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Differentials quiz #1 Flashcards

Differentials quiz #1
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  • Given the function f(x) = x^3 + x, find the differential dy when x = 2 and dx = 0.1.
    First, find the derivative: f'(x) = 3x^2 + 1. At x = 2, f'(2) = 3(2^2) + 1 = 13. Then, dy = f'(x)dx = 13 × 0.1 = 1.3.
  • How can differentials be used to approximate the value of a function, and what is the approximate value of f(2.1) for f(x) = x^3 + x?
    Differentials approximate f(x + dx) ≈ f(x) + dy, where dy = f'(x)dx. For f(x) = x^3 + x, x = 2, dx = 0.1: f(2) = 10, dy = 1.3, so f(2.1) ≈ 10 + 1.3 = 11.3.
  • Explain how to calculate absolute error and relative error when using differentials to approximate a function value, and compute both for the approximation of f(2.1) for f(x) = x^3 + x.
    Absolute error = |exact - approximate|. Relative error = absolute error / exact. For f(2.1), exact = 11.361, approximate = 11.3. Absolute error = |11.361 - 11.3| = 0.061. Relative error = 0.061 / 11.361 ≈ 0.00537.
  • What is the formula for the differential dy in terms of f'(x) and dx?
    The formula is dy = f'(x)dx, where f'(x) is the derivative of the function and dx is the change in x.
  • Given f(x) = x^3 + x, what is f'(x)?
    f'(x) = 3x^2 + 1.
  • How do you calculate dy for f(x) = x^3 + x when x = 2 and dx = 0.1?
    First, find f'(2) = 13, then multiply by dx: dy = 13 × 0.1 = 1.3.
  • How can differentials be used to approximate the value of a function at x + dx?
    You can use f(x + dx) ≈ f(x) + dy, where dy = f'(x)dx.
  • Using differentials, what is the approximate value of f(2.1) for f(x) = x^3 + x?
    f(2) = 10, dy = 1.3, so f(2.1) ≈ 10 + 1.3 = 11.3.
  • How do you calculate the absolute error when using differentials to approximate a function value?
    Absolute error is the absolute value of the difference between the exact value and the approximate value: |exact - approximate|.
  • What is the relative error for the approximation of f(2.1) for f(x) = x^3 + x, given the exact value is 11.361 and the approximate value is 11.3?
    Relative error = 0.061 / 11.361 ≈ 0.00537.