Untitled Flashcards Set

Trigonometric Identities

Q: Pythagorean Identity (sin and cos)
A: sin²x + cos²x = 1

Q: Pythagorean Identity (sec and tan)
A: sec²x - tan²x = 1

Q: Reciprocal Identity for secant
A: sec x = 1 / cos x

Q: Reciprocal Identity for cosecant
A: csc x = 1 / sin x

Q: Double Angle Formula for sine
A: sin 2x = 2 sin x cos x

Q: Double Angle Formula for cosine
A: cos 2x = cos²x - sin²x

Q: Sum & Difference Formula for sin(A ± B)
A: sin(A ± B) = sin A cos B ± cos A sin B

Q: Sum & Difference Formula for cos(A ± B)
A: cos(A ± B) = cos A cos B ∓ sin A sin B

Limits & Continuity

Q: Definition of a Limit
A: lim(x→a) f(x) = L if and only if lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L

Q: One-Sided Limit (approaching from the left)
A: lim(x→a⁻) f(x)

Q: One-Sided Limit (approaching from the right)
A: lim(x→a⁺) f(x)

Q: Limit Laws (Sum/Difference)
A: lim(x→a) [f(x) ± g(x)] = lim(x→a) f(x) ± lim(x→a) g(x)

Q: Definition of a Vertical Asymptote
A: x = a is a vertical asymptote if lim(x→a⁻) f(x) = ±∞ or lim(x→a⁺) f(x) = ±∞

Q: Definition of a Horizontal Asymptote
A: y = L is a horizontal asymptote if lim(x→±∞) f(x) = L

Q: Squeeze Theorem
A: 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

Derivatives

Q: Definition of the Derivative
A: f'(x) = lim(h→0) (f(x+h) - f(x)) / h

Q: d/dx [xⁿ]
A: n xⁿ⁻¹

Q: d/dx [e^x]
A: e^x

Q: d/dx [ln x]
A: 1 / x

Q: d/dx [a^x]
A: ln(a) ⋅ a^x

Q: d/dx [sin x]
A: cos x

Q: d/dx [cos x]
A: -sin x

Q: d/dx [tan x]
A: sec²x

Q: d/dx [arcsin x]
A: 1 / √(1 - x²)

Q: d/dx [arctan x]
A: 1 / (1 + x²)

Q: d/dx [arcsec x]
A: 1 / |x|√(x² - 1)

Q: Product Rule
A: d/dx [f(x)g(x)] = f'(x) g(x) + f(x) g'(x)

Q: Quotient Rule
A: d/dx [f(x) / g(x)] = (f'(x) g(x) - f(x) g'(x)) / g²(x)

Q: Chain Rule
A: d/dx f(g(x)) = f'(g(x)) ⋅ g'(x)

Q: L'Hôpital’s Rule
A: 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)

Integrals

Q: ∫ xⁿ dx (for n ≠ -1)
A: (xⁿ⁺¹) / (n+1) + C

Q: ∫ e^x dx
A: e^x + C

Q: ∫ 1/x dx
A: ln|x| + C

Q: ∫ sin x dx
A: -cos x + C

Q: ∫ cos x dx
A: sin x + C

Q: ∫ sec²x dx
A: tan x + C

Q: Fundamental Theorem of Calculus (Part 1)
A: ∫[a,b] f(x)dx = F(b) - F(a), where F is an antiderivative of f

Q: Fundamental Theorem of Calculus (Part 2)
A: d/dx ∫[c,x] f(t) dt = f(x)

Applications: Motion, Areas, and Volumes

Q: Position, Velocity, Acceleration Relationships
A:

• s(t) = position

• v(t) = s'(t) = velocity

• a(t) = v'(t) = s''(t) = acceleration

Q: Total Distance Traveled Formula
A: ∫[a,b] |v(t)| dt

Q: Volume of Solid (Disk Method)
A: V = π ∫[a,b] [R(x)]² dx

Q: Volume of Solid (Washer Method)
A: V = π ∫[a,b] (R²(x) - r²(x)) dx

Q: Volume of Solid (Cross-Sectional Area)
A: V = ∫[a,b] A(x) dx

Riemann Sums & Approximation Methods

Q: LRAM (Left Riemann Sum)
A: Uses left endpoints of subintervals for f(x)

Q: RRAM (Right Riemann Sum)
A: Uses right endpoints of subintervals for f(x)

Q: MRAM (Midpoint Riemann Sum)
A: Uses midpoints of subintervals for f(x)

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

Other Theorems

Q: Mean Value Theorem
A: 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)

Q: Squeeze Theorem
A: 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