Calculus 1 Formula Sheet

CALC 1 FORMULA SHEET

Limits

• Limit Definitions:

  • \lim_{x \to a} k = k

  • \lim_{x \to a} x = a

  • \lim_{x \to a} (f + g) = \lim f + \lim g

  • \lim_{x \to a} (fg) = (\lim f)(\lim g)

  • \lim_{x \to a} \frac{f}{g} = \frac{\lim f}{\lim g}

• Standard Limits:

  • \lim_{x \to 0} \frac{\sin x}{x} = 1

  • \lim_{x \to 0} \frac{1 - \cos x}{x} = 0

Continuity

• Definition of Continuity:

  • A function is continuous at a point if \lim_{x \to a} f(x) = f(a)

Derivative Rules

• Basic Derivative Formulas:

  • \frac{d}{dx}(c) = 0

  • \frac{d}{dx}(x^n) = n x^{n-1}

  • \frac{d}{dx}(cf) = c f'

  • \frac{d}{dx}(f + g) = f' + g'

  • \frac{d}{dx}(fg) = f'g + fg'

  • \frac{d}{dx}(\frac{f}{g}) = \frac{f'g - fg'}{g^2}

  • (f \circ g)' = f'(g(x)) g'(x)

Trigonometric Derivatives

• Derivative Formulas:

  • (\sin x)' = \cos x

  • (\cos x)' = -\sin x

  • (\tan x)' = \sec^2 x

  • (\cot x)' = -\csc^2 x

  • (\sec x)' = \sec x \tan x

  • (\csc x)' = -\csc x \cot x

Inverse Trigonometric Derivatives

• Derivative Formulas:

  • (\sin^{-1} x)' = \frac{1}{\sqrt{1 - x^2}}

  • (\cos^{-1} x)' = -\frac{1}{\sqrt{1 - x^2}}

  • (\tan^{-1} x)' = \frac{1}{1 + x^2}

Exponential and Logarithmic Derivatives

• Derivative Formulas:

  • (e^x)' = e^x

  • (a^x)' = a^x \ln a

  • (\ln x)' = \frac{1}{x}

Implicit Differentiation

• Formula:

  • \frac{d}{dx}(y^n) = n y^{n-1} y'

Related Rates

• Formula:

  • \frac{dy}{dt} = \left(\frac{dy}{dx}\right)\left(\frac{dx}{dt}\right)

Linear Approximation

• Formula:

  • L(x) = f(a) + f'(a)(x - a)

Differentials

• Definition:

  • dy = f'(x) dx

Antiderivatives

• Integral Formulas:

  • \int x^n dx = \frac{x^{n+1}}{n + 1} + C

  • \int \frac{1}{x} dx = \ln|x| + C

  • \int e^x dx = e^x + C

  • \int a^x dx = \frac{a^x}{\ln(a)} + C

Trigonometric Integrals

• Integral Formulas:

  • \int \sin x dx = -\cos x + C

  • \int \cos x dx = \sin x + C

  • \int \sec^2 x dx = \tan x + C

  • \int \csc^2 x dx = -\cot x + C

  • \int \sec x \tan x dx = \sec x + C

  • \int \csc x \cot x dx = -\csc x + C

Definite Integrals

• Formulas:

  • \int_a^b f(x) dx = F(b) - F(a)

  • \int_a^a f(x) dx = 0

  • \inta^b f(x) dx = -\intb^a f(x) dx

Riemann Sums

• Left Riemann Sum:

  • Ln = \sum f(x{i-1}) \Delta x

• Right Riemann Sum:

  • Rn = \sum f(xi) \Delta x

• Delta x Calculation:

  • \Delta x = \frac{b - a}{n}

Average Value of a Function

• Formula:

  • f{avg} = \frac{1}{b - a} \inta^b f(x) dx

Fundamental Theorem of Calculus (FTC)

• First Part:

  • \frac{d}{dx} \int_a^x f(t) dt = f(x)

• Second Part:

  • \int_a^b f'(x) dx = f(b) - f(a)

Optimization

• Critical Points:

  • Occur where f'(x) = 0 or is undefined.

• Mean Value Theorem (MVT):

  • Given f'(c) = \frac{f(b) - f(a)}{b - a}

Motion

• Relationship Between Position, Velocity, and Acceleration:

  • v(t) = s'(t)

  • a(t) = v'(t) = s''(t)

  • s(t) = \int v(t) dt