calculus sheet cheat

Guri Bee

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19-10-2024

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Calculus Cheat Sheet: Essential Formulas and Concepts for Success

Calculus, the study of continuous change, can be a challenging subject. Mastering the core concepts and formulas is essential for achieving success. This comprehensive cheat sheet provides a concise overview of key calculus concepts and formulas, designed to help you navigate your journey through this fascinating field.

1. Limits and Continuity

What is a limit?

• A limit describes the behavior of a function as its input approaches a certain value.
• Example: The limit of f(x) = x² as x approaches 2 is 4. This means that as x gets closer and closer to 2, the value of f(x) gets closer and closer to 4.

What is continuity?

• A function is continuous at a point if its limit exists at that point, and the value of the function at that point is equal to the limit.
• Example: f(x) = x² is continuous at x = 2 because the limit of f(x) as x approaches 2 is 4, and f(2) = 4.

Key Formulas:

• Limit Laws: These laws help to calculate limits of combinations of functions.
• Squeeze Theorem: If two functions "squeeze" a third function, and both the "squeezing" functions have the same limit, then the "squeezed" function also has that limit.

2. Derivatives

What is a derivative?

• The derivative of a function represents its instantaneous rate of change at a specific point.
• Example: The derivative of f(x) = x² is f'(x) = 2x. This means that the instantaneous rate of change of f(x) at x = 2 is 4.

Key Formulas:

• Power Rule: d/dx (x^n) = nx^(n-1)
• Product Rule: d/dx (u(x)v(x)) = u'(x)v(x) + u(x)v'(x)
• Quotient Rule: d/dx (u(x)/v(x)) = (v(x)u'(x) - u(x)v'(x)) / (v(x))²
• Chain Rule: d/dx (f(g(x))) = f'(g(x)) * g'(x)

3. Integrals

What is an integral?

• An integral is the opposite of a derivative. It represents the area under a curve.
• Example: The integral of f(x) = 2x is F(x) = x², plus a constant of integration (C).

Key Formulas:

• Power Rule for Integration: ∫x^n dx = (x^(n+1))/(n+1) + C (n ≠ -1)
• Substitution Rule: Allows you to change variables within an integral.
• Integration by Parts: Used when the integrand is a product of two functions.

4. Applications of Calculus

Key Applications:

• Optimization: Finding maximum or minimum values of functions.
• Motion Analysis: Understanding velocity, acceleration, and displacement.
• Economics: Calculating marginal cost, revenue, and profit.
• Physics: Modeling forces, work, and energy.

Additional Resources:

• Khan Academy: Provides free video tutorials and practice exercises for calculus.
• Paul's Online Math Notes: Offers detailed notes and examples on various calculus topics.
• Wolfram Alpha: A powerful computational engine that can help you solve calculus problems and visualize solutions.

Remember: This cheat sheet serves as a starting point. For a deeper understanding of calculus, it's crucial to practice regularly and consult comprehensive textbooks and online resources.