Calc Formulas

Pre-Calculus & Trigonometry

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

Applications: Motion, Areas, and Volumes

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

Total Distance Traveled
∫[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

Riemann Sums & Approximation Methods

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
Approximates area using trapezoids:
A ≈ (Δx / 2) [f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]