Calculus 1

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

2 Limits and Continuity


2.1 The Limit of a Function

The limit of a function ff is a tool for investigating the behavior of f(x)f(x) as xx approaches some number cc.

limxcf(x)\lim\limits_{x \to c} f(x)

Read as “the limit of the function ff as xx approaches cc.” 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 xcx \to c below lim\lim means we are taking the limit of ff as values of xx approach cc.

One-sided limits

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.

Formal definition of a limit

limxcf(x)=L\lim\limits_{x \to c} f(x) = L

means that to establish a specific limit, a number ϵ>0\epsilon > 0 is chosen first to establish a desired degree of proximity to LL, then a number δ>0\delta > 0 is found that determines how close xx must be to cc to ensure that f(x)f(x) is within ϵ\epsilon units of LL.

2.2 Algebraic Computation of Limits

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

Limit laws

More at Avidemia - Theorems for Calculating Limits.

Basic limit results

For any real number aa and any constant cc,

Constant function rule

limxax=a\lim\limits_{x \to a} x = a

Identity function rule

limxac=c\lim\limits_{x \to a} c = c

Basic algebraic operations

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.

The Squeeze Theorem

Let f(x)f(x), g(x)g(x), and h(x)h(x) be defined for all xax \neq a over an open interval containing aa. If

f(x)g(x)g(x)f(x) \leq g(x) \leq g(x)

for all xax \neq a in an open interval containing aa and

limxaf(x)=L=limxah(x)\lim\limits_{x \to a} f(x) = L = \lim\limits_{x \to a} h(x)

where LL is a real number, then

limxag(x)=L\lim\limits_{x \to a} g(x) = L

2.3 Continuity

2.4 Exponential and logarithmic functions

3 Differentiation

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}

Procedural rules for finding derivatives

See differentiation formulas.

Study cards

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

4 Applications of Derivatives

4.5 L’Hôpital’s Rule

L’Hôpital’s rule is used to evaluate indeterminate forms. L’Hôpital’s rule states if the limxcf(x)g(x)\lim\limits_{x \to c} \frac{f(x)}{g(x)} yields one of the indeterminate forms of the type 00\frac{0}{0} of \frac{\infin}{\infin}, then you can evaluate the limit by replacing ff with ff^{\prime} and gg with gg^{\prime}. Because the rule can be applied repeatedly, you can keep applying it until you get something other than an indeterminate form.

5 Integration

5.1 Antidifferentiation


A function GG is an antiderivative of ff on an interval II if

G(x)=f(x)G^{\prime}(x) = f(x)

for all xx in II.

If GG is an antiderivative of ff, then so is G+CG + C for any constant CC given that the derivative of any constant is 00.

Indefinite integral notation

f(x)dx=G(x)+C\int f(x)dx = G(x) + C

The antiderivative (integral) of ff is G+CG + C.

  • ff is the integrand
  • dxdx is the variable of integration

Integration formulas

See integration formulas

To obtain the power rule, for nn is any number and n1n \neq -1

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

5.5 Integration by Substitution

Integration by substitution or u-substitution is the inverse of the chain rule for diferrentiation.

Let ff, gg, and uu be differentiable functions of xx such that

f(x)=g(u)dudxf(x) = g(u) \frac{du}{dx}


f(x)=g(u)dudx=g(u)du=G(u)+C\int f(x) = \int g(u) \frac{du}{dx} = \int g(u)du = G(u) + C

Where GG is an antiderivative of gg.

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