Average Velocity | MIT 18.01SC Single Variable Calculus, Fall 2010
TLDRIn this recitation, the professor discusses the application of integration to find the average velocity of a jogger's sprint. Christine, initially sprinting away from what she thought was a bear, accelerates and then decelerates back to her original speed within 10 seconds. The professor guides through the process of calculating the average velocity by integrating the velocity function over the interval and dividing by the time span. The integral of a complex function is simplified with a substitution, leading to an arctangent-based solution. The average velocity is approximately 6.9 meters per second, illustrating a practical use of integration in real-life scenarios.
Takeaways
- 🏃♀️ The story involves Christine jogging and encountering a bear, which turns out to be an unkempt math graduate student, highlighting the context of the problem.
- 📚 The lecture discusses the application of integration to compute average values of a function, specifically focusing on average velocity.
- 🔢 Christine's velocity function is given by \( v(t) = \frac{1500}{100 + (t - 5)^2} - 7 \) meters per second, representing her speed at any time t seconds after sprinting.
- ⏱ The problem spans a 10-second interval, starting from when Christine begins sprinting until she returns to her original velocity.
- 🚀 Christine's initial velocity at t=0 is 5 m/s, and she reaches a peak velocity of 8 m/s at t=5 seconds.
- 📉 After realizing the 'bear' is a person, she slows down and returns to her initial velocity by t=10 seconds.
- 🧩 To find the average velocity, the professor suggests integrating the velocity function over the interval and dividing by the interval's length.
- 📐 The integral of the velocity function gives the total distance traveled, which is then divided by the total time to find the average velocity.
- 📘 The professor introduces a substitution method to simplify the integral, setting \( u = t - 5 \) to make the expression more manageable.
- 📚 The integral involves an arctangent function, hinting at the use of trigonometric substitution or recognition of an arctangent integral form.
- 🔍 The final step involves evaluating the definite integral using the fundamental theorem of calculus and simplifying the result.
- 📊 The average velocity is calculated to be approximately 6.9 m/s, which is the key result of the problem.
Q & A
What is the context of the problem presented in the script?
-The script presents a problem involving Christine, who starts sprinting upon seeing what she thinks is a bear and later realizes it's just an unkempt math graduate student. The context is an application of integration to find the average velocity of Christine over a 10-second sprint.
What is the given velocity function for Christine's sprint?
-The velocity function is given as \( v(t) = \frac{1500}{(100 + (t - 5)^2) - 7} \) meters per second, where \( t \) represents the time in seconds after she started sprinting.
What is the initial velocity of Christine when she starts sprinting at t=0?
-At \( t=0 \), the initial velocity is \( v(0) = \frac{1500}{125 - 7} = 12 \) meters per second.
How does Christine's velocity change after 5 seconds of sprinting?
-After 5 seconds, the velocity function simplifies to \( v(5) = \frac{1500}{0} \), which is undefined. However, the script mentions that she reaches 8 meters per second at this point, indicating a mistake in the script. The correct calculation should consider the velocity function as it approaches t=5.
What is the average velocity formula used in the script?
-The average velocity formula used is the total distance traveled divided by the time interval, represented mathematically as \( \text{avg}(v) = \frac{1}{\text{length of interval}} \times \int_{\text{start time}}^{\text{end time}} v(t) \, dt \).
What substitution is suggested in the script to simplify the integral?
-The substitution suggested is \( u = t - 5 \), which simplifies the integral by transforming the denominator into a form that resembles a standard arctangent function.
What is the significance of the arctangent function in solving the integral?
-The arctangent function is significant because it is the antiderivative of the form \( \frac{1}{a^2 + u^2} \), which appears in the integral after the substitution.
How does the script handle the negative part of the integral?
-The script simplifies the negative part by using the property of the arctangent function that \( \text{arctan}(-x) = -\text{arctan}(x) \), which allows for combining terms with opposite signs.
What is the final expression for the average velocity based on the script?
-The final expression for the average velocity is \( 30 \times \text{arctan}(1/2) - 7 \) meters per second.
What is the approximate value of Christine's average velocity over the 10-second sprint?
-The approximate value of the average velocity is 6.9 meters per second.
What is the main takeaway from the script regarding the process of finding the average value of a function?
-The main takeaway is that to find the average value of a function over an interval, one should integrate the function over that interval and then divide by the length of the interval, applying the fundamental theorem of calculus to evaluate the definite integral.
Outlines
🏃♀️ Christine's Jogging and Bear Encounter
The script begins with a professor introducing an average value problem involving integration. The scenario described is of a jogger named Christine who, upon seeing what she thinks is a bear, starts sprinting. Her velocity function is given by a formula involving a quadratic expression. The professor explains how Christine's velocity changes from 5 meters per second to 8 meters per second and back to 5 meters per second over a 10-second period, and poses the question of finding her average velocity during this time. The audience is encouraged to solve the problem before the solution is revealed in the subsequent part of the script.
📚 Calculating Christine's Average Velocity
In this paragraph, the professor proceeds to solve the problem of finding Christine's average velocity over the 10-second sprint. The approach involves integrating the velocity function over the time interval from 0 to 10 seconds and then dividing by the length of the interval to find the average. The professor outlines a method to simplify the integral by using a substitution and hints at using an arctangent function for the solution. The integral is evaluated, and the result is simplified to an expression involving the arctangent of 1/2, with the final average velocity calculated to be approximately 6.9 meters per second. The professor concludes by summarizing the steps taken to solve the problem and emphasizes the standard method for calculating the average value of a function.
Mindmap
Keywords
💡Integration
💡Average Value
💡Velocity Function
💡Arctangent
💡Substitution
💡Definite Integral
💡Fundamental Theorem of Calculus
💡Anti-derivative
💡Odd Function
💡Sprint
💡Meters per Second
Highlights
Introduction to the concept of computing average values of a function.
A real-world scenario involving Christine's velocity while jogging and encountering a bear.
The mathematical expression for Christine's velocity as a function of time, v(t).
Calculation of Christine's initial velocity at t=0 and its value.
Christine's velocity at t=5 seconds and the significance of the bear being a math graduate student.
Christine's velocity returning to her original speed after 10 seconds.
The problem posed: finding Christine's average velocity over the 10-second sprint.
Explanation of the method to compute the average value of any function by integrating over an interval.
The formula for average velocity involving total distance traveled and time interval.
Setting up the integral to find the average velocity of Christine.
Simplification of the integral using substitution to make it more manageable.
Identification of the integral as an arctangent type due to its form.
Computation of the anti-derivative of the velocity function.
Application of the fundamental theorem of calculus to evaluate the definite integral.
Simplification of the result using properties of the arctangent function.
Final calculation of Christine's average velocity, approximately 6.9 meters per second.
Recap of the process used to find the average velocity, emphasizing the method's generality.
Transcripts
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