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Calculus Final Exam Study Guide Flashcards

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  • Relation between Cartesian and polar coordinates

    x = r cos θ, y = r sin θ, r = √(x² + y²), θ = tan⁻¹(y/x)

  • Slope of tangent line for polar curve r = f(θ)

    dy/dx = (f'(θ) sin θ + f(θ) cos θ) / (f'(θ) cos θ - f(θ) sin θ)

  • Arc length of a polar curve r = f(θ) from α to β

    L = ∫αβ √{(f'(θ))² + (f(θ))²} dθ

  • Area enclosed by a polar curve r = f(θ) from α to β

    A = ½ ∫αβ (f(θ))² dθ

  • Polar form of a complex number z = a + bi

    z = r (cos θ + i sin θ) where r = √(a² + b²), a = r cos θ, b = r sin θ

  • Geometric interpretation of complex multiplication and division

    Multiplication corresponds to multiplying magnitudes and adding angles; division corresponds to dividing magnitudes and subtracting angles in polar form.

  • Dot product orthogonality condition

    Two vectors ⃗u and ⃗v are orthogonal if and only if ⃗u · ⃗v = 0

  • Cross product parallelism condition

    Two vectors ⃗u and ⃗v are parallel if and only if ⃗u × ⃗v = ⃗0

  • Formula for surface area generated by revolving y = f(x) about x-axis

    S = ∫ab 2π f(x) √{1 + (f'(x))²} dx

  • Integration by parts formula

    ∫ u dv = uv - ∫ v du

  • Trigonometric substitution for √(a² - x²)

    Use x = a sin θ, then dx = a cos θ dθ

  • Trigonometric substitution for √(a² + x²)

    Use x = a tan θ, then dx = a sec² θ dθ

  • Trigonometric substitution for √(x² - a²)

    Use x = a sec θ, then dx = a sec θ tan θ dθ

  • Alternating Series Estimation Theorem

    The error |S - SN| in approximating an alternating series by its first N terms is less than or equal to the absolute value of the (N+1)th term.

  • Taylor series centered at a

    f(x) = Σl=0^∞ (f⁽ˡ⁾(a) / l!) (x - a)ˡ

  • Maclaurin series for tan⁻¹(x)

    tan⁻¹(x) = Σk=0^∞ (-1)^k x^(2k+1) / (2k+1)

  • Volume of solid with base region and cross sections

    Volume = ∫ (area of cross section) dx or dy depending on orientation

  • Arc length formula for y = f(x) over [a, b]

    L = ∫ab √{1 + (f'(x))²} dx

  • Partial fraction decomposition

    Express a rational function as a sum of simpler fractions to facilitate integration.

  • Radius and interval of convergence for power series

    Use ratio or root test to find radius; interval includes all x where series converges.