FORMULAS AND THEOREMS - Appendixes - 5 Steps to a 5 AP Calculus AB & BC

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)

Image

2. Distance Formula:

Image

3. Equation of a Circle:

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

4. Equation of an Ellipse:

Image center at (0, 0).

Image center at (h, k).

5. Area and Volume Formulas:

Image

Image

6. Special Angles:

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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.

Image

Image

8. Pythagorean Identities:

• sin2 θ + cos2 θ = 1

• 1 + tan2 θ = sec2θ

• 1 + cot2 θ = csc2 θ

9. Limits:

Image

10. Rules of Differentiation:

a. Definition of the Derivative of a Function:

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b. Power Rule: Image

c. Sum & Difference Rules:

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d. Product Rule:

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e. Quotient Rule:

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

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f. Chain Rule:

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11. Inverse Function and Derivatives:

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12. Differentiation and Integration Formulas: Integration Rules

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Differentiation Formulas:

Image

Image

Integration Formulas:

Image

Image

More Integration Formulas:

Image

Image

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 Image (and follow LIPET Rule).

14. The Fundamental Theorems of Calculus

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where F′(x) = f(x).

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15. Trapezoidal Approximation:

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16. Average Value of a Function:

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17. Mean Value Theorem:

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Mean Value Theorem for Integrals:

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in (a, b).

18. Area Bounded by 2 Curves:

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where f(x)g(x).

19. Volume of a Solid with Known Cross Section:

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where A(x) is the cross section.

20. Disc Method:

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21. Using the Washer Method:

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

22. Distance Traveled Formulas:

• Position Function: Image

• Velocity: Image; Image

• Acceleration: Image

• Speed: |v(t)|

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

• Total Distance Traveled from t1 to

Image

23. Business Formulas:

Image

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

24. Exponential Growth/Decay Formulas:

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25. Logistic Growth Models

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26. Integration by Parts

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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 Image and Image, or Image and Image, then Image.

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

28. Derivatives of Parametric Functions

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and

Image

29. Vector Functions

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

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Image

30. Arc Length of a Curve

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(b) Parametric Equations

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

(c) Polar Equations

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31. Polar Curves

(a) Slope of r = f θ at (r,θ)

Image

or written as Image.

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

Image

(c) Area between two Polar Curves:

Given r1 = f (θ) and r2 = g (θ), 0 ≤ r1r2 and αθβ, the area between r1 and r2:

Image

32. Series and Convergence

(a) Geometric Series

Image

if |r| ≥ 1, series diverges;

if |r| < 1, series converges and the Image

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

(b) p-Series Image

if p > 1, series converges;

if 0 < p ≤ 1, series diverges.

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

Series converges if

(1) a1a2a3 … ≥ ak ≥ … and

(2) Image

(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 − Sn | ≤ an+1.

(d) Harmonic Series

Image diverges.

Alternating Harmonic Series

Image converges.

Image

33. Convergence Tests for Series

(a) Divergence Test

Given a series Image, if Image then the series diverges.

(b) Ratio Test for Absolute Convergence

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

(1) converges absolutely if p < 1;

(2) diverges if p > 1;

(3) needs more testing if p = 1.

(c) Comparison Test

Given Image and Image with ak > 0, bk > 0 for all ks, and a1b1, a2b2, … ak for all ks:

(1) If Image converges, then Image converges.

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

(2) If Image diverges, then Image diverges.

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

(d) Limit Comparison Test

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

(e) Integral Test

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

34. Maclaurin Series

Image

Image

Image

Image

Image

Image

Image

Image

35. Taylor Series

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Partial Sum

Image

Rn (error for Image, 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

Image

(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.