Calculus 2 Exam 2

tan(θ)

sin(θ)/cos(θ)

sec(θ)

1/cos(θ)

sin²(θ) + cos²(θ)

1

tan²(θ) + 1

sec²(θ)

sin(2θ)

2sin(θ)cos(θ)

cos(2θ)

cos²(θ) - sin²(θ)

cos²(θ)

½(1 + cos(2θ))

sin²(θ)

½(1 - cos(2θ))

sin(0)

0

cos(0)

1

tan(0)

0

sin(π/6)

1/2

cos(π/6)

√3/2

tan(π/6)

1/√3

sin(π/4)

1/√2

cos(π/4)

1/√2

tan(π/4)

1

sin(π/3)

√3/2

cos(π/3)

1/2

tan(π/3)

√3

sin(π/2)

1

cos(π/2)

0

tan(π/2)

undefined

d/dt (sin(t))

cos(t)

d/dt (cos(t))

-sin(t)

d/dt (tan(t))

sec²(t)

d/dt (sec(t))

sec(t)tan(t)

d/dt (arctan(t))

1/(1+t²)

d/dt (tᵏ)

ktᵏ⁻¹

d/dt (ln|t|)

1/t

d/dt (eᵏᵗ)

keᵏᵗ

∫ tan(t) dt

ln|sec(t)| + C

∫ sec(t) dt

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

∫[a to b] f(g(x))g'(x) dx

∫[g(a) to g(b)] f(u) du (u-substitution)

∫ u dv

uv - ∫ v du (Integration by Parts)

Substitution for √(b² - u²)

u = b sin(θ)

Substitution for √(b² + u²)

u = b tan(θ)

Substitution for √(u² - b²)

u = b sec(θ)

Volume by Cross-sections

V = ∫a to b(width)

Volume (Washer Method)

V = ∫[a to b] π((RADIUS)² - (radius)²)(width)

Volume (Shell Method)

V = ∫[a to b] 2π(radius)(height)(width)

Arc Length (Cartesian)

L = ∫[a to b] √(1 + (f'(x))²) dx

Arc Length (Parametric)

L = ∫[a to b] √((x'(t))² + (y'(t))²) dt

Surface Area of Revolution

SA = ∫[a to b] 2π(radius)(length)

Work (Constant/Variable Force)

W = ∫a to b(distance)

Work (Pumping Liquid)

W = ∫a to b(distance)(area)(width)

Hydrostatic Force

F = ∫a to b(depth)(width)(height)

Mass

M = ∫a to b(measure)

Moment

Mₓ = ∫[a to b] (distance to x-axis)(density)(measure) OR M

Center of Mass (x̄, ȳ)

(M

Polar to Cartesian Conversion (x)

x = r cos(θ)

Polar to Cartesian Conversion (y)

y = r sin(θ)

Polar to Cartesian Conversion (r²)

x² + y² = r²

Area in Polar Coordinates

A = ∫[α to β] ½ ((f(θ))² - (g(θ))²) dθ OR A = ∫[α to β] ½ r² dθ

Arc Length in Polar Coordinates

L = ∫[α to β] √((f(θ))² + (f'(θ))²) dθ OR L = ∫[α to β] √(r² + (dr/dθ)²) dθ

Trapezoidal Rule Error Bound (E

≤ M(b - a)³ / (12n²), where |f''(x)| ≤ M

Simpson's Rule Error Bound (E

≤ M(b - a)⁵ / (180n⁴), where |f⁽⁴⁾(x)| ≤ M

Trapezoidal Rule Formula Pattern

Δx/2 (1-2-2…2-1)

Simpson's Rule Formula Pattern

Δx/3 (1-4-2-4…2-4-1)