Calculus AB and Calculus BC
Appendix: Formulas and Theorems for Reference
ALGEBRA
1. QUADRATIC FORMULA. The roots of the quadratic equation
ax^{2} + bx + c = 0 (a ≠ 0)
are given by
2. BINOMIAL THEOREM. If n is a positive integer, then
3. REMAINDER THEOREM. If the polynomial Q(x) is divided by (x − a) 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
In a right triangle, c^{2} = a^{2} + b^{2}. |
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:
5. Trapezoid:
6. Parallelogram: A = bh.
7. Circle: C = 2πr, A = πr^{2}.
8. Circular sector:
9. Circular arc: l = rθ.
10. Cylinder:
11. Cone:
12. Sphere:
TRIGONOMETRY
BASIC IDENTITIES
13. sin^{2} θ + cos^{2} θ = 1.
14. 1 + tan^{2} θ = sec^{2} θ.
15. 1 + cot^{2} θ = csc^{2} 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 sin α sin β.
18.
DOUBLE-ANGLE FORMULAS
19. sin 2α = 2 sin α cos α.
20. cos 2α = cos^{2} α − sin^{2} α = 2 cos^{2} α − 1 = 1 − 2 sin^{2} α.
21.
HALF-ANGLE FORMULAS
22.
23.
REDUCTION FORMULAS
24. sin (−α) = −sin α: cos (−α) = cos α.
25.
26.
27. sin (π − α) = sin α; cos (π − α) = −cos α.
28. sin (π + α) = −sin α; cos (π + α) = −cos α.
29. LAW OF COSINES. c^{2} = a^{2} + b^{2} − 2ab cos C.
30. LAW OF SINES.
31. The area sin C.
GRAPHS OF TRIGONOMETRIC FUNCTIONS
The four functions sketched above, sin, cos, csc, and sec, all have period 2π.
ANALYTIC GEOMETRY
RECTANGULAR COORDINATES
DISTANCE
32. The distance d between two points, P_{1} (x_{1}, y_{1}) and P_{2} (x_{2}, y_{2}), is given by
EQUATIONS OF THE STRAIGHT LINE
33. POINT-SLOPE FORM. Through P_{1} (x_{1}, y_{1}) and with slope m:
y − y_{1} = m(x − x_{1}).
34. SLOPE-INTERCEPT FORM. With slope m and y-intercept b:
y = mx + b.
35. TWO-POINT FORM. Through P_{1} (x_{1}, y_{1}) and P_{2} (x_{2}, y_{2}):
36. INTERCEPT FORM. With x- and y-intercepts of a and b, respectively:
37. GENERAL FORM. Ax + By + C = 0, where A and B are not both zero. If B ≠ 0, the slope is the y-intercept, the x-intercept,
DISTANCE FROM POINT TO LINE
38. Distance d between a point P(x_{1}, y_{1}) and the line Ax + By + C = 0 is
EQUATIONS OF THE CONICS
CIRCLE
39. With center at (0, 0) and radius r: x^{2} + y^{2} = r^{2}.
40. With center at (h, k) and radius r: (x − h)^{2} + (y − k)^{2} = r^{2}.
PARABOLA
41. With vertex at (0, 0) and focus at (p, 0): y^{2} = 4px.
42. With vertex at (0, 0) and focus at (0, p): x^{2} = 4py.
With vertex at (h, k) and axis
43. parallel to x-axis, focus at (h + p, k): (y − k)^{2} = 4p(x − h).
44. parallel to y-axis, focus at (h, k + p): (x − h)^{2} = 4p(y − k).
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):
46. Center at (0, 0), foci at (0, ±c), and vertices at (0, ±a):
47. Center at (h, k), major axis horizontal, and vertices at (h ± a, k):
48. Center at (h, k), major axis vertical, and vertices at (h, k ± a):
For the ellipse, a^{2} = b^{2} + c^{2}, and the eccentricity 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):
50. Center at (0, 0), foci at (0, ±c), and vertices at (0, ±a):
51. Center at (h, k), real axis horizontal, vertices at (h ± a, k):
52. Center at (h, k), real axis vertical, vertices at (h, k± a):
For the hyperbola, c^{2} = a^{2} + b^{2}, and eccentricity which is greater than 1.
POLAR COORDINATES
RELATIONS WITH RECTANGULAR COORDINATES
53. x = r cos θ;
y = r sin θ;
r^{2} = x^{2} + y^{2};
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.
58.
roses (four leaves)
59. r = cos 2θ.
60. r = sin 2θ.
cardioids (specific examples below)
61. r = a (1 ± cos θ).
62. r = a (1 ± sin θ).
63. r^{2} = cos 2θ, lemniscate, symmetric to the x-axis.
64. r = θ, (double) spiral of Archimedes
65. rθ = a (θ > 0), hyperbolic (or reciprocal) spiral
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
PROPERTIES
INVERSE PROPERTIES
f (x) = e^{x} and f ^{−1}(x) = ln x are inverses of each other:
f ^{−1}(f (x)) = f (f ^{−1}(x)) = x;
ln e^{x} = e^{ln x} = x(x > 0).
GRAPHS