AP Calculus AB / Mr. Hansen 1/5/2005 [rev. 1/6/2005, 1/9/2005]

Name: _______________________

The Fairly Complete Midterm Study Guide

The exam is cumulative, covering through §6.1. Topics not covered in those sections, but which you are expected to know for the exam, include the “u substitution method” for antidifferentiation, Euler’s Method, and the concept of adaptive quadrature. If there is an Euler’s Method problem on the exam, you can rest assured that no more than 3 or 4 steps will be required. However, the Euler’s Method formula (namely y_{n + 1} = y_n + y′_nDx) will not be furnished. You are expected to know it, since it is a consequence of the concept of linearization of a function, which you are also expected to know.

If there is a Newton’s Method problem, the Newton’s Method formula (namely x_{n + 1} = x_n – y_n/y′_n) will be furnished for you on the exam. Again, no more than 3 or 4 steps would be required.

Simpson’s Rule will not be on the midterm. However, the left endpoint, right endpoint, midpoint, and trapezoid methods for quadrature will be fair game. Formulas will not be provided; you will have to memorize them. However, derivation will not be required.

You should know all the derivative formulas we studied this semester, as well as the corresponding antiderivative formulas that descend from them. For example, since we learned that d/dx (sin x) = cos x, it must be true that ∫ cos x dx = sin x + C. Problems #1-42 in §6.1 give you a good review of most of these. Remember that although d/dx (ln x) = 1/x, it is also true that d/dx (ln |x|) = 1/x; hence the conventional textbook formula for ∫ 1/x dx is written as ∫ 1/x dx = ∫ dx/x = ln |x| + C. In class, we went even further and discussed why the conventional formula does not tell the full story, since you could have a valid antiderivative using C₁ to the left of the origin and C₂ to the right.

Don’t forget the product rule, the quotient rule, and the chain rule. Students forget these so often that I use the abbreviations PR, QR, and CR to save time when marking papers.

Some terminology will be tested. For example, you should know the two main branches of our course (the differential calculus, the integral calculus) and the purpose of each. You should be able to state EVT, IVT, MVT, and both forms of FTC accurately and completely. You should know both forms of the definition of derivative (i.e., the form that has x – c in the denominator, as well as the form that has h in the denominator) and, more importantly, how to recognize and apply them. You should be able to define the terms solution, general solution, and particular solution as applied to differential equations and antiderivative-type problems. You should be able to define Riemann sum and, more importantly, be able to recognize one when it pops up. You should be able to define the term accumulator function and to explain what its rate of growth (a.k.a. derivative) is.

Given any continuous function, no matter how bizarre, you should be able to write an antiderivative for it. Not only that, but if you are also given an initial condition, you should be able to write the antiderivative that satisfies the requirements. The key to solving problems of this type (see #7 on the Chapter 5 test) is FTC2.

Applications: The main applications we have seen so far are area calculations (using definite integrals) and position/velocity/acceleration problems (easy if you have studied physics, not so easy otherwise).

Precal: There will be a good amount of Precal represented on the exam, specifically trig identities (sin² x + cos² x = 1, plus the corollaries that you get after dividing through by sin² x or cos² x, respectively) and knowledge of logs and exponentials. You should also know the change-of-base formula and the handy conversion q^r = e^{r ln q}.

The Rule of 72, interesting and useful though it is, will not be on the midterm exam. However, exponential growth and exponential decay are fair game. A typical problem might be something like this: “A radioactive substance has a half-life of 22 days. How long will it take for 90% of the original quantity present to decay?” Solution: Let M(t) represent mass present at time t, where t = elapsed time in days. By the underlying physical principal of exponential decay, M ′(t) is always proportional to M(t). That is the same as saying dM/dt = kM. In class we proved (painfully) that such a diff. eq. has the general solution M = Ce^kt, where C and k are constants that must be determined from the initial conditions. Let A denote the initial amount present. Let us plug in what we know:

1. We know something at time t = 0, namely that M (at time 0) equals A. Symbolically,
M(0) = Ce^kt = Ce^{k · 0} = C · 1 = A. From this we conclude that C = A.

2. We know something at time t = 22, namely that M (at time 22) equals A/2. Symbolically,
M(22) = Ce^kt = Ae^{k · 22} = Ae^22k = A/2
Þ e^22k = 0.5
Þ 22k = ln(0.5)
Þ k = ln(0.5)/22 » –0.03150669.
As you remember from Precal, the negative sign signifies exponential decay instead of growth.

Now that we know C and k, we know everything there is to know about function M. If we are using our calculator, we should store the value of k so that we don’t have to write it. As for A, any arbitrary value will do, and the easiest to choose is 1 (i.e., 1 chunk of matter).

Our goal is to find the time t such that M(t) = A/10. To do this, we store function M into Y₁, writing what we know in the slightly mangled form Y₁ = 1*e^(KX), Y₂ = 0.1. Then we use the root finder (or, if you prefer, 2nd CALC 5) to find the correct value of t. Answer: 73.082 days.

Proofs? There will be a few. Here are some examples of possible proof problems:

• Given an incomplete proof that each version of FTC implies the other, fill in the missing steps and/or the reasons that justify each step.
• Prove the change-of-base formula: b > 0, b ¹ 1 Þ log_b x = (ln x)/(ln b).
• Prove the handy exponential conversion formula: q > 0 Þ q^r = e^{r ln q}.
• Prove that the derivative of any even function is odd, or that the derivative of any odd function is even.
• Without using a calculator, prove that if f (x) = 3 – 2x^½, and if g(x) = f ^–1(x),
  then g′(3/2) = –3/4.
• Prove that no linear function can be a solution of the differential equation y ′ = 1/(2xy).
• Prove that in the first quadrant, the differential equation y ′ = 1/(2xy) has solutions that are all concave downward.

How about diff. eqs.? Remember that the only type you would ever be asked to solve analytically (i.e., without a calculator) would have to be separable. For non-separable diff. eqs., the best you can do for now is to use Euler’s Method to generate an approximate solution. (In general, the smaller the step size, the better the estimate.)

Try solving the separable diff. eq. y ′ = 1/(2xy) without using your calculator. Do the general solution first; then find the particular solution that passes through (1, 1). (What is the point (1, 1) called?) Then, with the assistance of your calculator, sketch the slope field, with your solution overlaid, in order to demonstrate that your solution is believable.

Then repeat the entire process using SLOPES and EULER. If you take 4 steps of size 0.1, what y value does Euler’s Method predict when x = 1.4? How close is that to the exact value? Why are the Euler’s Method estimates—at least with this differential equation—always going to be too high for points in the first quadrant?

And, finally . . . you must be able to pronounce “Euler” correctly.