Questions and Answers
What is the McLaurin Series for 1/(1-x)?
What is the McLaurin Series for e^x?
∑xⁿ/n!=1+x/1!+x²/2!+x³/3!+...R=infinity
What is the McLaurin Series for sin(x)?
∑(-1)ⁿ(x²ⁿ⁺¹)/(2n+1)!=x-x³/3!+x⁵/5!-x⁷/7!+...R=infinity
What is the McLaurin Series for cos(x)?
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What does it mean for a series to be conditionally convergent?
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What is the definition of Absolutely Convergent?
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What does the Root Test indicate when L=1?
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What does the Ratio Test indicate when L=1?
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What is the condition for the Alternating Series Test?
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What does the Integral Test evaluate?
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What is the formula for Surface Area?
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What is the formula for the Length of a Polar Curve?
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What is the derivative of sin(x)?
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What is the integral of sin(x)?
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What is the formula for sin(2x)?
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What does √(a²-x²) transform to?
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What does √(x²-a²) transform to?
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What does √(a²+x²) transform to?
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What is the derivative of cos(x)?
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What is the integral of cos(x)?
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What is the average value of a function f?
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What is the formula for Area between two curves?
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What is the definition of the Divergence Test?
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What is the limit of arctan(n) as n approaches infinity?
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Study Notes
McLaurin Series
- 1/(1-x): Series expansion is ∑xⁿ = 1 + x + x² + x³ + ... with a radius of convergence R = 1.
- e^x: Series expansion is ∑(xⁿ/n!) = 1 + x/1! + x²/2! + x³/3! + ... with infinite radius of convergence.
- sin(x): Series expansion is ∑((-1)ⁿ(x²ⁿ⁺¹)/(2n+1)!) = x - x³/3! + x⁵/5! - x⁷/7! + ... with infinite radius of convergence.
- cos(x): Series expansion is ∑((-1)ⁿ(x²ⁿ)/(2n)!) = 1 - x²/2! + x⁴/4! - x⁶/6! + ... with infinite radius of convergence.
Convergence Tests
- Conditionally Convergent: A series ∑a_n is conditionally convergent if it converges but is not absolutely convergent.
- Absolutely Convergent: A series ∑a_n is absolutely convergent if ∑|a_n| converges, implying ∑a_n also converges.
- Root Test: If L = 1, the test is inconclusive; if L < 1, the series converges; if L > 1, the series diverges.
- Ratio Test: Similar to the root test; inconclusive when L = 1, converges when L < 1, diverges when L > 1.
- The Alternating Series Test: Applies to alternating series and determines convergence based on the terms' behavior (details needed).
- Integral Test: If ∑f(n) is a positive, decreasing function, the convergence of ∑f(n) can be determined by the integral of f(x).
- Divergence Test: If lim(a_n) ≠ 0, then ∑a_n diverges.
- P-Series Test: A p-series ∑1/n^p converges if p > 1 and diverges if p ≤ 1.
- Comparison Test: If ∑b_n converges and a_n ≤ b_n, then ∑a_n also converges; if ∑b_n diverges and a_n ≥ b_n, then ∑a_n diverges.
Surface Area and Volume
- Surface Area Rotation about the x-axis: Given by integrating π times the difference of the squares of outer and inner radii.
- Surface Area Rotation about the y-axis: Calculated as 2π∫rh dy using shell method where r is the radius and h is the height.
- Washer Method: Used to find volume when revolving around an axis, defined as π∫(outer radius)² - (inner radius)² dx.
- Shell Method: Volume determined with 2π∫rh dx, useful for rotating regions around the y-axis.
Integrals and Derivatives
- Integration by Parts: Formula is ∫u dv = uv - ∫v du.
- Arc Length Formula: The length of a curve defined by ∫√(1+(dy/dx)²) dx.
-
Standard Integrals: Basic integrals include:
- ∫ln(u) = uln(u) - u.
- ∫sec(x) dx = ln|sec(x) + tan(x)|.
- ∫sin(x) = -cos(x).
- ∫cos(x) = sin(x).
Trigonometric Identities
- sin(2x): Expressed as 2sin(x)cos(x).
- sin²(x): Translates to (1 - cos(2x))/2.
- cos²(x): Can be represented as (1 + cos(2x))/2.
- Sin(A) Sin(B) Identity: Given by 1/2[Cos(A-B) - Cos(A+B)].
- Cos(A) Cos(B) Identity: Given by 1/2[Cos(A-B) + Cos(A+B)].
- Sin(A) Cos(B) Identity: Given by 1/2[Sin(A-B) + Sin(A+B)].
Average and Area
- Average Value of Function: Given by (1/(B-A)) ∫ f(x) dx over the interval [A, B].
- Area Between Curves: A = ∫(f(x) - g(x)) dx across the interval [a, b].
Substitutions for Integrals
- To simplify integrals involving square roots:
- For √(a² - x²) use x = a sin(θ).
- For √(x² - a²) use x = a sec(θ).
- For √(a² + x²) use x = a tan(θ).
Polar Curves
- Length of a Polar Curve: Length calculated by ∫√(r² + (dr/dθ)²) dθ.
Derivatives
- Derivative of sin(x): Result is cos(x).
- Derivative of cos(x): Result is -sin(x).
- Derivative of sec(x): Result is sec(x)tan(x).
- Derivative of tan(x): Result is sec²(x).
- Derivative of csc(x): Result is -csc(x)cot(x).
- Derivative of cot(x): Result is -csc²(x).
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Prepare for your Calculus 2 final exam with these comprehensive flashcards focusing on McLaurin Series. Each card provides a specific series along with its definition, ensuring you grasp the foundational concepts required for success in calculus. Use these to reinforce your understanding and practice for your upcoming exam.
