Calculus 2 Final Exam Review Flashcards
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Questions and Answers

What is the McLaurin Series for 1/(1-x)?

  • ∑xⁿ=1+x+x²+x³+x⁴+...R=1 (correct)
  • ∑xⁿ/n!=1+x/1!+x²/2!+x³/3!+...R=infinity
  • ∑(-1)ⁿ(x²ⁿ+1)/(2n+1)!=x-x³/3!+x⁵/5!-x⁷/7!+...R=infinity
  • ∑(-1)ⁿ(x²ⁿ)/(2n)!=1-x²/2!+x⁴/4!-x⁶/6!+...R=infinity

    • 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)?

    <p>∑(-1)ⁿ(x²ⁿ)/(2n)!=1-x²/2!+x⁴/4!-x⁶/6!+...R=infinity</p> Signup and view all the answers

    What does it mean for a series to be conditionally convergent?

    <p>A series is ∑ a_n is called conditionally convergent if it is convergent but not absolutely convergent.</p> Signup and view all the answers

    What is the definition of Absolutely Convergent?

    <p>A series ∑ a_n is called Absolutely convergent if the series of absolute values ∑ |a_n| is convergent.</p> Signup and view all the answers

    What does the Root Test indicate when L=1?

    <p>Nothing</p> Signup and view all the answers

    What does the Ratio Test indicate when L=1?

    <p>Nothing</p> Signup and view all the answers

    What is the condition for the Alternating Series Test?

    Signup and view all the answers

    What does the Integral Test evaluate?

    Signup and view all the answers

    What is the formula for Surface Area?

    <p>S=∫ 2π f(x) √(1+[f'(x)]²) dx</p> Signup and view all the answers

    What is the formula for the Length of a Polar Curve?

    <p>∫√(r²+(dr/d∅)²)d∅</p> Signup and view all the answers

    What is the derivative of sin(x)?

    <p>cos(x)</p> Signup and view all the answers

    What is the integral of sin(x)?

    <p>-cos(x)</p> Signup and view all the answers

    What is the formula for sin(2x)?

    <p>2sin(x)cos(x)</p> Signup and view all the answers

    What does √(a²-x²) transform to?

    <p>x=aSinθ</p> Signup and view all the answers

    What does √(x²-a²) transform to?

    <p>x=aSecθ</p> Signup and view all the answers

    What does √(a²+x²) transform to?

    <p>x=aTanθ</p> Signup and view all the answers

    What is the derivative of cos(x)?

    <p>-sin(x)</p> Signup and view all the answers

    What is the integral of cos(x)?

    <p>sin(x)</p> Signup and view all the answers

    What is the average value of a function f?

    <p>(1/B-A)∫f(x)dx</p> Signup and view all the answers

    What is the formula for Area between two curves?

    <p>∫[f(x)-g(x)]dx</p> Signup and view all the answers

    What is the definition of the Divergence Test?

    Signup and view all the answers

    What is the limit of arctan(n) as n approaches infinity?

    <p>π/2</p> Signup and view all the answers

    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.

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