As we wrap up Unit 5 and begin the final unit, we quickly approach the actual AP exam.  We will fortunately have a couple of weeks to prepare for the AP exam, but the majority of that time will be spent focusing on question style as opposed to actual content review.

For this week’s blog, I want you to reflect back on the AP Calculus course so far and give a summary (list form is fine) of as many of the topics covered in class as you can possibly recall.  This is really just an exercise to begin “free associating” different concepts we’ve covered in class. Hopefully, as you think of topics, you’ll also consider what topics you still recall and which ones you’ll need to review more.

As you probably know, Pi Day is coming up on Sunday, March 14 (3/14).  As announced in class, there will be a series of activities and events taking place during the next couple of weeks.  I encourage you to participate in as many of these as possible.  To build up the excitement for Pi Day, this week’s blog is devoted to that crazy little Greek symbol.

RESPONSE DUE: Monday, March 15:
The Exploratorium in San Francisco has gathered a list of just a FEW websites about pi (the links are at the bottom of the page), ranging from interesting facts to silly songs. Visit a few of the sites from this list (or find your own via any search engine) and find out some interesting and/or fun pi-related facts.  Share your findings and briefly describe why you found them fun/interesting.  Please include a link to the website(s) so others will be able to check them out as well!

For YouTube enthusiasts (and aren’t we all?), here’s a man typing the first 1000 digits of pi from memory (he holds the national record: over 15000 digits).

You do NOT need to submit a blog response this week since (a) you have a test on Monday and (b) you have a midterm coming up.

Midterms are quickly approaching (I know, you didn’t want to be reminded already).  Your midterm for AP Calculus will be on Thursday, March 4.  The final exam will have a 30-question multiple choice section (non-calculator) and a brief 5-question fill-in-the-blank section (calculator-active).

Here is a reminder of the topics we’ve covered this term:

Unit 1: Limits & Continuity

• Evaluate limits analytically (direct substitution, rationalizing the numerator/denominator, simplifying complex fractions, simplifying rational expressions, etc.)
• One-sided limits and the existence of a double-sided limit
• Evaluating limits at infinity and horizontal asymptotes
• Infinite limits and vertical asymptotes
• Continuity

Unit 2: Derivatives

• Differentiate a wide variety of functions applying a wide variety of rules and techniques (e.g. power rule, product rule, quotient rule, etc.)
• Concept of differentiability
• Differentiability implies continuity (but not necessarily the reverse)
• Evaluate a derivative at a given value
• The derivative interpreted as the slope of the tangent line to a point on a graph
• Implicit Differentiation
• Linear Approximation (using a tangent line equation to approximate a function value)

Unit 3: Applications of the Derivative

• Relationship between behavior of f, f’ and f”
• Critical numbers
• Points of inflection
• Relationship between position, velocity and acceleration (specific application of f, f’ and f”)
• Related rates
• Optimization/finding max. or min. of a function
• Mean-Value Theorem

Unit 4: Area & The Definite Integral

• Convert from limit of a Riemann Sum to definite integral notation (and vice versa)
• Evaluate definite integrals using geometric area formulas

Calculator Skills

You should be able to use your graphing calculator to find the following:

• Value of a derivative at a specific point
• Value of a definite integral
• Roots/zeroes/x-intercepts of a function

I’m so pleased that so many of you seemed to enjoy Strogatz’ columns.  He publishes them once per week, so there is a new column (entitled Division and Its Discontents) that was posted on Sunday, Feb. 21.

If you would like to keep up on his columns, you can bookmark his Opinionator Blog page using this link.

This week I’m giving you a small break from Calculus for the weekly blog.  Instead, I’m expanding your horizons!

Steven Strogatz is an Applied Mathematics professor at Cornell University.  At the end of January, he introduced a new column in the New York Times about mathematics, trying to convey the beauty and abstract concepts of mathematics to “ordinary” folks.  So far he has written three columns:

• From Fish to Infinity (1/31)
• Rock Groups (2/7)
• The Enemy of My Enemy (2/14)

I’d like you read through at LEAST one of them (although I would love for you to read through all 3).  Just respond with a (minimum) one-paragraph response to something in the column that made you think or elicited some kind of reaction from you.

When considering the graph of a function f, we can look at a variety of pieces of information.

The domain of f allows us to note any discontinuities (holes, jumps or vertical asymptotes). The intercepts of f give us specific “key” points through which our graph must be drawn.

Using the sign of f’, we can analyze the increasing/decreasing behavior of f. When f’ is positive, f is increasing on that interval; when f’ is negative, f is decreasing on that interval.

The critical numbers of f (obtained by finding values in the domain of f that make f’ equal to zero or undefined) can also give us the location of relative extrema.  If f’ changes from positive to negative at x=c, then f has a relative maximum there; if f’ changes from negative to positive at x=c, then f has a relative minimum there.

The increasing/decreasing behavior of f’ also informs us about the concavity of f.  If f’ increases on an interval, then f” (the second derivative of f) is positive and f is concave upward.  If f’ decreases on an interval, then f” is negative and f is concave downward.  If f is differentiable at a point where f” changes sign or f’ has a relative extremum, then f has a point of inflection there.

For the past week, we have looked at how the first and second derivatives of a function f can inform us about the behavior of the graph of f.

Summarize what you’ve learned!  In your summary, make sure you include (but don’t limit yourself to) the following: connections between f, f’ and f”; finding relative extrema; and finding points of inflection, among other things.  There isn’t necessarily a single right answer, but the more thorough and complete your summary is, the more fully you probably understand these connections and the more likely you will be to adapt to a new-style question that might appear on the AP exam.

First, since s(t) = 4·sin(t) – cos(t), we know that

v(t) = s’(t) = 4·cos(t) + sin(t)

and

a(t) = v’(t) = s”(t) = -4·sin(t) + cos(t)

(a) average velocity on [pi/2, pi] = -6/pi
(b) v(pi) = s’(pi) = -4
(c) At time t=pi, the bug is moving to the left because its velocity is negative (and motion is in the direction of the x-axis).
(d) a(pi) = v’(pi) = s”(pi) = -1
(e) At time t=pi, the bug’s speed is increasing because velocity and acceleration have the same sign.

In the last blog, we worked with average rate of change and instantaneous rate of change.  This is a general interpretation.  One of the most common specific interpretations involves velocity (which is a rate of change of distance over time).

Given a position function s(t) which gives the position of a moving object (relative to some origin) after time t, we can find the average velocity of the object on the closed interval [a, b] by using the slope formula again:

s(b) – s(a)
b – a

We can find that object’s instantaneous velocity at specific time t=c by evaluating s’(c) (remember, velocity IS the derivative of the position function: v(t)=s’(t)).

Question:
A bug crawls along the x-axis such that its position at time t is given by the function

s(t) = 4·sin(t) – cos(t)

(a) What is the bug’s average velocity on the time interval [pi/2, pi]?
(b) What is the bug’s velocity (implying instantaneous velocity) at time t=pi?
(c) At time t=pi, which direction is the bug moving?  Why?
(d) What is the bug’s acceleration at time t=pi?
(e) At time t=pi, is the bug’s speed increasing or decreasing?  Why?

RESPONSE DUE: Tuesday, February 9, 2010

(a)  6/pi

(b)  3/2