1 Algebra (vii) |xn | = |x|n 2 Functions · Horizontal aysmptote (if domain is unlimited at
x |x| ±∞) if:
(viii) =
1.1 Exponential Properties y |y| 2.1 Domain limx→+∞ f (x) = k (right y = k)
(ix) |a − b| = b − a, if a ≤ b limx→−∞ f (x) = h (left y = h).
(i) x0 = 1 · Fractions denominator 6= 0.
(x) |a + b| ≤ |a| + |b| · Oblique aysmptote (if domain is unlimited at
(ii) xn xm = xn+m
(xi) |a| − |b| ≤ |a − b| · Logarithms if the base is a number, the argument ±∞) if:
xn 1
(iii) xm
= xn−m = xm−n
must be > 0, if the base depends on a variable, the f (x)
limx→+∞ x = m ∧ limx→+∞ [f (x) − mx] = q
(iv) (xn )m = xnm base must be > 0∧ 6= 1. (right at y = mx + q)
1.5 Factorization
n f (x)
x xn · Roots with even index, the argument must be limx→−∞ x = m ∧ limx→−∞ [f (x) − mx] = q
(v) y
= yn (i) x2 − a2 = (x + a)(x − a) ≥ 0, for roots with odd index the domain is R. (left at y = mx + q).
(vi) x−n = 1 (ii) x2 + 2ax + a2 = (x + a)2
xn · Arccos/Arcsin the agrument must be ∈ [−1, 1].
(vii) 1
= xn (iii) x − 2ax + a2 = (x − a)2
2
2.6 Monotonicity
x−n For other trig functions we use trig properties to
−n (iv) x2 + (a + b)x + ab = (x + a)(x + b) change them to cos and sin.
x y n yn A funciton f is:
(viii) y
= x = xn (v) x3 + 3ax2 + 3a2 x + a3 = (x + a)3
1 n √ · Exponential base > 0. · Monotonically increasing if:
(ix) x
n
m = xm
1
= (xn ) m = m
xn (vi) x3 − 3ax2 + 3a2 x − a3 = (x − a)3
∀x, y : x ≤ y ⇒ f (x) ≤ f (y)
(vii) x3 + a3 = (x + a)(x2 − ax + a2 )
2.2 Parity · Monotonically decreasing if:
(viii) x3 − a3 = (x − a)(x2 + ax + a2 )
1.2 Logarithm Properties ∀x, y : x ≤ y ⇒ f (x) ≥ f (y)
(ix) x2n − a2n = (xn − an )(xn + an ) We consider the partiy of the function only if
(i) logn (0) = Undefined Dom(f ) is mirrored on the origin: · Strictly increasing if:
(ii) logn (1) = 0 (Dom(f ) = [−2, 2] ∨ (−∞, ∞) ∨ (−∞, −1] ∪ [1, ∞]). ∀x, y : x < y ⇒ f (x) < f (y)
1.6 Complete The Square
(iii) logn (n) = 1 · Strictly decreasing if:
ax2 + bx + c = 0 ⇒ a(x + d)2 + e = 0 · Even function (with respect to the y axis) if:
∀x, y : x < y ⇒ f (x) > f (y)
(iv) logn (nx ) = x f (−x) = f (x).
b
(v) nlogn (x) = x · d= 2a · Odd function (with respect to the origin) if: 2.7 Max, Min
(vi) logn (xr ) = r logn (x) 6= logrn (x) = (logn (x))r 2 f (−x) = −f (x).
b
(vii) logn (xy) = logn (x) + logn (y)
· e=c− 4a Calculate f 0 (x) = 0, then all the solutions xi are
· In every other case the function is neither even nor our candidates, where for a small > 0:
(viii) logn x y
= logn (x) − logn (y) 1.7 Quadratic Formula
odd.
· Max if: f 0 (xi − ) > 0 ∧ f 0 (xi + ) < 0.
(ix) − logn (x) = logn x1
√
ax2 + bx + c = 0 ⇒ x=
−b± b2 −4ac
2.3 Axis Intercept · Min if: f 0 (xi − ) < 0 ∧ f 0 (xi + ) > 0.
log(x) 2a
(x) log(n)
= logn (x) · Inflection if (use sign table):
· If b2 − 4ac > 0 ⇒ Two real unequal solutions. · X intercept can be many; is calculated by solving
g(x) f 0 (xi − ) < 0 ∧ f 0 (xi + ) < 0, or
f (x) = 0. If f (x) = h(x) we solve just g(x) = 0. f 0 (xi − ) > 0 ∧ f 0 (xi + ) > 0
1.3 Radical Properties · If b2 − 4ac = 0 ⇒ Two repeated real solutions. The points are then (xi , 0).
√
n
1 If f 0 (x) > 0, then f is strictly increasing.
(i) x=x n
· If b2 − 4ac < 0 ⇒ Two complex solutions.
√ √ √ · Y intercept can be just one; is calculated by If f 0 (x) < 0, then f is strictly decreasing.
(ii) n xy = n x n y setting x = 0, the point is then (0, f (0)). If If f 0 (x) = 0 f is constant.
p √ √ x=0∈ / Dom(f ) there is no Y intercept.
(iii) m n x = mn x
√
q
(iv) n x
nx
= n
√
2.8 Convexity
y y
√ 2.4 Sign · Convex (∪) if: f 00 (x) > 0
n n
(v) x = x, if n is odd
√ The sign can only change when there is a x inter- · Concave (∩) if: f 00 (x) < 0
(vi) n xn = |x|, if n is even cept (if the function is continuous), thus if we solve
f (x) ≥ 0 we get both the X intercepts and where
2.9 Inflection Points
1.4 Absolute Value Properties the function is positive.
( Calculate f 00 (x) = 0, then all the solutions xi are
x if x ≥ 0 our candidates (except where f (x) is not defined),
(i) |x| = 2.5 Asymptotes/Holes
−x if x < 0 where for a small > 0:
(ii) |x| ≥ 0 · Hole at point (x0 , fsemplified (x0 )) if plugging the · Increasing Inflection if:
(iii) | − x| = |x| critical point x0 in the numerator of f gives 00 . f 00 (xi − ) < 0 ∧ f 00 (xi + ) > 0
(iv) |ca| = c|a|, if c > 0 · Vertical asymptote at a critical point x0 if: · Decreasing Inflection if:
(v) |xy| = |x||y| limx→x− f (x) = ±∞ (left at x = x0 ) f 00 (xi − ) > 0 ∧ f 00 (xi + ) < 0
0
(vi) |x2 | = x2 limx→x+ f (x) = ±∞ (right at x = x0 ). · Otherwise nothing happens on xi .
0
Flavio Schneider · Cheat Sheet Page 1
AP
, 3 Trigonometry 3.5 Reciprocal Identities 3.12 Half-Angle Identities 3.19 Degrees
1 q
(i) cot(x) = x 1−cos(x)
3.1 Unit Circle
tan(x) (i) sin 2
=± 2
y
1
(ii) csc(x) = sin(x)
q
1+cos(x)
x
(ii) cos 2
=± 2
(0, 1)
1
(iii) sec(x) = cos(x) q
− 12 ,
√
2
3
1
√
2, 2
3
x
1−cos(x)
(iii) tan 2
=± 2
−
√
2
√
2
√
2
√
2
2 , 2 π 2 , 2
2
3.6 Quotient Identities (iv) tan x
=
1−cos(x) √
2π
3
π
3 √
2 sin(x) − 3 1
2 , 2
3π
4 90◦
π
4
3 1
2 , 2
sin(x) sin(x)
120◦ 60◦
(i) tan(x) = cos(x) (v) tan x
= 5π
6
π
6
2 1+cos(x) 150◦ 30◦
cos(x)
(ii) cot(x) = sin(x) (−1, 0) (1, 0)
3.13 Sum-to-Product Formulas π 180◦ 0◦ ◦
360 2π x
3.7 Sum Identities (i) sin(x) + sin(y) = 2 sin x+y
2
cos x−y
2 210◦ 330◦
7π 11π
(i) sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
6 6
x−y x+y
(ii) sin(x) − sin(y) = 2 sin cos 2 2
−
√
3 1
5π
240 ◦
270◦
300 ◦
7π
√
3 1
2 , −2 2 , −2
(ii) cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
4
4π 5π
4
3 3
(iii) tan(x + y) =
tan(x)+tan(y) (iii) cos(x) + cos(y) = 2 cos x+y2
cos x−y 2
−
√
2
√
2 ,− 2
2
3π
2
√
2
√
2 ,− 2
2
1−tan(x) tan(y) √ √
− 12 , − 3 1 3
(iv) cos(x) − cos(y) = −2 sin x+y 2
cos x−y 2
2 2, − 2
(0, −1)
3.8 Difference Identities
(i) sin(x − y) = sin(x) cos(y) − cos(x) sin(y) 3.14 Product-to-Sum Formulas
(ii) cos(x − y) = cos(x) cos(y) + sin(x) sin(y) (i) sin(x) sin(y) = 1
[cos(x − y) − cos(x + y)]
2
tan(x)−tan(y)
(iii) tan(x − y) = 1+tan(x) tan(y) (ii) 1
cos(x) cos(y) = 2 [cos(x − y) + cos(x + y)]
3.2 Domain and Range
(iii) sin(x) cos(y) = 21 [sin(x + y) + sin(x − y)]
· sin : R −→ [−1, 1] 3.9 Double Angle Identities
(iv) cos(x) sin(y) = 21 [sin(x + y) − sin(x − y)]
· cos : R −→ [−1, 1] (i) sin(2x) = 2 sin(x) cos(x)
· tan : x ∈ R x 6= π
+ kπ −→ R (ii) cos(2x) = cos2 (x) − sin2 (x) 3.15 Tangent expression
2
cos(2x)+1
(iii) cos(2x) = 2 cos2 (x) − 1 ⇒ cos2 (x) = h i
· cot : {x ∈ R | x 6= kπ} −→ R 2
If u = tan( x2 ) : dx = 2
du
1−cos(2x) 1+u2
(iv) cos(2x) = 1 − 2 sin2 (x) ⇒ sin2 (x) = 2
· csc : {x ∈ R | x 6= kπ} −→ R \ (−1, 1)
2 tan(x) 1−u2
(v) tan(2x) = (i) cos(x) = 1+u2
6 π2 + kπ −→ R \ (−1, 1) 1−tan2 (x)
· sec : x ∈ R x =
2u
(ii) sin(x) = 1+u 2
· sin−1 : [−1, 1] −→ − π2 , π2
3.10 Co-Function Identities (iii) 2u
tan(x) = 1−u2
π
· cos−1 : [−1, 1] −→ [0, π]
(i) sin 2
− x = cos(x)
π
3.16 Hyperbolic Functions
· tan−1 : R −→ − π2 , π2
(ii) cos 2
− x = sin(x)
π
(iii) tan − x = cot(x) ex −e−x
2 (i) sinh(x) = 2
3.3 Pythagorean Identities (iv) cot π
− x = tan(x)
2 ex +e−x
(ii) cosh(x) =
(i) sin2 (x) + cos2 (x) = 1 π 2
(v) csc 2
− x = sec(x)
ex −e−x
(ii) tan2 (x) + 1 = sec2 (x) π
(iii) tanh(x) =
(vi) sec 2
− x = csc(x) ex +e−x
(iii) 1 + cot2 (x) = csc2 (x)
3.11 Even-Odd Identities 3.17 Laws of Sines
3.4 Periodicity Identities sin(α) sin(β) sin(γ)
(i) sin(−x) = − sin(x) (i) a
= b
= c
(i) sin(x ± 2π) = sin(x)
(ii) cos(−x) = cos(x)
(ii) cos(x ± 2π) = cos(x)
(iii) tan(−x) = − tan(x) 3.18 Laws of Cosines
(iii) tan(x ± π) = tan(x)
(iv) cot(x ± π) = cot(x) (iv) cot(−x) = − cot(x) (i) a2 = b2 + c2 − 2bc cos(α)
(v) csc(x ± 2π) = csc(x) (v) csc(−x) = − csc(x) (ii) b2 = a2 + c2 − 2ac cos(β)
(vi) sec(x ± 2π) = sec(x) (vi) sec(−x) = sec(x) (iii) c2 = a2 + b2 − 2ab cos(γ)
Flavio Schneider Analysis I · Cheat Sheet Page 2
x |x| ±∞) if:
(viii) =
1.1 Exponential Properties y |y| 2.1 Domain limx→+∞ f (x) = k (right y = k)
(ix) |a − b| = b − a, if a ≤ b limx→−∞ f (x) = h (left y = h).
(i) x0 = 1 · Fractions denominator 6= 0.
(x) |a + b| ≤ |a| + |b| · Oblique aysmptote (if domain is unlimited at
(ii) xn xm = xn+m
(xi) |a| − |b| ≤ |a − b| · Logarithms if the base is a number, the argument ±∞) if:
xn 1
(iii) xm
= xn−m = xm−n
must be > 0, if the base depends on a variable, the f (x)
limx→+∞ x = m ∧ limx→+∞ [f (x) − mx] = q
(iv) (xn )m = xnm base must be > 0∧ 6= 1. (right at y = mx + q)
1.5 Factorization
n f (x)
x xn · Roots with even index, the argument must be limx→−∞ x = m ∧ limx→−∞ [f (x) − mx] = q
(v) y
= yn (i) x2 − a2 = (x + a)(x − a) ≥ 0, for roots with odd index the domain is R. (left at y = mx + q).
(vi) x−n = 1 (ii) x2 + 2ax + a2 = (x + a)2
xn · Arccos/Arcsin the agrument must be ∈ [−1, 1].
(vii) 1
= xn (iii) x − 2ax + a2 = (x − a)2
2
2.6 Monotonicity
x−n For other trig functions we use trig properties to
−n (iv) x2 + (a + b)x + ab = (x + a)(x + b) change them to cos and sin.
x y n yn A funciton f is:
(viii) y
= x = xn (v) x3 + 3ax2 + 3a2 x + a3 = (x + a)3
1 n √ · Exponential base > 0. · Monotonically increasing if:
(ix) x
n
m = xm
1
= (xn ) m = m
xn (vi) x3 − 3ax2 + 3a2 x − a3 = (x − a)3
∀x, y : x ≤ y ⇒ f (x) ≤ f (y)
(vii) x3 + a3 = (x + a)(x2 − ax + a2 )
2.2 Parity · Monotonically decreasing if:
(viii) x3 − a3 = (x − a)(x2 + ax + a2 )
1.2 Logarithm Properties ∀x, y : x ≤ y ⇒ f (x) ≥ f (y)
(ix) x2n − a2n = (xn − an )(xn + an ) We consider the partiy of the function only if
(i) logn (0) = Undefined Dom(f ) is mirrored on the origin: · Strictly increasing if:
(ii) logn (1) = 0 (Dom(f ) = [−2, 2] ∨ (−∞, ∞) ∨ (−∞, −1] ∪ [1, ∞]). ∀x, y : x < y ⇒ f (x) < f (y)
1.6 Complete The Square
(iii) logn (n) = 1 · Strictly decreasing if:
ax2 + bx + c = 0 ⇒ a(x + d)2 + e = 0 · Even function (with respect to the y axis) if:
∀x, y : x < y ⇒ f (x) > f (y)
(iv) logn (nx ) = x f (−x) = f (x).
b
(v) nlogn (x) = x · d= 2a · Odd function (with respect to the origin) if: 2.7 Max, Min
(vi) logn (xr ) = r logn (x) 6= logrn (x) = (logn (x))r 2 f (−x) = −f (x).
b
(vii) logn (xy) = logn (x) + logn (y)
· e=c− 4a Calculate f 0 (x) = 0, then all the solutions xi are
· In every other case the function is neither even nor our candidates, where for a small > 0:
(viii) logn x y
= logn (x) − logn (y) 1.7 Quadratic Formula
odd.
· Max if: f 0 (xi − ) > 0 ∧ f 0 (xi + ) < 0.
(ix) − logn (x) = logn x1
√
ax2 + bx + c = 0 ⇒ x=
−b± b2 −4ac
2.3 Axis Intercept · Min if: f 0 (xi − ) < 0 ∧ f 0 (xi + ) > 0.
log(x) 2a
(x) log(n)
= logn (x) · Inflection if (use sign table):
· If b2 − 4ac > 0 ⇒ Two real unequal solutions. · X intercept can be many; is calculated by solving
g(x) f 0 (xi − ) < 0 ∧ f 0 (xi + ) < 0, or
f (x) = 0. If f (x) = h(x) we solve just g(x) = 0. f 0 (xi − ) > 0 ∧ f 0 (xi + ) > 0
1.3 Radical Properties · If b2 − 4ac = 0 ⇒ Two repeated real solutions. The points are then (xi , 0).
√
n
1 If f 0 (x) > 0, then f is strictly increasing.
(i) x=x n
· If b2 − 4ac < 0 ⇒ Two complex solutions.
√ √ √ · Y intercept can be just one; is calculated by If f 0 (x) < 0, then f is strictly decreasing.
(ii) n xy = n x n y setting x = 0, the point is then (0, f (0)). If If f 0 (x) = 0 f is constant.
p √ √ x=0∈ / Dom(f ) there is no Y intercept.
(iii) m n x = mn x
√
q
(iv) n x
nx
= n
√
2.8 Convexity
y y
√ 2.4 Sign · Convex (∪) if: f 00 (x) > 0
n n
(v) x = x, if n is odd
√ The sign can only change when there is a x inter- · Concave (∩) if: f 00 (x) < 0
(vi) n xn = |x|, if n is even cept (if the function is continuous), thus if we solve
f (x) ≥ 0 we get both the X intercepts and where
2.9 Inflection Points
1.4 Absolute Value Properties the function is positive.
( Calculate f 00 (x) = 0, then all the solutions xi are
x if x ≥ 0 our candidates (except where f (x) is not defined),
(i) |x| = 2.5 Asymptotes/Holes
−x if x < 0 where for a small > 0:
(ii) |x| ≥ 0 · Hole at point (x0 , fsemplified (x0 )) if plugging the · Increasing Inflection if:
(iii) | − x| = |x| critical point x0 in the numerator of f gives 00 . f 00 (xi − ) < 0 ∧ f 00 (xi + ) > 0
(iv) |ca| = c|a|, if c > 0 · Vertical asymptote at a critical point x0 if: · Decreasing Inflection if:
(v) |xy| = |x||y| limx→x− f (x) = ±∞ (left at x = x0 ) f 00 (xi − ) > 0 ∧ f 00 (xi + ) < 0
0
(vi) |x2 | = x2 limx→x+ f (x) = ±∞ (right at x = x0 ). · Otherwise nothing happens on xi .
0
Flavio Schneider · Cheat Sheet Page 1
AP
, 3 Trigonometry 3.5 Reciprocal Identities 3.12 Half-Angle Identities 3.19 Degrees
1 q
(i) cot(x) = x 1−cos(x)
3.1 Unit Circle
tan(x) (i) sin 2
=± 2
y
1
(ii) csc(x) = sin(x)
q
1+cos(x)
x
(ii) cos 2
=± 2
(0, 1)
1
(iii) sec(x) = cos(x) q
− 12 ,
√
2
3
1
√
2, 2
3
x
1−cos(x)
(iii) tan 2
=± 2
−
√
2
√
2
√
2
√
2
2 , 2 π 2 , 2
2
3.6 Quotient Identities (iv) tan x
=
1−cos(x) √
2π
3
π
3 √
2 sin(x) − 3 1
2 , 2
3π
4 90◦
π
4
3 1
2 , 2
sin(x) sin(x)
120◦ 60◦
(i) tan(x) = cos(x) (v) tan x
= 5π
6
π
6
2 1+cos(x) 150◦ 30◦
cos(x)
(ii) cot(x) = sin(x) (−1, 0) (1, 0)
3.13 Sum-to-Product Formulas π 180◦ 0◦ ◦
360 2π x
3.7 Sum Identities (i) sin(x) + sin(y) = 2 sin x+y
2
cos x−y
2 210◦ 330◦
7π 11π
(i) sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
6 6
x−y x+y
(ii) sin(x) − sin(y) = 2 sin cos 2 2
−
√
3 1
5π
240 ◦
270◦
300 ◦
7π
√
3 1
2 , −2 2 , −2
(ii) cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
4
4π 5π
4
3 3
(iii) tan(x + y) =
tan(x)+tan(y) (iii) cos(x) + cos(y) = 2 cos x+y2
cos x−y 2
−
√
2
√
2 ,− 2
2
3π
2
√
2
√
2 ,− 2
2
1−tan(x) tan(y) √ √
− 12 , − 3 1 3
(iv) cos(x) − cos(y) = −2 sin x+y 2
cos x−y 2
2 2, − 2
(0, −1)
3.8 Difference Identities
(i) sin(x − y) = sin(x) cos(y) − cos(x) sin(y) 3.14 Product-to-Sum Formulas
(ii) cos(x − y) = cos(x) cos(y) + sin(x) sin(y) (i) sin(x) sin(y) = 1
[cos(x − y) − cos(x + y)]
2
tan(x)−tan(y)
(iii) tan(x − y) = 1+tan(x) tan(y) (ii) 1
cos(x) cos(y) = 2 [cos(x − y) + cos(x + y)]
3.2 Domain and Range
(iii) sin(x) cos(y) = 21 [sin(x + y) + sin(x − y)]
· sin : R −→ [−1, 1] 3.9 Double Angle Identities
(iv) cos(x) sin(y) = 21 [sin(x + y) − sin(x − y)]
· cos : R −→ [−1, 1] (i) sin(2x) = 2 sin(x) cos(x)
· tan : x ∈ R x 6= π
+ kπ −→ R (ii) cos(2x) = cos2 (x) − sin2 (x) 3.15 Tangent expression
2
cos(2x)+1
(iii) cos(2x) = 2 cos2 (x) − 1 ⇒ cos2 (x) = h i
· cot : {x ∈ R | x 6= kπ} −→ R 2
If u = tan( x2 ) : dx = 2
du
1−cos(2x) 1+u2
(iv) cos(2x) = 1 − 2 sin2 (x) ⇒ sin2 (x) = 2
· csc : {x ∈ R | x 6= kπ} −→ R \ (−1, 1)
2 tan(x) 1−u2
(v) tan(2x) = (i) cos(x) = 1+u2
6 π2 + kπ −→ R \ (−1, 1) 1−tan2 (x)
· sec : x ∈ R x =
2u
(ii) sin(x) = 1+u 2
· sin−1 : [−1, 1] −→ − π2 , π2
3.10 Co-Function Identities (iii) 2u
tan(x) = 1−u2
π
· cos−1 : [−1, 1] −→ [0, π]
(i) sin 2
− x = cos(x)
π
3.16 Hyperbolic Functions
· tan−1 : R −→ − π2 , π2
(ii) cos 2
− x = sin(x)
π
(iii) tan − x = cot(x) ex −e−x
2 (i) sinh(x) = 2
3.3 Pythagorean Identities (iv) cot π
− x = tan(x)
2 ex +e−x
(ii) cosh(x) =
(i) sin2 (x) + cos2 (x) = 1 π 2
(v) csc 2
− x = sec(x)
ex −e−x
(ii) tan2 (x) + 1 = sec2 (x) π
(iii) tanh(x) =
(vi) sec 2
− x = csc(x) ex +e−x
(iii) 1 + cot2 (x) = csc2 (x)
3.11 Even-Odd Identities 3.17 Laws of Sines
3.4 Periodicity Identities sin(α) sin(β) sin(γ)
(i) sin(−x) = − sin(x) (i) a
= b
= c
(i) sin(x ± 2π) = sin(x)
(ii) cos(−x) = cos(x)
(ii) cos(x ± 2π) = cos(x)
(iii) tan(−x) = − tan(x) 3.18 Laws of Cosines
(iii) tan(x ± π) = tan(x)
(iv) cot(x ± π) = cot(x) (iv) cot(−x) = − cot(x) (i) a2 = b2 + c2 − 2bc cos(α)
(v) csc(x ± 2π) = csc(x) (v) csc(−x) = − csc(x) (ii) b2 = a2 + c2 − 2ac cos(β)
(vi) sec(x ± 2π) = sec(x) (vi) sec(−x) = sec(x) (iii) c2 = a2 + b2 − 2ab cos(γ)
Flavio Schneider Analysis I · Cheat Sheet Page 2
