Final Exam Flashcards
sin²(x)
1 - cos²(x)
1/2[1 - cos(2x)]
cos²(x)
1 - sin²(x)
1/2[1 + cos(2x)]
sec²(x)
1 + tan²(x)
tan²(x)
sec²(x) - 1
1 - cos(2x))/(1 + cos(2x)
csc²(x)
1 + cot²(x)
cot²(x)
csc²(x) - 1
Pythagorean Identities
sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)
sin(a + b)
sin(a)cos(b) + sin(b)cos(a)
sin(a - b)
sin(a)cos(b) - sin(b)cos(a)
cos(a + b)
cos(a)cos(b) - sin(a)sin(b)
cos(a - b)
cos(a)cos(b) + sin(a)sin(b)
tan(a + b)
(tan(a) + tan(b))/(1 - tan(a)tan(b))
tan(a - b)
(tan(a) - tan(b))/(1 + tan(a)tan(b))
Sum and Difference Formulas
sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
sin(a - b) = sin(a)cos(b) - sin(b)cos(a)
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))
tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)tan(b))
sin(2x)
2sin(x)cos(x)
cos(2x)
cos²(x) - sin²(x)
2cos²(x) - 1
1 - 2sin²(x)
tan(2x)
(2tan(x))/(1 - tan²x)
Double Angle Formulas
sin(2x) = 2sin(x)cos(x) cos(2x) = cos²(x) - sin²(x) cos(2x) = 2cos²(x) - 1 cos(2x) = 1 - 2sin²(x) tan(2x) = (2tan(x))/(1 - tan²x)
Product to Sum Formulas
sin(a)sin(b) = 1/2[cos(a - b) - cos(a + b)] cos(a)cos(b) = 1/2[cos(a - b) + cos(a + b)] sin(a)cos(b) = 1/2[sin(a + b) + sin(a - b)] cos(a)sin(b) = 1/2[sin(a + b) - sin(a - b)]
Circular Function Definitions
0 < Ɵ < π/2 sin(Ɵ) = y/r cos(Ɵ) = x/r tan(Ɵ) = y/x csc(Ɵ) = r/y sec(Ɵ) = r/x cot(Ɵ) = x/y
Right Triangle Definitions
sin(Ɵ) = opp/hyp cos(Ɵ) = adj/hyp tan(Ɵ) = opp/adj csc(Ɵ) = hyp/opp sec(Ɵ) = hyp/adj cot(Ɵ) = adj/opp
√(a² - b²x²)
x = (a/b)sin(Ɵ)
1 - sin²(Ɵ) = cos²(Ɵ)
√(a² + b²x²)
x = (a/b)tan(Ɵ)
1 + tan²(Ɵ) = sec²(Ɵ)
√(b²x² - a²)
x = (a/b)sec(Ɵ)
sec²(Ɵ) -1 = tan²(Ɵ)
d/dx(sin(x))
cos(x)
d/dx(cos(x))
-sin(x)
d/dx(tan(x))
sec²(x)
d/dx(cot(x))
-csc²(x)
d/dx(sec(x))
sec(x)tan(x)
d/dx(csc(x))
-csc(x)cot(x)
∫cos(x)dx
sin(x) + C
∫sin(x)dx
-cos(x) + C
∫sec²(x)dx
tan(x) + C
∫csc²(x)dx
-cot(x) + C
∫sec(x)tan(x)dx
sec(x) + C
∫csc(x)cot(x)dx
-csc(x) + C
∫tan(x)dx
-ln|cos(x)| + C
∫cot(x)dx
ln|sin(x)| + C
∫sec(x)dx
ln|sec(x) + tan(x)| + C
∫csc(x)dx
ln|csc(x) - cot(x)| + C
Half Angle Formulas
sin²(x) = [1 - cos(2x)]/2 cos²(x) = [1 + cos(2x)]/2 tan²(x) = [1 - cos(2x)]/[1 + cos(2x)]
sin(x/2) = ± √[(1 - cos(x))/2)] cos(x/2) = ± √[1 + cos(x)/2)] tan(x/2) = ± √[(1 - cos(x))/(1 + cos(x))] tan(x/2) = (1 - cos(x))/(sin(x)) tan(x/2) = (sin(x))/(1 + cos(x))
sin²(x)cos²(x)
[1/2sin(2x)]²
Divergence Test
lim n→ ∞ aₙ = L
- if L ≠ 0 Σ aₙ diverges
- if L = 0 test is inconclusive
P-Series
aₙ = 1/(nᴾ), n ≥ 1
if p > 1 Σ aₙ converges
if p ≤ 1 Σ aₙ diverges
Geometric Series
aₙ = arⁿ⁻¹, n ≥ 1
if |r| < 1 Σ (n = 1, ∞) aₙ = a/ (1-r)
if |r| ≥ 1 Σ aₙ diverges
Alternating Series
aₙ = (-1)ⁿbₙ or aₙ = (-1)ⁿ⁺¹bₙ, b ≥ 0 Requirements: 1. bₙ₊₁ ≤ bₙ 2. lim n→ ∞ bₙ = 0 (Divergence Test) Σ aₙ converges
Telescoping Series
If subsequent terms cancel out previous terms in the sum. You may have to use partial fractions, properties of logarithms, etc. to put in appropriate form.
lim n→ ∞ sₙ = s
1. if s is finite Σ aₙ = s
2. if s isn’t finite Σ aₙ diverges
Comparison Test
Pick {bₙ}
- if Σ bₙ converges and 0 ≤ aₙ ≤ bₙ then Σ aₙ converges
- if Σ bₙ diverges and 0 ≤ bₙ ≤ aₙ then Σ aₙ diverges
Limit Comparison Test
Pick {bₙ}
1. lim n→ ∞ aₙ / bₙ = c where c > 0 and c is finite
2. aₙ, bₙ > 0
If Σ (n = 1, ∞) bₙ converges Σ aₙ converges
If Σ (n = 1, ∞) bₙ diverges Σ aₙ diverges
Integral Test
aₙ = f(n) Requirements for [a,∞): 1. f(x) is continuous 2. f(x) is positive 3. f(x) is decreasing if ∫ (a,∞) f(x) converges Σ (n = a, ∞) aₙ converges if ∫ (a,∞) f(x) diverges Σ aₙ diverges
Ratio Test
lim n→ ∞ |aₙ₊₁/aₙ| = L
- If L < 1 Σ aₙ absolutely converges
- if L = 1 test is inconclusive
- if L > 1 Σ aₙ diverges
Root Test
lim n→ ∞ ⁿ√|aₙ| = L
- If L < 1 Σ aₙ absolutely converges
- if L = 1 test is inconclusive
- if L > 1 Σ aₙ diverges
If c is a real, positive number, that the limit of the sequence c¹/ ⁿ →
1
If c is a real, positive number, then 1/nᶜ →
0
cⁿ/n! →
0
n¹/ ⁿ →
1
(1 + c/n)ⁿ →
eᶜ
tan⁻¹(x)
Σ((-1)ⁿx²ⁿ⁺¹)/(2n+1) from n=0 to ∞
[-1,1]
ln(1+x)
Σ((-1)ⁿxⁿ⁺¹)/(n+1) from n=0 to ∞
(-1,1]
cos(x)
Σ((-1)ⁿx²ⁿ)/(2n)! from n=0 to ∞
-∞,∞
Work
W = integral from a to b (density)(height)(area) W = 1/2kx^2 (for spring)
Average Value of a Function
f(c) = 1/(b-a) integral from a to b f(x)dx
Integration by Parts
∫udv = uv - ∫vdu
