Calc Formulas

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 Formula for Sine

sin(A + B) = sin A cos B + cos A sin B

Difference Formula for Sine

sin(A - B) = sin A cos B - cos A sin B

Sum Formula for Cosine

cos(A + B) = cos A cos B - sin A sin B

Difference Formula for Cosine

cos(A - B) = cos A cos B + sin A sin B

Definition of a Limit

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

One-Sided Limit from the left

lim(x→a⁻) f(x) = L (approaching from the left).

One-Sided Limit from the right

lim(x→a⁺) f(x) = L (approaching from the right).

Limit Law for sum

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

Limit Law for constant multiplication

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.

Definition of the Derivative

f'(x) = lim(h→0) (f(x+h) - f(x)) / h.

Power Rule for Derivatives

d/dx [xⁿ] = n xⁿ⁻¹.

Product Rule for Derivatives

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

Quotient Rule for Derivatives

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

Chain Rule for Derivatives

d/dx f(g(x)) = f'(g(x)) ⋅ g'(x).

Derivative of sin x

d/dx [sin x] = cos x.

Derivative of cos x

d/dx [cos x] = -sin x.

Derivative of tan x

d/dx [tan x] = sec²x.

Derivative of arcsin x

d/dx [arcsin x] = 1 / √(1 - x²).

Derivative of arctan x

d/dx [arctan x] = 1 / (1 + x²).

Derivative of arcsec x

d/dx [arcsec x] = 1 / |x|√(x² - 1).

Derivative of e^x

d/dx [e^x] = e^x.

Derivative of ln x

d/dx [ln x] = 1 / x.

Derivative of a^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).

First Derivative Test

If f'(x) changes from + to -, f(c) is a local max; if it changes from - to +, f(c) is a local min.

Second Derivative Test for local minima

If f''(c) > 0, f(c) is a local min.

Second Derivative Test for local maxima

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).

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.

Integral of sin x

∫ sin x dx = -cos x + C.

Integral of cos x

∫ cos x dx = sin x + C.

Integral of sec²x

∫ sec²x dx = tan x + C.

Position function in motion

s(t) = position.

Velocity function in motion

v(t) = s'(t) = velocity.

Acceleration function in motion

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.

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)].