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Calc 1 Cheat Sheet for Your 10X Faster Calculus Mastery

Introduction

Calculus 1 is known to be one of the toughest math subjects, so it's understandable that you're looking for a cheat sheet to help you ace it. With that said, what you're looking for is not a cheat sheet, it's a guide. This guide will provide you with the essential concepts, formulas, and techniques you need to know for Calculus 1.

Essential Concepts

calc 1 cheat sheet

Limits: A limit is a value that a function approaches as the input approaches some value.
Derivatives: A derivative is the rate of change of a function with respect to its input.
Integrals: An integral is the area under the curve of a function.

Key Formulas

Limit Laws:
- $\lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
- $\lim_{x \to a} (f(x) - g(x)) = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
- $\lim_{x \to a} (f(x)g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
- $\lim_{x \to a} (f(x)/g(x)) = \lim_{x \to a} f(x) / \lim_{x \to a} g(x), \quad \text{if} \quad \lim_{x \to a} g(x) \neq 0$
Derivative Rules:
- $\frac{d}{dx} (x^n) = nx^{n-1}$
- $\frac{d}{dx} (e^x) = e^x$
- $\frac{d}{dx} (\sin x) = \cos x$
- $\frac{d}{dx} (\cos x) = -\sin x$
Integral Rules:
- $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
- $\int e^x dx = e^x + C$
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$

Techniques

Chain Rule: Used to find the derivative of a composite function.
Product Rule: Used to find the derivative of the product of two functions.
Quotient Rule: Used to find the derivative of the quotient of two functions.
Integration by Substitution: Used to evaluate integrals of functions that can be rewritten as a composition of two functions.
Integration by Parts: Used to evaluate integrals of products of two functions.

Common Mistakes to Avoid

Calc 1 Cheat Sheet for Your 10X Faster Calculus Mastery

Forgetting to add the constant of integration when finding integrals.
Mixing up the power rule and the product rule when finding derivatives.
Not taking the absolute value when finding the derivative of a function that is not differentiable at a point.

How to Use This Guide

This guide is not meant to be a substitute for your textbook or class notes. It is meant to be a supplement that you can use to review the material and practice your skills. The best way to use this guide is to:

Read the material in your textbook or class notes.
Review the corresponding sections in this guide.
Practice solving problems using the formulas and techniques provided in this guide.

Conclusion

Calculus 1 is a challenging subject, but it is also very rewarding. By following the tips and using the resources in this guide, you can master Calculus 1 and succeed in your math courses.

Tables

Introduction

Function	Derivative	Integral
$x^n$	$nx^{n-1}$	$\frac{x^{n+1}}{n+1} + C$
$e^x$	$e^x$	$e^x + C$
$\sin x$	$\cos x$	$-\cos x + C$
$\cos x$	$-\sin x$	$\sin x + C$

Rule	Formula	Example
Chain Rule	$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$	$\frac{d}{dx} (\sin x^2) = \cos x^2 \cdot \frac{d}{dx} (x^2) = 2x \cos x^2$
Product Rule	$\frac{d}{dx} (f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$	$\frac{d}{dx} (x^2 \sin x) = 2x \sin x + x^2 \cos x$
Quotient Rule	$\frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}$	$\frac{d}{dx} \left(\frac{x^2}{x+1}\right) = \frac{(x+1)(2x) - x^2(1)}{(x+1)^2} = \frac{x^2+2x-x^2}{(x+1)^2} = \frac{2x}{(x+1)^2}$
Integration by Substitution	$\int f(g(x))g'(x) dx = \int f(u) du$	$\int x^2 \cos x^3 dx = \int x^2 \cos u \frac{d}{du} (x^3) du = \int x^2 \cos u (3x^2) dx = \int 3x^4 \cos u du$
Integration by Parts	$\int u dv = uv - \int v du$	$\int x \sin x dx = x(-\cos x) - \int (-\cos x) dx = -x \cos x + \int \cos x dx = -x \cos x + \sin x + C$