BASIC CAL

• 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.