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Calculus - Study - Notes

These calculus study notes cover key concepts including limits, derivatives, and integrals. It details definitions, rules, techniques, and applications for each topic, such as continuity, optimization, and area under curves. Additionally, it introduces sequences and series, along with convergence tests.

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Steven Wolfe
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39 views2 pages

Calculus - Study - Notes

These calculus study notes cover key concepts including limits, derivatives, and integrals. It details definitions, rules, techniques, and applications for each topic, such as continuity, optimization, and area under curves. Additionally, it introduces sequences and series, along with convergence tests.

Uploaded by

Steven Wolfe
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as TXT, PDF, TXT or read online on Scribd
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Calculus Study Notes

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1. Limits and Continuity


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- Definition of a Limit: lim(x→a) f(x) = L
- Left-hand and Right-hand limits
- Properties of limits (sum, product, quotient)
- Techniques: Direct substitution, factoring, rationalizing
- Continuity: A function f(x) is continuous at x=a if:
1. f(a) is defined
2. lim(x→a) f(x) exists
3. lim(x→a) f(x) = f(a)

2. Derivatives
--------------
- Definition: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
- Basic Rules:
- Power Rule: d/dx[x^n] = nx^(n-1)
- Product Rule: d/dx[uv] = u'v + uv'
- Quotient Rule: d/dx[u/v] = (u'v - uv') / v^2
- Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
- Applications:
- Tangent lines
- Motion (velocity, acceleration)
- Optimization problems
- Related rates

3. Integrals
------------
- Indefinite Integrals:
- ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
- ∫e^x dx = e^x + C
- ∫1/x dx = ln|x| + C
- Definite Integrals:
- ∫[a to b] f(x) dx = F(b) - F(a), where F is an antiderivative of f
- Fundamental Theorem of Calculus:
- Part 1: If F is an antiderivative of f, then d/dx ∫[a to x] f(t) dt = f(x)
- Part 2: ∫[a to b] f(x) dx = F(b) - F(a)

4. Techniques of Integration
----------------------------
- Substitution Method
- Integration by Parts: ∫u dv = uv - ∫v du
- Trigonometric Integrals and Substitution
- Partial Fractions (for rational functions)

5. Applications of Integration
------------------------------
- Area under a curve
- Volume of solids of revolution (Disk and Shell methods)
- Arc length
- Surface area of revolution

6. Sequences and Series (Introductory)


--------------------------------------
- Convergence of sequences
- Series and partial sums
- Geometric Series: ∑ ar^n = a / (1 - r), |r| < 1
- p-Series: ∑ 1/n^p converges if p > 1
- Tests for convergence: nth-term test, comparison test, ratio test

End of Notes

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