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:

  • $\int_a^b f(x) dx = F(b) - F(a)$

  • $\int_a^a f(x) dx = 0$

  • $\int<em>a^b f(x) dx = -\int</em>b^a f(x) dx$

Riemann Sums

• Left Riemann Sum:

  • $L<em>n = \sum f(x</em>{i-1}) \Delta x$

• Right Riemann Sum:

  • $R<em>n = \sum f(x</em>i) \Delta x$

• Delta x Calculation:

  • $\Delta x = \frac{b - a}{n}$

Average Value of a Function

• Formula:

  • $f<em>{avg} = \frac{1}{b - a} \int</em>a^b f(x) dx$

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:

  • $v(t) = s'(t)$

  • $a(t) = v'(t) = s''(t)$

  • $s(t) = \int v(t) dt$