BASIC CAL

Subdecks (6)

Cards (83)

  • is a line that passing through two points of a curve.
    Secant Line
  • Secant line
    A line passing through two points of a curve
  • Slope of secant line
    (-f(x2) - f(x1)) / (x2 - x1)
  • Tangent line
    A straight line that passes through a point on a curve and has the same slope as the curve at that point
  • Slope of tangent line
    lim(Δx→0) (f(x0 + Δx) - f(x0)) / Δx, where Δx ≠ 0
  • Not all functions have a tangent line at all points
  • Function with no tangent line at a point

    • f(x) = |x| has no tangent line at (0,0)
  • Derivative
    lim(Δx→0) = (f(x0 + Δx) - f(x0)) / Δx
  • If the derivative exists, the function is differentiable at that point
  • Derivative computation
    • f(x) = 3x^2 + 2x - 1, f'(1) = 11
  • Leibniz notation for derivative

    df/dx, d/dx f(x), f'(x)
  • If a function is differentiable at a point, it is also continuous at that point
  • Proof that if a function is differentiable at a point, it is also continuous at that point
  • Upper bound
    A number ! such that ! ∀ % ∈
  • Lower bound
    A number ' such that % ≥ ' ∀ % ∈
  • Supremum
    The least upper bound of a set
  • Infimum
    The greatest lower bound of a set
  • If a nonempty set of real numbers has a lower bound, then it has a greatest lower bound or an infimum. Equivalently, if it has an upper bound, then it has a least upper bound or supremum.
  • Intermediate Value Theorem
    If a function ) is continuous on a closed interval [a,b] and )(a) ≠ )(b), then for any number E between )(a) and )(b), there exists a point c in (a,b) such that )(c) = E.
  • Bisection Method
    1. Set a margin of error M
    2. Define x1 and x2
    3. Define xm = (x1 + x2)/2
    4. If |f(xm)| ≤ M, take x = xm and end. Else, proceed to step 5
    5. If f(xm) < 0, let x1 = xm. Else, let x2 = xm and go back to step 3.
  • If a sequence {xn} is monotone (increasing or decreasing) and bounded, then it has a finite limit.
  • Absolute maximum
    A point x0 such that f(x) f(x0) for all x in the domain
  • Absolute minimum
    A point x0 such that f(x) ≥ f(x0) for all x in the domain
  • Local maximum
    A point x0 such that f(x) f(x0) for all x in a small neighbourhood around x0
  • Local minimum
    A point x0 such that f(x)f(x0) for all x in a small neighbourhood around x0
  • If a sequence {xn} is convergent, then it is bounded.
  • and by definition of M, |%S| − |%| ≤ |%S − %| < 1. This implies that |%S| < 1 + |%|
  • Define ε = max{1 + |%|, |%R|, |%\|, … , |%|}
    Then ∀ n ∈ ℕ, |%S| ε
  • References:
    Coburn, J. (2016). Pre-Calculus. McGraw Hill Education.
    Minton, R. & Smith, R. (2016). Basic Calculus. McGraw Hill Education.