Formulas

Polar area: ½ ò r² dq

Exponential growth: Diffeq. y' = ky has solution y = ce^kx

Logistic growth: Diffeq. y' = ky(A – y) has solution y = A / (1 + ce^–Akx)

Volume by disks: ò pr² dx if axis of rotation is parallel to x-axis (use dy if parallel to y-axis)

Volume by shells: ò 2pr dx if axis of rotation is parallel to y-axis (use dy if parallel to x-axis)

Average value of f on [a, b] is ò_a^b f(x) dx / (b – a).

cos² x = ½(1 + cos 2x)

sin² x = ½(1 – cos 2x)

f(x) » f(a) + f ' (a)(x – a) + [f '' (a)/2!] (x – a)² + . . . + [f ⁽ⁿ⁾(a) / n!] (x – a)ⁿ

Arc Length

Regular: ò Ö (1+(dy/dx)²) dx

Parametric: ò Ö ((dx/dt)² + (dy/dt)²) dt

Polar: ò Ö (r² + (dr/dq )²) dq
(If needed, can also be derived from parametric using parameter q , with x = r cos q and y = r sin q.)

MVT

If f is differentiable on (a, b) and continuous on [a, b],

then $ c Î (a, b) ' f ' (c) = (f(b) – f(a)) / (b – a).

In words: There is at least one place where the slope of the tangent equals the average slope between a and b. Conditions are crucial: f differentiable on (a, b) and continuous on [a, b].

Definitions

Derivative at a point: f ' (c) = lim_x®c [ (f(x) – f(c)) / (x – c) ]

Derivative function: f ' (x) = lim_h®0 [ (f(x + h) – f(x)) / h ]

Maclaurin and Taylor Series

e^x = 1 + x + x²/2! + x³/3! + . . . [converges for all x]

sin x = x – x³/3! + x⁵/5! – x⁷/7! + x⁹/9! – . . . [converges for all x]

cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + x⁸/8! – . . . [converges for all x]

ln x = (x – 1) – (x – 1)²/2 + (x – 1)³/3 – (x – 1)⁴/4 + . . . [converges if 0 < x £ 2]

1 / (1–x) = 1 + x + x² + x³ + x⁴ + . . . [converges if |x| < 1 since geometric series]

AST Error Bound

|R_n| < |t_n+1|
In words: In a convergent alternating series with terms of decreasing absolute value, the magnitude of the error is bounded by the first term not taken.

Lagrange Error Bound

If M is the maximum absolute value of f ^{(n + 1)}(x) on the interval between a and x, then the nth-degree Taylor polynomial that approximates f(x) has a remainder (error) bounded by |R_n| £ M |x – a|ⁿ + 1 / (n + 1)!

Techniques for Multiple Choice

Techniques for Free Response

1. Pace yourself. Keep brainpower in reserve for free response.
2. Get the answer any way you can. Work is not graded for multiple choice.
3. Circle the hard ones and come back to them later.
4. If you can positively rule out one or more choices, choose randomly from those that remain. Do not make an educated guess, since you will probably fall into a trap.
5. In an integral problem where a common mistake would be to be off by a factor of 2, look closely at the two choices that differ by a factor of 2. The correct answer is probably one of these.

1. If you can’t get part (a), skip it and do the others. Part (a) may be worth only a point.
2. A few lines of accurate work are usually enough. Long, tedious problems are rare.
3. Keep intermediate results in full precision (can use STO to save to a variable). Write ". . ." on paper if you are omitting some digits.
4. Round final answers to 3 decimal places.
5. Show all steps. Don’t make leaps of logic. You may use Þ and \ symbols as transitions (e.g., "f diff. at x (given) Þ f cont. at x"), but it’s easier just to put one thought on each line.
6. Don’t waste time erasing large areas. Just mark them out with a quick X.
7. Avoid using the word it.