assignment related to the motion of an object.
Link to important videos: Project problem
Here is the pdf of the notes on the project like problems: Section 3.6 and project like problems
Here is the link to the submission folder: project submission folder
One one pdf file per group is required. Please attach all the graphs (excel only) to the solutions of the problem in one pdf file.
Introduction
In Calculus I, we learned about derivative defined as rate of change, rules for finding derivatives, graphing functions using derivatives and other important applications of derivatives. This assignment combines the conceptual knowledge of the derivatives along with interpretation of the graphs of functions using derivatives.
Describing the motion of object and making deductions about the nature of motion is crucial in applications. Calculus I provide us with tools and ways to calculate displacement, velocity, acceleration for a given function of a particle in motion.
The position function tells us where an object is at a certain point in time. Let’s say an object has a position function f = s(t), where:
s = position (e.g. feet, meters, miles)
t = time (e.g. seconds, minutes, days)
then the velocity function is v(t) = s?(t). The single prime (?) indicates the derivative
You can take this one step further: taking the derivative of the velocity function gives you the acceleration function.
If you want to find acceleration from a position function, then take the derivative twice.
The following problem is about motion of a given particle. This problem will help you analyze and understand the calculus behind the one dimensional motion of an object. At various steps, you need to draw and interpret the graphs, perform calculations, make deductions and explain the results.
Problem: The position function of a particle is given by
where t is the time in seconds between [0, 6] and the distance is measured in meters.
Answer the following questions
Graph the position function in the domain [0, 6] by stating the critical points and points of infection.
What is happening at the critical numbers in terms of motion of the particle? Explain.
What is happening at the point of inflection in terms of motion of the particle? Explain.
Write the velocity and acceleration functions of the particle.
At what time is the velocity of the particle zero? When is the velocity maximum?
What is the velocity of the particle at 
?
What is the acceleration of the particle at 
?
Interpret your answers from Parts f) and g) and make a judgement about the speed of the particle at 
.
What is the average velocity of the particle between the given time frame?
Will there be a point on the graph of f where average velocity is same as the instantaneous velocity? What theorem verifies this? Find the location of that point.
Write the equation of the secant line between the initial and the final point and the tangent line at the point you got in Part j).
Write observations about the two lines in Part k)
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