Calculus I

The 00\frac{0}{0} form is the basis of differentiation; the ×0\infty \times 0 form is the basis of integration.

Limits and continuity

The limit of a function is the value that f(x)f(x) gets closer to as xx approaches some number.

limxaf(x)\lim\limits_{x \to a} f(x)

"The limit of the function ff as xx approaches aa." The symbol lim\lim means we are taking the limit of something. The expression to the right is the expression we are taking the limit of. The expression xax \to a below lim\lim means we are taking the limit of ff as values of xx approach aa.

Approach

As values increase towards aa, this is called approaching from the left (notated xax \to a^{-}).

As values decrease towards aa, this is called approaching from the right (notated xa+x \to a^{+}).

When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.

Limit theorems

Arithmetic

The limit of a sum is the sum of the individual limits.

The limit of a difference is the difference of the individual limits.

The limit of a product is the product of the individual limits.

The limit of a quotient is the quotient of the individual limits, provided each individual limit exists and the limit of the denominator is nonzero.

Limits using direct substitution

Limits using algebraic manipulation

Limits by factoring

Limits by rationalizing

Two special limits

  • limθ0sin(θ)θ=1\lim\limits_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1
  • limθ0cos(θ)θ=0\lim\limits_{\theta \to 0} \frac{\cos(\theta)}{\theta} = 0

Derivatives

Definition of derivative

f(x)=limh0f(x+h)f(x)hf^{\prime}(x) = \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}

Derivatives of trig functions

  • ddxsin(x)=cos(x)\frac{d}{dx} \sin(x) = \cos(x)
  • ddxcos(x)=sin(x)\frac{d}{dx} \cos(x) = -\sin(x)
  • ddxtan(x)=sec2(x)\frac{d}{dx} \tan(x) = \sec^2(x)
  • ddxcot(x)=csc2(x)\frac{d}{dx} \cot(x) = -\csc^2(x)
  • ddxsec(x)=sec(x)tan(x)\frac{d}{dx} \sec(x) = \sec(x)\tan(x)
  • ddxcsc(x)=csc(x)cot(x)\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)

Derivatives of inverse trig functions

  • ddxsin1(u)=du/dx1u2\frac{d}{dx} \sin^{-1}(u) = \frac{du/dx}{\sqrt{1 - u^2}}
  • ddxcos1(u)=du/dx1u2\frac{d}{dx} \cos^{-1}(u) = \frac{-du/dx}{\sqrt{1 - u^2}}
  • ddxtan1(u)=du/dx1+u2\frac{d}{dx} \tan^{-1}(u) = \frac{du/dx}{1 + u^2}
  • ddxcot1(u)=du/dx1+u2\frac{d}{dx} \cot^{-1}(u) = \frac{-du/dx}{1 + u^2}
  • ddxsec1(u)=du/dxuu21\frac{d}{dx} \sec^{-1}(u) = \frac{du/dx}{\lvert u \rvert \sqrt{u^2 - 1}}
  • ddxcsc1(u)=du/dxuu21\frac{d}{dx} \csc^{-1}(u) = \frac{-du/dx}{\lvert u \rvert \sqrt{u^2 - 1}}

Derivatives of exponential functions

General rule for the derivative of exponential function: write it down x derivative of the exponent x natural log of the base

  • ddx8e2x=8e2x2ln(e)\frac{d}{dx} 8e^{2x} = 8e^{2x} \cdot 2 \cdot \ln(e)

Derivatives of logarithmic functions

General rule for the derivative of natural log functions: 1 over the object x derivative of the object

  • ddxln(u)=1ududx\frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx}
  • ddxlogax\frac{d}{dx} \log_{a} x use the base-changing formula first

Integrals

Trigonometric power reduction identities

  • sin2(θ)=1cos(2θ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}
  • cos2(θ)=1+cos(2θ)2\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}

Integration of logarithmic trigonometric functions

  • tan(θ)dx=sin(θ)cos(θ)dx\int \tan(\theta) dx = \int \frac{\sin(\theta)}{\cos(\theta)} dx
  • sec(θ)dxsec(θ)+tan(θ)sec(θ)+tan(θ)\int \sec(\theta) dx \cdot \frac{\sec(\theta) + \tan(\theta)}{\sec(\theta) + \tan(\theta)}
  • csc(θ)dxcsc(θ)+cot(θ)csc(θ)+cot(θ)\int \csc(\theta) dx \cdot \frac{\csc(\theta) + \cot(\theta)}{\csc(\theta) + \cot(\theta)}

Integration resulting in inverse trionometric functions

Essential formulas

  • dua2+u2=sin1(ua)+c\int \frac{du}{\sqrt{a^2 + u^2}} = \sin^{-1}(\frac{u}{a}) + c
  • dua2+u2=1atan1(ua)+c\int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1}(\frac{u}{a}) + c
  • duuu2a2=1asec1(ua)+c\int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1}(\frac{\lvert u \rvert}{a}) + c