Untitled Flashcard Set

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

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

Reciprocal Identities
sec x = 1 / cos x
csc x = 1 / sin x

Double Angle Formulas
sin 2x = 2 sin x cos x
cos 2x = cos²x - sin²x

Sum & Difference Formulas (Sine)
sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A sin B

Sum & Difference Formulas (Cosine)
cos(A + B) = cos A cos B - sin A sin B
cos(A - B) = cos A cos B + sin A sin B

Limits & Continuity

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 Limits
lim(x→a⁻) f(x) = L (approaching from the left)
lim(x→a⁺) f(x) = L (approaching from the right)

Limit Laws
lim(x→a) [f(x) ± g(x)] = lim(x→a) f(x) ± lim(x→a) g(x)
lim(x→a) [c⋅f(x)] = c⋅lim(x→a) f(x)

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

Definition of 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

Derivatives & Differentiation Rules

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

Power Rule
d/dx [xⁿ] = n xⁿ⁻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)

Trigonometric Derivatives
d/dx [sin x] = cos x
d/dx [cos x] = -sin x
d/dx [tan x] = sec²x

Inverse Trig Derivatives
d/dx [arcsin x] = 1 / √(1 - x²)
d/dx [arctan x] = 1 / (1 + x²)
d/dx [arcsec x] = 1 / |x|√(x² - 1)

Logarithmic & Exponential Derivatives
d/dx [e^x] = e^x
d/dx [ln x] = 1 / x
d/dx [a^x] = ln(a) ⋅ a^x

L'Hôpital’s Rule
If lim(x→a) f(x)/g(x) is in an indeterminate form (0/0 or ∞/∞), then:
lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x)

Critical Points & Extrema

First Derivative Test
If f'(x) changes from + to -, f(c) is a local max.
If f'(x) changes from - to +, f(c) is a local min.

Second Derivative Test
If f''(c) > 0, f(c) is a local min.
If f''(c) < 0, f(c) is a local max.

Test for Concavity
If f''(x) > 0, f(x) is concave up.
If f''(x) < 0, f(x) is concave down.

Point of Inflection
A point where f''(x) changes sign.

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)

Integrals & Fundamental Theorem of Calculus

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)

Integral of xⁿ
∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C, for n ≠ -1

Integral of e^x
∫ e^x dx = e^x + C

Integral of 1/x
∫ (1/x) dx = ln|x| + C

Trigonometric Integrals
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C