ams 161 trig identies midterm 2

d/dx tan(x)

sec²(x)

d/dx cot(x)

-csc²(x)

sin²x + cos²x

1

tan²x + 1

sec²x

sin(2x)

2(sinx * cos x)

cos(2x)

2cos²x - 1

d/dx (sec x)

sec (x) * tan(x)

divergent test

lim_{n → ∞} b_n = DNE, or ≠ 0 - series diverges

lim_{n → ∞} b_n = 0 - inconclusive

alternating series test

if ∑(-1)ⁿ * b_n

b_n is decreasing

lim_{n → ∞} b_n = 0

then series converges

absolute ratio test

L = lim_{n → ∞} (b_{n + 1} * (1/b_n)).

If L < 1 convergent, L > 1 divergent, L = 1 inconclusive

integration by parts

∫u dv = uv - ∫v du

Integral Test

criteria: continuous, positive, and decreasing over interval

L = lim_{n → ∞} (b_n). if finite, converges. If not finite, diverges

d/dx (csc x)

-cscx * cot x

arc length formula

^b_a∫√(1 + f’(x)²)

average value formula

1(b-a) * ^b_a∫f(x)dx

demoivres theorem

if z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos (nθ) + i sin (nθ))

euler’s formula

r * e^iθ = r(cos(θ) + i sin(θ)) = r cis θ

θ - argument, r - modulus