Journal #2: Derivatives
In this journal you are to create a detailed graph of a power function using methods
learned through the last couple of units. The M2I2ACIDS poster and explorations 8.1-8.2
will come in handy for this journal. You must demonstrate your understanding of graphing
functions using derivatives without the use of a calculator (though your function may
require its use for simple computations). You must choose a function that helps you
describe the critical aspects of derivatives. Power functions will include fractional
exponents. When you select a function to use, clear it with me first. Do not graph the
function on the calculator! You are to use purely derivative techniques to find an accurate
representation of the graph!
A rough draft will be due the Monday we get back from break. We will critique your work
in class shortly thereafter. You will then take these comments and suggestions, refine your
work and resubmit by the following Wednesday.
Information to be included in the journal is as follows:

Give two definitions of a derivative – in your own common sense words and the
technical definition: lim 𝐷(𝑥). You will need to show the latter process with a
ℎ→0







different function if your function is too difficult (it may be!).
You may need to provide definitions of other terms that are vital to the
understanding of the definition of derivative.
Show connections between limits, derivatives and integrals to the best of your
ability.
Discuss the purpose of both first and second derivatives (discuss with examples –
algebraic and real life if possible).
When explaining the graphical context, show relationship of tangent and secant
lines.
When developing graphical context, make connections between 𝑓(𝑥), 𝑓 ′ (𝑥) and
𝑓"(𝑥) graphs. Within this process demonstrate your understanding of the derivative
as a function.
Demonstrate process of writing the tangent line equation.
Find the derivative at specific locations on a function as necessary to create a solid
representation of the graph of your function.
o You may use the derivative properties we have developed, i.e., you do not
need to state or develop these unless they are critical to your discussion.

o Show how the derivative can represented from multiple perspectives –
graphical, numerical and algebraic – when you show a derivative at a point.
o Identify where derivatives do not exist in your function and why.
Discuss explicit versus implicit differentiation and the benefits of implicit.
Graphs must be large enough to show detail necessary to help in the explanation process.
Annotation of the graphs is highly recommended.
Make sure your journal “tells the story” and you have a quality introduction and
conclusion. Please limit this journal to 4 or 5 pages. Don’t let your graphs/tables take up
too much space. Make them readable.
Remember:
 A quality journal will state the problem, give reason to do the problem, have a
verbal component that complements the algebraic and graphic components, and
have a flow between parts that makes your thoughts clear to you and the reader.
 This journal should be written from your point of view and should be a window into
how you perceive this topic.
 Journals help with metacognition (thinking about your thinking) by having you
reflect on your understanding from multiple perspectives.