Free Calculus 1 Cheatsheet

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Calculus 1 Cheatsheet

Definitions, Symbols, Formulas, and Notes — All in One Place.

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Limits

Term		Definition		Formula
Limit		Describes the behavior of a function as the input approaches a certain value or as the input moves towards a specific point or infinity. The provided formula means that as $x$ gets arbitrarily close to $c$ (but not necessarily equal to $c$ ), the values of $f (x)$ approach $L$ .		$lim_{x \to c} f (x) = L$

Evaluating Limits

Term	Definition	Formula
Sum of Limits	The limit of the sum of two functions $f (x)$ and $g (x)$ as $x$ approaches $a$ is equal to the sum of the limits of $f (x)$ and $g (x)$ individually as $x$ approaches $a$ .	$lim_{x \to a} (f (x) + g (x)) = L_{f} + L_{g}$
Difference of Limits	The limit of the difference between two functions $f (x)$ and $g (x)$ as $x$ approaches $a$ is equal to the difference between the limits of $f (x)$ and $g (x)$ individually as $x$ approaches $a$ .	$lim_{x \to a} (f (x) - g (x)) = L_{f} - L_{g}$
Product of Limits	The limit of the product of two functions $f (x)$ and $g (x)$ as $x$ approaches $a$ is equal to the product of the limits of $f (x)$ and $g (x)$ individually as $x$ approaches $a$ .	$lim_{x \to a} (f (x) \cdot g (x)) = L_{f} \cdot L_{g}$
Quotient of Limits	The limit of the quotient of two functions $f (x)$ and $g (x)$ as $x$ approaches $a$ is equal to the quotient of the limits of $f (x)$ and $g (x)$ individually as $x$ approaches $a$ , as long as the limit of $g (x)$ as $x$ approaches $a$ is not zero.	$lim_{x \to a} \frac{f (x)}{g (x)} = \frac{L_{f}}{L_{g}}, if L_{g} \neq 0$

Special Limits

Term		Limit
$lim_{x \to 0} \frac{\sin (x)}{x}$		$1$
$lim_{x \to 0} \frac{\tan (x)}{x}$		$1$
$lim_{x \to 0} \frac{1 - \cos (x)}{x}$		$0$
$lim_{x \to 0} \frac{e^{x} 1}{x}$		$1$
$lim_{x \to \infty} {(1 + \frac{a}{x})}^{x}$		$e^{a}$
$lim_{x \to \infty} \frac{\ln (x)}{x}$		$0$
$lim_{x \to 0} \frac{\sinh (x)}{x}$		$1$
$lim_{x \to 0} \frac{\cosh (x) 1}{x^{2}}$		$\frac{1}{2}$
$lim_{x \to 0} \frac{\tanh (x)}{x}$		$1$
$lim_{x \to a} \frac{\sqrt[n]{x} \sqrt[n]{a}}{x a}$		$\frac{1}{n \cdot a^{\frac{n 1}{n}}}$ (for $n$ even, and $a > 0$ )
$lim_{x \to 0} \frac{\sin (x)}{x}$		$1$

Derivatives

Term		Definition		Symbol		Formula
Derivative	Represents the rate of change of the function at any given point.	$f^{'} (x)$ $\frac{d f}{d x}$	$f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$
Mean Value Theorem	If a function $f (x)$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$ , then there exists a point $c$ in $(a, b)$ such that $f^{'} (c) = \frac{f (b) - f (a)}{b - a}$ .
Rolle's Theorem	A special case of the Mean Value Theorem where if a function $f (x)$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $f (a) = f (b)$ , then there exists at least one point $c$ in $(a, b)$ such that $f^{'} (c) = 0$ .

Term

Definition

Symbol

Formula

Derivative

Represents the rate of change of the function at any given point.

$f^{'} (x)$

$\frac{d f}{d x}$

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

Mean Value Theorem

If a function

f (x)

is continuous on the closed interval

[a, b]

and differentiable on the open interval

(a, b)

, then there exists a point

c

(a, b)

such that

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

Rolle's Theorem

A special case of the Mean Value Theorem where if a function

f (x)

is continuous on

[a, b]

, differentiable on

(a, b)

, and

f (a) = f (b)

, then there exists at least one point

c

(a, b)

such that

f^{'} (c) = 0

Derivative Rules

Term	Definition	Formula
Power Rule	To find the derivative of a function $f (x)$ where $f (x) = x^{n}$ , where $n$ is a constant, differentiate the function by multiplying the constant $n$ with the coefficient of $x$ and then subtracting 1 from the exponent $n$	$\frac{d}{d x} (x^{n}) = n x^{n - 1}$
Product Rule	To find the derivative of the product of two functions $u (x)$ and $v (x)$ , differentiate the first function $u (x)$ , multiply it by the second function $v (x)$ , then add to it the product of the derivative of the second function $v (x)$ and the first function $u (x)$ .	$\frac{d}{d x} (u (x) \cdot v (x)) = u^{'} (x) v (x) + u (x) v^{'} (x)$
Quotient Rule	To find the derivative of the quotient of two functions $u (x)$ and $v (x)$ , differentiate the numerator $u (x)$ , multiply it by the denominator $v (x)$ , then subtract from it the product of the numerator $u (x)$ and the derivative of the denominator $v (x)$ , all divided by the square of the denominator $v (x)$ .	$\frac{d}{d x} (\frac{u (x)}{v (x)}) = \frac{u^{'} (x) v (x) - u (x) v^{'} (x)}{v (x)^{2}}$
Chain Rule	To find the derivative of a composite function $f (g (x))$ , differentiate the outer function $f$ with respect to its argument $g (x)$ , and then multiply it by the derivative of the inner function $g^{'} (x)$ .	$\frac{d}{d x} (f (g (x))) = f^{'} (g (x)) \cdot g^{'} (x)$

Common Derivatives

Term		Derivative
$\frac{d}{d x} (c)$ (where $c$ is a constant)		$0$
$\frac{d}{d x} (x)$		$1$
$\frac{d}{d x} (x^{n})$		$n x^{n - 1}$
$\frac{d}{d x} (e^{x})$		$e^{x}$
$\frac{d}{d x} (\sin (x))$		$\cos (x)$
$\frac{d}{d x} (\cos (x))$		$- \sin (x)$
$\frac{d}{d x} (\tan (x))$		$\sec^{2} (x)$
$\frac{d}{d x} (\csc (x))$		$- \csc (x) \cot (x)$
$\frac{d}{d x} (\sec (x))$		$\sec (x) \tan (x)$
$\frac{d}{d x} (\cot (x))$		$- \csc^{2} (x)$

Applications of Derivatives

Term	Definition	Formula
Tangent Lines	A straight line that touches the curve of a function at that point, sharing the same slope as the function at that point.	$y = f^{'} (a) (x - a) + f (a)$
Maximum Points	Occur where a function reaches its highest value in a specific domain	To find the maximum of a function $f (x)$ over a closed interval $[a, b]$ , we evaluate $f (x)$ at critical points and endpoints and compare the values to determine the maximum.
Minimum Points	Occur where a function reaches its lowest value in a specific domain.	To find the minimum of a function $f (x)$ over a closed interval $[a, b]$ , we evaluate $f (x)$ at critical points and endpoints and compare the values to determine the minimum.
Critical Point	Points on the graph of a function where its derivative is either zero or undefined.	A point $(a, f (a))$ is a critical point of a function $f (x)$ if $f^{'} (a) = 0$ or $f^{'} (a)$ is undefined.
Interval of Increase	An interval where a function's values are increasing. This occurs where the derivative of the function is positive.	An interval $(a, b)$ is an interval of increase for a function $f (x)$ if $f^{'} (x) > 0$ for all $x$ in the interval.
Interval of Decrease	An interval where a function's values are decreasing. This occurs where the derivative of the function is negative.	An interval $(a, b)$ is an interval of decrease for a function $f (x)$ if $f^{'} (x) < 0$ for all $x$ in the interval.
Inflection Point	Point on the graph of a function where the concavity changes. It's where the second derivative of the function changes sign.	A point $(a, f (a))$ is an inflection point of a function $f (x)$ if $f^{″} (a) = 0$ or $f^{″} (a)$ is undefined.
L'Hopital's Rule	A mathematical technique used to evaluate limits of indeterminate forms by applying derivatives.	If $lim_{x \to a} f (x) = lim_{x \to a} g (x) = 0$ or $lim_{x \to a} f (x) = lim_{x \to a} g (x) = \pm \infty$ , and $lim_{x \to a} \frac{f^{'} (x)}{g^{'} (x)}$ exists, then: $lim_{x \to a} \frac{f (x)}{g (x)} = lim_{x \to a} \frac{f^{'} (x)}{g^{'} (x)}$

Integration

Term		Definition		Symbol
Fundamental Theorem of Calculus	States that if $f (x)$ is a continuous function on the closed interval $[a, b]$ , and $F (x)$ is an antiderivative of $f (x)$ on $[a, b]$ , then: $\int_{a}^{b} f (x) d x = F (b) - F (a)$
Indefinite Integral Antiderivative	Represents the family of functions whose derivative is equal to a given function. An indefinite integral does not have specified limits of integration and represents a set of functions differing only by a constant.	$\int f (x) d x$
Definite Integral	Represents the accumulation of the quantity described by the integrand over a specified interval [ $a$ , $b$ ] where $a$ and $b$ are the lower and upper bounds of integration, respectively.	$\int_{a}^{b} f (x) d x$

Term

Definition

Symbol

Fundamental Theorem of Calculus

States that if

f (x)

is a continuous function on the closed interval

[a, b]

, and

F (x)

is an antiderivative of

f (x)

[a, b]

, then:

\int_{a}^{b} f (x) d x = F (b) - F (a)

Indefinite Integral

Antiderivative

Represents the family of functions whose derivative is equal to a given function. An indefinite integral does not have specified limits of integration and represents a set of functions differing only by a constant.

\int f (x) d x

Definite Integral

Represents the accumulation of the quantity described by the integrand over a specified interval [

a

b

] where

a

and

b

are the lower and upper bounds of integration, respectively.

\int_{a}^{b} f (x) d x

Evaluating Integrals

Term		Definition		Example
U-Substitution Integration by Substitution	A technique used to simplify integrals by substituting a new variable for part of the integrand.	Consider the integral $\int x \cos (x^{2}) d x$ . Identify the Inner Function: In this case, the inner function is $u = x^{2}$ . Choose a Suitable Substitution: We choose $u = x^{2}$ because its derivative, $d u = 2 x d x$ , is present in the integrand. Compute the Differential: $d u = 2 x d x$ . Rewrite the Integrand: We rewrite the integrand as $\frac{1}{2} \cos (u) d u$ . Evaluate the Integral: The integral becomes $\frac{1}{2} \int \cos (u) d u$ , which is $\frac{1}{2} \sin (u) + C$ . Undo the Substitution: Substitute back $u = x^{2}$ to get $\frac{1}{2} \sin (x^{2}) + C$ . Adjust the Limits (if necessary): If there were limits of integration, they would be adjusted accordingly based on the substitution.

Term

Definition

Example

U-Substitution

Integration by Substitution

A technique used to simplify integrals by substituting a new variable for part of the integrand.

Consider the integral $\int x \cos (x^{2}) d x$ .

Identify the Inner Function: In this case, the inner function is $u = x^{2}$ .
Choose a Suitable Substitution: We choose $u = x^{2}$ because its derivative, $d u = 2 x d x$ , is present in the integrand.
Compute the Differential: $d u = 2 x d x$ .
Rewrite the Integrand: We rewrite the integrand as $\frac{1}{2} \cos (u) d u$ .
Evaluate the Integral: The integral becomes $\frac{1}{2} \int \cos (u) d u$ , which is $\frac{1}{2} \sin (u) + C$ .
Undo the Substitution: Substitute back $u = x^{2}$ to get $\frac{1}{2} \sin (x^{2}) + C$ .
Adjust the Limits (if necessary): If there were limits of integration, they would be adjusted accordingly based on the substitution.

Common Integrals

Term		Integral
$\int 1 d x$		$x + C$
$\int x^{n} d x$		$\frac{x^{n + 1}}{n + 1} + C$ (where $n \neq - 1$ )
$\int e^{x} d x$		$e^{x} + C$
$\int \sin (x) d x$		$- \cos (x) + C$
$\int \cos (x) d x$		$\sin (x) + C$
$\int \sec^{2} (x) d x$		$\tan (x) + C$
$\int \csc^{2} (x) d x$		$- \cot (x) + C$
$\int \sec (x) \tan (x) d x$		$\sec (x) + C$
$\int \csc (x) \cot (x) d x$		$- \csc (x) + C$

Applications of Integrals

Term	Definition	Formula
Area Under A Curve	Represents the region enclosed between the curve and the x-axis over a specified interval.	The area $A$ under the curve of a function $f (x)$ from $x = a$ to $x = b$ is given by: $A = \int_{a}^{b} f (x) d x$
Area Between Curves	Refers to the region enclosed between two curves over a specified interval. It's calculated by finding the difference between the integrals of the upper and lower curves.	The area $A$ between two curves $f (x)$ and $g (x)$ from $x = a$ to $x = b$ is given by: $A = \int_{a}^{b} \| f (x) - g (x) \| d x$
Average Value	The average "height" of a function over an interval, calculated by finding the total area under the curve divided by the length of the interval.	The average value $\bar{f}$ of a function $f (x)$ over the interval $[a, b]$ is given by: $\bar{f} = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

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