interval narrows down to a single point. This gives a better understanding of how quickly or slowly something is moving at a particular instant.
You can picture it as looking at the speedometer at that moment. In calculus, instantaneous velocity is determined using the concept of a limit, which mathematically estimates the speed as the time interval narrows down to a single point. This gives a better understanding of how quickly or slowly something is moving at a particular instant.
Position Function
A position function describes the location of an object as a function of time. For example, if you have a formula like \(p(t) = e^t\), it tells you where the object is at any given time \(t\).
This function is essential for understanding motion. For instance:
This function is essential for understanding motion. For instance:
- At \(t = 1\), the position is about 2.718.
- At \(t = 2\), the position is \(e^2\), or approximately 7.389.
Derivative
In calculus, a derivative gives a way to understand how a function changes. When talking about motion, the derivative of a position function with respect to time is the velocity function, showing how fast an object moves.
For a function \(p(t) = e^t\), its derivative is itself, \(e^t\). This property of the exponential function makes it a neat example to work with. At any point \(t\), the slope or rate of change of \(p(t)\) is simply its value at that time.
Understanding derivatives is crucial for calculating instantaneous velocity, as they give the slope of the tangent to the position curve, describing how position changes over infinitesimally small time intervals.
For a function \(p(t) = e^t\), its derivative is itself, \(e^t\). This property of the exponential function makes it a neat example to work with. At any point \(t\), the slope or rate of change of \(p(t)\) is simply its value at that time.
Understanding derivatives is crucial for calculating instantaneous velocity, as they give the slope of the tangent to the position curve, describing how position changes over infinitesimally small time intervals.
Tangent Line
A tangent line is a straight line that touches a curve at one single point without crossing it. It's like a beam of light grazing the surface of the curve. In the context of a position function, this line represents the object's instantaneous velocity at that specific point in time.
To find the tangent line, you need its slope, which is the derivative of the position function at that point. For example, if the position at \(t = 1\) is \(2.718281828\) and its derivative, the slope \(m\), is also \(2.718281828\), the tangent line at \(t = 1\) would reflect the velocity at that instant.
This line is vital for visually confirming the instantaneous velocity, as it shows how closely the line follows the position function around the point of tangency.
To find the tangent line, you need its slope, which is the derivative of the position function at that point. For example, if the position at \(t = 1\) is \(2.718281828\) and its derivative, the slope \(m\), is also \(2.718281828\), the tangent line at \(t = 1\) would reflect the velocity at that instant.
This line is vital for visually confirming the instantaneous velocity, as it shows how closely the line follows the position function around the point of tangency.
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