Untitled Flashcards Set

Pythagorean Identity (sin and cos)

sin²x + cos²x = 1

Pythagorean Identity (sec and tan)

sec²x - tan²x = 1

Reciprocal Identity for secant

sec x = 1 / cos x

Reciprocal Identity for cosecant

csc x = 1 / sin x

Double Angle Formula for sine

sin 2x = 2 sin x cos x

Double Angle Formula for cosine

cos 2x = cos²x - sin²x

Sum & Difference Formula for sin(A ± B)

sin(A ± B) = sin A cos B ± cos A sin B

Sum & Difference Formula for cos(A ± B)

cos(A ± B) = cos A cos B ∓ sin A sin B

Definition of a Limit

lim(x→a) f(x) = L if and only if lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L

One-Sided Limit (approaching from the left)

lim(x→a⁻) f(x)

One-Sided Limit (approaching from the right)

lim(x→a⁺) f(x)

Limit Laws (Sum/Difference)

lim(x→a) [f(x) ± g(x)] = lim(x→a) f(x) ± lim(x→a) g(x)

Definition of a Vertical Asymptote

x = a is a vertical asymptote if lim(x→a⁻) f(x) = ±∞ or lim(x→a⁺) f(x) = ±∞

Definition of a Horizontal Asymptote

y = L is a horizontal asymptote if lim(x→±∞) f(x) = L

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L

Definition of the Derivative

f'(x) = lim(h→0) (f(x+h) - f(x)) / h

d/dx [xⁿ]

n xⁿ⁻¹

d/dx [e^x]

e^x

d/dx [ln x]

1 / x

d/dx [a^x]

ln(a) ⋅ a^x

d/dx [sin x]

cos x

d/dx [cos x]

-sin x

d/dx [tan x]

sec²x

d/dx [arcsin x]

1 / √(1 - x²)

d/dx [arctan x]

1 / (1 + x²)

d/dx [arcsec x]

1 / |x|√(x² - 1)

Product Rule

d/dx [f(x)g(x)] = f'(x) g(x) + f(x) g'(x)

Quotient Rule

d/dx [f(x) / g(x)] = (f'(x) g(x) - f(x) g'(x)) / g²(x)

Chain Rule

d/dx f(g(x)) = f'(g(x)) ⋅ g'(x)

L'Hôpital’s Rule

If lim(x→a) f(x)/g(x) is 0/0 or ∞/∞, then lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x)

∫ xⁿ dx (for n ≠ -1)

(xⁿ⁺¹) / (n+1) + C

∫ e^x dx

e^x + C

∫ 1/x dx

ln|x| + C

∫ sin x dx

-cos x + C

∫ cos x dx

sin x + C

∫ sec²x dx

tan x + C

Fundamental Theorem of Calculus (Part 1)

∫[a,b] f(x)dx = F(b) - F(a), where F is an antiderivative of f

Fundamental Theorem of Calculus (Part 2)

d/dx ∫[c,x] f(t) dt = f(x)

Position, Velocity, Acceleration Relationships

s(t) = position, v(t) = s'(t) = velocity, a(t) = v'(t) = s''(t) = acceleration

Total Distance Traveled Formula

∫[a,b] |v(t)| dt

Volume of Solid (Disk Method)

V = π ∫[a,b] [R(x)]² dx

Volume of Solid (Washer Method)

V = π ∫[a,b] (R²(x) - r²(x)) dx

Volume of Solid (Cross-Sectional Area)

V = ∫[a,b] A(x) dx

LRAM (Left Riemann Sum)

Uses left endpoints of subintervals for f(x)

RRAM (Right Riemann Sum)

Uses right endpoints of subintervals for f(x)

MRAM (Midpoint Riemann Sum)

Uses midpoints of subintervals for f(x)

Trapezoidal Rule

A ≈ (Δx / 2) [f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]

Mean Value Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a)

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L