A Function and its Derivatives

Directions: The goal of this paper is to enhance your thinking about derivatives which are one of the major topics in Calculus. Also, this paper should demonstrate that you have a solid understanding of the relationship between the graph of a function and its derivatives and how derivatives represent rates of change.

For each set of graphs given, write a few paragraphs to discuss functions and first and second derivatives. You should include answers to the following in your discussion of each situation.
Identify which curve represents the original function, the first derivative function, and the second derivative function. Explain your choices.
How does the shape of the graph of f(x) give us information about the graph of f^’ (x)?
Where is f^’ (x)   positive, negative, or zero? How do we determine this information from the graph of f(x)?
How does the shape of the graph of  f^’ (x)  give us information about the graph of f”(x)?
Where is f^” (x) positive, negative, or zero? How can you tell?
In the each situation, f^’ (x)  and f^” (x)  represent the rate of change of what?
Compare what is meant by the following in each situation: f(3),f^’ (3),and f”(3). (For example, if you are given a population function, f(3) represents the population at time t=3, while f’(3) represents the rate of change of the population at time t=3).
If the first derivative is positive (or negative), explain what this would imply about what is happening in the given scenario. (For example, if you are given a population function, the first derivative is positive when the population is increasing.)

Requirements: Your finished paper should…
Be approximately 2 pages in length
Be typed in size 11 font
Be well-written  and organized, including a short introduction, transitions, and a conclusion
Use complete sentences and avoid using bullet points to answer the discussion questions
Include graphs if necessary to clarify your ideas

Grading Rubric: 25 points total possible (5 points for each of the following)
Form – Are all the given questions addressed? Does the paper explain ideas in enough detail?
Clarity – Is the paper well-written and easy to follow? Is it written for a Calculus level audience?
Mathematical accuracy – Are mathematical concepts explained and used correctly?
Mathematical grammar – Are mathematical notation and vocabulary used correctly?
Presentation – Does the paper use correct spelling, grammar, and punctuation? Is the paper organized, typed, and neat?

Tips for Mathematical Writing:
Ask yourself the following questions: Could someone learn and understand this topic by reading my paper? Is there anything that I can explain in more detail? Remember that the reader will see only what you wrote, not what you meant to say.

Avoid using “it,” “this” or “that” when the function or term referred to is not clear. Be clear when you are referencing a function or idea. (For instance write, “Function A has a horizontal tangent line around x=3” instead of “It has a tangent line around that point”).

Give reasons for your answers and explain your steps. Keep the reader informed of what you do to arrive at each conclusion.

Proofread to check how well you have addressed the intended audience. You might want to ask another calculus student to assess your work and identify where explanations are confusing.

Reread and edit your paper before turning in a final draft. Look for mistakes in your mathematical thinking or notation as well as for mistakes in grammar, spelling, or punctuation.

Other Notes:
If you would like to include the graphs of each situation in your paper, they can be found on our class Niihka page in the “Writing Project” folder under the Resources tab.

If you would like feedback on a rough draft of your paper, you and your group can make an appointment to meet with me.  This appointment has to be by Tuesday March 10, 2015 to allow you to make changes before the final deadline.

Situation One: The graph shows the position (in meters) of a car after t seconds and the first and second derivatives of the position function.

Situation Two: The graph shows the volume (in liters) of water in a tank after draining for x hours and the first and second derivatives of the volume function.