cheat sheet for calculus 1

Guri Bee

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21-10-2024

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Calculus 1 Cheat Sheet: A Quick Guide to Essential Concepts

Calculus 1 is a foundational course in mathematics that introduces the fundamental concepts of limits, derivatives, and integrals. Mastering these concepts is crucial for understanding more advanced mathematics and its applications in various fields like physics, engineering, and economics.

This cheat sheet aims to provide a concise overview of key concepts and formulas from Calculus 1, helping you quickly refresh your understanding or prepare for exams.

I. Limits

• What are limits? Limits describe the behavior of a function as its input approaches a certain value.
• Key Notation: lim_(x->a) f(x) = L
  • This reads: "The limit of f(x) as x approaches a is L."
• Important Limit Laws:
  • lim_(x->a) [f(x) + g(x)] = lim_(x->a) f(x) + lim_(x->a) g(x) (Sum Law)
  • lim_(x->a) [f(x) * g(x)] = lim_(x->a) f(x) * lim_(x->a) g(x) (Product Law)
  • lim_(x->a) [f(x) / g(x)] = lim_(x->a) f(x) / lim_(x->a) g(x) (Quotient Law, provided lim_(x->a) g(x) ≠ 0)
• Special Limits:
  • lim_(x->0) sin(x)/x = 1
  • lim_(x->0) (1-cos(x))/x = 0
  • lim_(x->0) (e^x - 1)/x = 1

Example: Consider the function f(x) = (x^2 - 1)/(x - 1). We cannot simply plug in x = 1 because it would result in division by zero. However, the limit as x approaches 1 exists: lim_(x->1) (x^2 - 1)/(x - 1) = lim_(x->1) (x + 1) = 2. This tells us that as x gets closer and closer to 1, the function's output gets closer and closer to 2.

II. Derivatives

• What are derivatives? Derivatives measure the instantaneous rate of change of a function.
• Key Notation: f'(x) or dy/dx
• Basic Differentiation Rules:
  • d/dx (x^n) = n*x^(n-1) (Power Rule)
  • d/dx (sin(x)) = cos(x)
  • d/dx (cos(x)) = -sin(x)
  • d/dx (e^x) = e^x
  • d/dx (ln(x)) = 1/x
• Chain Rule: d/dx [f(g(x))] = f'(g(x)) * g'(x)
• Product Rule: d/dx [f(x) * g(x)] = f'(x)g(x) + f(x)g'(x)
• Quotient Rule: d/dx [f(x) / g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]^2

Example: Find the derivative of f(x) = x^2 * sin(x). Applying the Product Rule, we get f'(x) = 2x * sin(x) + x^2 * cos(x).

III. Integrals

• What are integrals? Integrals represent the area under a curve.
• Key Notation: ∫f(x)dx
• Fundamental Theorem of Calculus: This theorem establishes the connection between derivatives and integrals.
  • Part 1: d/dx ∫[a,x] f(t) dt = f(x)
  • Part 2: ∫[a,b] f(x) dx = F(b) - F(a), where F(x) is an antiderivative of f(x)
• Basic Integration Formulas:
  • ∫x^n dx = (x^(n+1))/(n+1) + C (where n ≠ -1)
  • ∫sin(x) dx = -cos(x) + C
  • ∫cos(x) dx = sin(x) + C
  • ∫e^x dx = e^x + C
  • ∫1/x dx = ln|x| + C

Example: Calculate the integral of f(x) = 2x from x = 0 to x = 2. The antiderivative of 2x is x^2. Applying the Fundamental Theorem of Calculus, we get: ∫[0,2] 2x dx = (2^2) - (0^2) = 4.

Further Exploration

• Applications of Calculus: Explore how Calculus 1 is used in real-world applications such as physics, engineering, economics, and computer science.
• Calculus 2: Dive deeper into integration techniques, sequences, and series.
• Multivariable Calculus: Expand your understanding of calculus to functions of multiple variables.

References:

• Calculus by Khan Academy: Offers a comprehensive and free online resource for calculus concepts and exercises.
• Calculus: Early Transcendentals by James Stewart: A widely-used textbook for calculus courses.

Note: This cheat sheet is intended as a quick reference and does not substitute for a thorough understanding of the subject. It's important to consult your textbook, lecture notes, or online resources for a more detailed explanation of the concepts and their applications.