Final Exam

Question 1

sin²(x)

Answer 1

1 - cos²(x)

1/2[1 - cos(2x)]

Question 2

cos²(x)

Answer 2

1 - sin²(x)

1/2[1 + cos(2x)]

Question 3

sec²(x)

Answer 3

1 + tan²(x)

Question 4

tan²(x)

Answer 4

sec²(x) - 1

1 - cos(2x))/(1 + cos(2x)

Question 5

csc²(x)

Answer 5

1 + cot²(x)

Question 6

cot²(x)

Answer 6

csc²(x) - 1

Question 7

Pythagorean Identities

Answer 7

sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)

Question 8

sin(a + b)

Answer 8

sin(a)cos(b) + sin(b)cos(a)

Question 9

sin(a - b)

Answer 9

sin(a)cos(b) - sin(b)cos(a)

Question 10

cos(a + b)

Answer 10

cos(a)cos(b) - sin(a)sin(b)

Question 11

cos(a - b)

Answer 11

cos(a)cos(b) + sin(a)sin(b)

Question 12

tan(a + b)

Answer 12

(tan(a) + tan(b))/(1 - tan(a)tan(b))

Question 13

tan(a - b)

Answer 13

(tan(a) - tan(b))/(1 + tan(a)tan(b))

Question 14

Sum and Difference Formulas

Answer 14

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))

Question 15

sin(2x)

Answer 15

2sin(x)cos(x)

Question 16

cos(2x)

Answer 16

cos²(x) - sin²(x)
2cos²(x) - 1
1 - 2sin²(x)

Question 17

tan(2x)

Answer 17

(2tan(x))/(1 - tan²x)

Question 18

Double Angle Formulas

Answer 18

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)

Question 19

Product to Sum Formulas

Answer 19

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)]

Question 20

Circular Function Definitions

Answer 20

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

Question 21

Right Triangle Definitions

Answer 21

sin(Ɵ) = opp/hyp
cos(Ɵ) = adj/hyp
tan(Ɵ) = opp/adj
csc(Ɵ) = hyp/opp
sec(Ɵ) = hyp/adj
cot(Ɵ) = adj/opp

Question 22

√(a² - b²x²)

Answer 22

x = (a/b)sin(Ɵ)

1 - sin²(Ɵ) = cos²(Ɵ)

Question 23

√(a² + b²x²)

Answer 23

x = (a/b)tan(Ɵ)

1 + tan²(Ɵ) = sec²(Ɵ)

Question 24

√(b²x² - a²)

Answer 24

x = (a/b)sec(Ɵ)

sec²(Ɵ) -1 = tan²(Ɵ)

Question 25

d/dx(sin(x))

Answer 25

cos(x)

Question 26

d/dx(cos(x))

Answer 26

-sin(x)

Question 27

d/dx(tan(x))

Answer 27

sec²(x)

Question 28

d/dx(cot(x))

Answer 28

-csc²(x)

Question 29

d/dx(sec(x))

Answer 29

sec(x)tan(x)

Question 30

d/dx(csc(x))

Answer 30

-csc(x)cot(x)

Question 31

∫cos(x)dx

Answer 31

sin(x) + C

Question 32

∫sin(x)dx

Answer 32

-cos(x) + C

Question 33

∫sec²(x)dx

Answer 33

tan(x) + C

Question 34

∫csc²(x)dx

Answer 34

-cot(x) + C

Question 35

∫sec(x)tan(x)dx

Answer 35

sec(x) + C

Question 36

∫csc(x)cot(x)dx

Answer 36

-csc(x) + C

Question 37

∫tan(x)dx

Answer 37

-ln|cos(x)| + C

Question 38

∫cot(x)dx

Answer 38

ln|sin(x)| + C

Question 39

∫sec(x)dx

Answer 39

ln|sec(x) + tan(x)| + C

Question 40

∫csc(x)dx

Answer 40

ln|csc(x) - cot(x)| + C

Question 41

Half Angle Formulas

Answer 41

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))

Question 42

sin²(x)cos²(x)

Answer 42

[1/2sin(2x)]²

Question 43

Divergence Test

Answer 43

lim n→ ∞ aₙ = L

1. if L ≠ 0 Σ aₙ diverges
2. if L = 0 test is inconclusive

Question 44

P-Series

Answer 44

aₙ = 1/(nᴾ), n ≥ 1
if p > 1 Σ aₙ converges
if p ≤ 1 Σ aₙ diverges

Question 45

Geometric Series

Answer 45

aₙ = arⁿ⁻¹, n ≥ 1
if |r| < 1 Σ (n = 1, ∞) aₙ = a/ (1-r)
if |r| ≥ 1 Σ aₙ diverges

Question 46

Alternating Series

Answer 46

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

Question 47

Telescoping Series

Answer 47

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

Question 48

Comparison Test

Answer 48

Pick {bₙ}

1. if Σ bₙ converges and 0 ≤ aₙ ≤ bₙ then Σ aₙ converges
2. if Σ bₙ diverges and 0 ≤ bₙ ≤ aₙ then Σ aₙ diverges

Question 49

Limit Comparison Test

Answer 49

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

Question 50

Integral Test

Answer 50

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

Question 51

Ratio Test

Answer 51

lim n→ ∞ |aₙ₊₁/aₙ| = L

1. If L < 1 Σ aₙ absolutely converges
2. if L = 1 test is inconclusive
3. if L > 1 Σ aₙ diverges

Question 52

Root Test

Answer 52

lim n→ ∞ ⁿ√|aₙ| = L

1. If L < 1 Σ aₙ absolutely converges
2. if L = 1 test is inconclusive
3. if L > 1 Σ aₙ diverges

Question 53

If c is a real, positive number, that the limit of the sequence c¹/ ⁿ →

Answer 53

1

Question 54

If c is a real, positive number, then 1/nᶜ →

Answer 54

0

Question 55

cⁿ/n! →

Answer 55

0

Question 56

n¹/ ⁿ →

Answer 56

1

Question 57

(1 + c/n)ⁿ →

Answer 57

eᶜ

Question 58

tan⁻¹(x)

Answer 58

Σ((-1)ⁿx²ⁿ⁺¹)/(2n+1) from n=0 to ∞

[-1,1]

Question 59

ln(1+x)

Answer 59

Σ((-1)ⁿxⁿ⁺¹)/(n+1) from n=0 to ∞

(-1,1]

Question 60

cos(x)

Answer 60

Σ((-1)ⁿx²ⁿ)/(2n)! from n=0 to ∞

-∞,∞

Question 61

Work

Answer 61

W = integral from a to b (density)(height)(area)
W = 1/2kx^2 (for spring)

Question 62

Average Value of a Function

Answer 62

f(c) = 1/(b-a) integral from a to b f(x)dx

Question 63

Integration by Parts

Answer 63

∫udv = uv - ∫vdu