Untitled Flashcard Set

Pythagorean Identity (sin and cos)

sin²x + cos²x = 1

Pythagorean Identity (sec and tan)

sec²x - tan²x = 1

Reciprocal Identity (sec)

sec x = 1 / cos x

Reciprocal Identity (csc)

csc x = 1 / sin x

Double Angle Formula (sin)

sin 2x = 2 sin x cos x

Double Angle Formula (cos)

cos 2x = cos²x - sin²x

Sum Formula (Sine)

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

Difference Formula (Sine)

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

Sum Formula (Cosine)

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

Difference Formula (Cosine)

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 (from the left)

lim(x→a⁻) f(x) = L

One-Sided Limit (from the right)

lim(x→a⁺) f(x) = L

Limit Law for Addition

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)

Vertical Asymptote Definition

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

Horizontal Asymptote Definition

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

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)

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|\sqrt{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 (max and min)

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 (min and max)

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)

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.