5 Steps to a 5 AP Calculus AB & BC, 2012-2013 Edition (2011)

Appendixes

FORMULAS AND THEOREMS

1. Quadratic Formula:

ax² + bx + c = 0(a ≠ 0)

2. Distance Formula:

3. Equation of a Circle:

x² + y² = r² center at (0, 0) and radius = r.

4. Equation of an Ellipse:

center at (0, 0).

center at (h, k).

5. Area and Volume Formulas:

6. Special Angles:

7. Double Angles:

• sin 2θ = 2 sin θ cos θ

• cos 2θ = cos² θ − sin² θ or 1 − 2 sin² θ or 2 cos² θ − 1.

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8. Pythagorean Identities:

• sin² θ + cos² θ = 1

• 1 + tan² θ = sec²θ

• 1 + cot² θ = csc² θ

9. Limits:

10. Rules of Differentiation:

a. Definition of the Derivative of a Function:

b. Power Rule:

c. Sum & Difference Rules:

d. Product Rule:

e. Quotient Rule:

Summary of Sum, Difference, Product, and Quotient Rules:

f. Chain Rule:

11. Inverse Function and Derivatives:

12. Differentiation and Integration Formulas: Integration Rules

Differentiation Formulas:

Integration Formulas:

More Integration Formulas:

Note: After evaluating an integral, always check the result by taking the derivative of the answer (i.e., taking the derivative of the antiderivative).

13. Intergration by parts (and follow LIPET Rule).

14. The Fundamental Theorems of Calculus

where F′(x) = f(x).

15. Trapezoidal Approximation:

16. Average Value of a Function:

17. Mean Value Theorem:

Mean Value Theorem for Integrals:

in (a, b).

18. Area Bounded by 2 Curves:

where f(x) ≥ g(x).

19. Volume of a Solid with Known Cross Section:

where A(x) is the cross section.

20. Disc Method:

21. Using the Washer Method:

where f(x) = outer radius and g(x) = inner radius.

22. Distance Traveled Formulas:

• Position Function:

• Velocity: ;

• Acceleration:

• Speed: |v(t)|

• Displacement from t1 to = s (t₂) − s (t₁).

• Total Distance Traveled from t1 to

23. Business Formulas:

P′(x), R′(x), C′(x) are the instantaneous rates of change of profit, revenue, and cost respectively.

24. Exponential Growth/Decay Formulas:

25. Logistic Growth Models

26. Integration by Parts

Note: When matching u and dv, begin with u and follow the order of the acronym LIPET (Logarithmic, Inverse Trigonometric, Polynomial, Exponential, and Trigonometric functions).

27. L’Hopital’s Rule

If and , or and , then .

(Note that “a” may represent a constant or ±∞.)

28. Derivatives of Parametric Functions

and

29. Vector Functions

Given r (t) = f (t) i + g (t) j:

30. Arc Length of a Curve

(b) Parametric Equations

x = f(t) and y = g(t)

(c) Polar Equations

31. Polar Curves

(a) Slope of r = f θ at (r,θ)

or written as .

(b) Given r = f (θ) and α ≤ θ ≤ β, the area of the region between the curve, the origin, θ = α and θ = β:

(c) Area between two Polar Curves:

Given r₁ = f (θ) and r₂ = g (θ), 0 ≤ r₁ ≤ r₂ and α ≤ θ ≤ β, the area between r₁ and r₂:

32. Series and Convergence

(a) Geometric Series

if |r| ≥ 1, series diverges;

if |r| < 1, series converges and the

(Partial sum of the first n terms: for all geometric series.)

(b) p-Series

if p > 1, series converges;

if 0 < p ≤ 1, series diverges.

(c) Alternating Series or , where a_k > = 0 for all ks.

Series converges if

(1) a₁ ≥ a₂ ≥ a₃ … ≥ a_k ≥ … and

(2)

(Note: Both conditions must be satisfied before the series converges.)

Error Approximation:

If S = sum of an alternating series, and Sn = partial sum of n terms, then | error | = | S − S_n | ≤ a_n+1.

(d) Harmonic Series

diverges.

Alternating Harmonic Series

converges.

33. Convergence Tests for Series

(a) Divergence Test

Given a series , if then the series diverges.

(b) Ratio Test for Absolute Convergence

Given where a_k ≠ 0 for all ks and let , then the series

(1) converges absolutely if p < 1;

(2) diverges if p > 1;

(3) needs more testing if p = 1.

(c) Comparison Test

Given and with a_k > 0, b_k > 0 for all ks, and a₁ ≤ b₁, a₂ ≤ b₂, … a_k for all ks:

(1) If converges, then converges.

(Note: that if the bigger series converges, then the smaller series converges)

(2) If diverges, then diverges.

(Note: that if the smaller series diverges, then the bigger series diverges)

(d) Limit Comparison Test

Given and with a_k > 0, b_k > 0 for all ks, and let , if 0 < p < ∞, then both series converge or both series diverge.

(e) Integral Test

Given for all ks, and a_k = f (k) for some function f(x), if the function f is positive, continuous, and decreasing for all x ≥ 1, then and either both converge or both diverge.

34. Maclaurin Series

35. Taylor Series

Partial Sum

R_n (error for , if x > a, c ∈ (a, x), or if x < a, c ∈ (x, a), of if x = a, c = a.

36. Testing a Power Series for Convergence Given

(1) Use Ratio Test to find values of x for absolute convergence.

(2) Exactly one of the following cases will occur:

(a) Series converges only at x = a.

(b) Series converges absolutely for all x ∈ R.

(c) Series converges on all x ∈ (a − R, a + R) and diverges for x < a − R or x > a+ R. At the endpoints x = a − R and x = a+ R, use an Integral Test, an Alternating Series Test, or a Comparison Test to test for convergence.