Calculus2

Question 1

tan²ø +1 =

Answer 1

sec²ø

Question 1

cot²ø + 1 =

Answer 1

csc²ø

Question 1

sin(2ø)

Answer 1

2sin(ø)cos(ø)

Question 1

cos(2ø)

Answer 1

cos²ø - sin²ø

2cos²ø - 1

1 - 2sin²ø

Question 1

Derivative of sin ø

Answer 1

f(ø) = sin ø

f^’(ø) = cos ø

Question 1

Derivative of cos ø

Answer 1

f(ø) = cos ø

f^’(ø) = -sin ø

Question 1

Derivative of tan ø

Answer 1

f(ø) = tan ø

f^’(ø) = sec²(ø)

Question 1

P test

Answer 1

Integral of 1/x^p from 1 - infity

Convergent P >1

Divergent P <= 1

Question 1

Q test

Answer 1

integeral of 1/x^q from 0 - 1

Convergent if q < 1

Divergent if q >= 1

Question 1

e test

Answer 1

integral of e^-ax from 0 - infinity

Convergent a > 0

Divergent a <= 0

Question 2

Sum of a finite series

Answer 2

a(1-xⁿ)/1-x

Question 3

Sum of infinite series

Answer 3

a/(1-x)

Question 4

Integral test

Answer 4

If

integral a_n

Converges then

series a_n

Converges

Question 5

Comparison test

Answer 5

If b_n is convergent then

_{<span>a</span>n} is convergent

with 0 < a_n < b_n for all n

Question 6

Limit comparison test

Answer 6

If c is positive (i.e. c > 0 ) and is finite (i.e. c < inf. ) then either both series converge or both series diverge.

Question 7

Convergence of absolute values

Answer 7

If

|a_n|

is convergent then

a_n

is convergent

Question 8

Ratio test

Answer 8

• If L < 1 the series is absolutley convergent
• if L > 1 the series is divergent
• if L = 1 the series may diverge or converge

Question 9

Alternating series test

Answer 9

if lim_{x -> inf} b_n = 0 and

[b_n] is a decreasing sequence then

the series is convergent

Question 10

Radius of convergence

Answer 10

if lim_{n -> inf} a_n+1 / a_n = infinity

then R = 0

if lim_{n -> inf} a_n+1 / a_n = 0

then R = inf

if lim_{n -> inf} a_n+1 / a_n = K|x-a|

then R = 1/K

Question 11

Taylor polynomial formula

Answer 11

Question 12

Maclaurin series

Answer 12

Question 13

Taylor expansion for sin(x)

Answer 13

x - x³/3! + x⁵/5! - x⁷/7! + x⁹/9! ….

Question 14

Taylor expansion for cos(x)

Answer 14

1 - x²/2! + x⁴/4! - x⁶/6! + x⁸/8! …

Question 15

Taylor expansion for e^x

Answer 15

1 + x + x²/2! + x³/3! + x⁴/4!…

Question 16

Taylor expansion for 1/1+x

Answer 16

1 - x + x² - x³…

Question 17

Taylor expansion for ln(1+x)

Answer 17

x - x²/2! + x³/3! - x⁴/4! …

Question 18

Mclauren expansion for cos(x)

Answer 18

Question 19

Mclauren expansion for sin(x)

Answer 19

Question 20

Mclauren expansion for e^x

Answer 20