Calculus AB and Calculus BC

Appendix: Formulas and Theorems for Reference

ALGEBRA

1. QUADRATIC FORMULA. The roots of the quadratic equation

ax2 + bx + c = 0 (a ≠ 0)

are given by

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2. BINOMIAL THEOREM. If n is a positive integer, then

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3. REMAINDER THEOREM. If the polynomial Q(x) is divided by (xa) until a constant remainder R is obtained, then R = Q(a). In particular, if a is a root of Q(x) = 0, then Q(a) = 0.

GEOMETRY

The sum of the angles of a triangle is equal to a straight angle (180°).

PYTHAGOREAN THEOREM

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In a right triangle,

c2 = a2 + b2.

In the following formulas,

A

is area

B

is area of base

S

surface area

r

radius

V

volume

C

circumference

b

base

l

arc length

h

height or altitude

θ

central angle (in radians)

s

slant height

4. Triangle: Image

5. Trapezoid: Image

6. Parallelogram: A = bh.

7. Circle: C = 2πr, A = πr2.

8. Circular sector: Image

9. Circular arc: l = rθ.

10. Cylinder:

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11. Cone:

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12. Sphere:
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TRIGONOMETRY

BASIC IDENTITIES

13. sin2 θ + cos2 θ = 1.

14. 1 + tan2 θ = sec2 θ.

15. 1 + cot2 θ = csc2 θ.

SUM AND DIFFERENCE FORMULAS

16. sin (α ± β) = sin α cos β ± cos α sin β.

17. cos (α ± β) = cos α cos β Image sin α sin β.

18. Image

DOUBLE-ANGLE FORMULAS

19. sin 2α = 2 sin α cos α.

20. cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α.

21. Image

HALF-ANGLE FORMULAS

22. Image

23. Image

REDUCTION FORMULAS

24. sin (−α) = −sin α: cos (−α) = cos α.

25. Image

26. Image

27. sin (π − α) = sin α; cos (π − α) = −cos α.

28. sin (π + α) = −sin α; cos (π + α) = −cos α.

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29. LAW OF COSINES. c2 = a2 + b2 − 2ab cos C.

30. LAW OF SINES. Image

31. The area Image sin C.

GRAPHS OF TRIGONOMETRIC FUNCTIONS

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The four functions sketched above, sin, cos, csc, and sec, all have period 2π.

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ANALYTIC GEOMETRY

RECTANGULAR COORDINATES

DISTANCE

32. The distance d between two points, P1 (x1, y1) and P2 (x2, y2), is given by

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EQUATIONS OF THE STRAIGHT LINE

33. POINT-SLOPE FORM. Through P1 (x1, y1) and with slope m:

yy1 = m(xx1).

34. SLOPE-INTERCEPT FORM. With slope m and y-intercept b:

y = mx + b.

35. TWO-POINT FORM. Through P1 (x1, y1) and P2 (x2, y2):

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36. INTERCEPT FORM. With x- and y-intercepts of a and b, respectively:

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37. GENERAL FORM. Ax + By + C = 0, where A and B are not both zero. If B ≠ 0, the slope is Image the y-intercept, Image the x-intercept, Image

DISTANCE FROM POINT TO LINE

38. Distance d between a point P(x1, y1) and the line Ax + By + C = 0 is

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EQUATIONS OF THE CONICS
CIRCLE

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

40. With center at (h, k) and radius r: (xh)2 + (yk)2 = r2.

PARABOLA

41. With vertex at (0, 0) and focus at (p, 0): y2 = 4px.

42. With vertex at (0, 0) and focus at (0, p): x2 = 4py.

With vertex at (h, k) and axis

43. parallel to x-axis, focus at (h + p, k): (yk)2 = 4p(xh).

44. parallel to y-axis, focus at (h, k + p): (xh)2 = 4p(yk).

ELLIPSE

With major axis of length 2a, minor axis of length 2b, and distance between foci of 2c:

45. Center at (0, 0), foci at (±c, 0), and vertices at (±a, 0):

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46. Center at (0, 0), foci at (0, ±c), and vertices at (0, ±a):

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47. Center at (h, k), major axis horizontal, and vertices at (h ± a, k):

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48. Center at (h, k), major axis vertical, and vertices at (h, k ± a):

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For the ellipse, a2 = b2 + c2, and the eccentricity Image which is less than 1.

HYPERBOLA

With real (transverse) axis of length 2a, imaginary (conjugate) axis of length 2b, and distance between foci of 2c:

49. Center at (0, 0), foci at (±c, 0), and vertices at (±a, 0):

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50. Center at (0, 0), foci at (0, ±c), and vertices at (0, ±a):

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51. Center at (h, k), real axis horizontal, vertices at (h ± a, k):

Image

52. Center at (h, k), real axis vertical, vertices at (h, k± a):

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For the hyperbola, c2 = a2 + b2, and eccentricity Image which is greater than 1.

POLAR COORDINATES

RELATIONS WITH RECTANGULAR COORDINATES

53. x = r cos θ;

y = r sin θ;

r2 = x2 + y2;

Image

Image

SOME POLAR EQUATIONS

54. r = a circle, center at pole, radius a.

55. r = 2a cos θ circle, center at (a, 0), radius a.

56. r = 2a sin θ circle, center at (0, a), radius a.

57. Image

58. Image

roses (four leaves)

59. r = cos 2θ.

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60. r = sin 2θ.

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cardioids (specific examples below)

61. r = a (1 ± cos θ).

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62. r = a (1 ± sin θ).

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63. r2 = cos 2θ, lemniscate, symmetric to the x-axis.

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64. r = θ, (double) spiral of Archimedes

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65. rθ = a (θ > 0), hyperbolic (or reciprocal) spiral

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EXPONENTIAL AND LOGARITHMIC FUNCTIONS

PROPERTIES

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INVERSE PROPERTIES

f (x) = ex and f −1(x) = ln x are inverses of each other:

f −1(f (x)) = f (f −1(x)) = x;

ln ex = eln x = x(x > 0).

GRAPHS

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