calc final s1


📘 CC1: Limits, Asymptotes & Continuity




Limits



  • Limit notation:
    \lim_{x \to a} f(x) → what f(x) approaches, not necessarily its value.

  • Ways to find limits:

    • Plug in (direct substitution)

    • Simplify (factor & cancel)

    • Rationalize

    • Use a graph


  • Limit exists if:
    Left-hand limit = right-hand limit.




Asymptotes



  • Vertical asymptote:
    \lim_{x \to a} f(x) = \pm\infty
    → denominator = 0 and does NOT cancel

  • Horizontal asymptote:
    \lim_{x \to \pm\infty} f(x) = L

    • Same degree → ratio of leading coefficients

    • Bottom degree bigger → y=0

    • Top degree bigger → no horizontal asymptote





Continuity



A function is continuous at x=a if:


  1. f(a) exists

  2. \lim_{x\to a} f(x) exists

  3. Limit = function value



⚠ Holes, jumps, and asymptotes = not continuous



Special Tools



  • Squeeze Theorem: If stuck between two functions with same limit → same limit

  • L’Hôpital’s Rule: Only for 0/0 or \infty/\infty






📗 CC2: The Derivative




What a Derivative Is



  • Slope of the tangent line

  • Instantaneous rate of change

  • Velocity = derivative of position




Derivative Notation



  • f'(x), \dfrac{dy}{dx}




Limit Definition



f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}



Differentiability



  • Differentiable ⇒ continuous

  • Continuous ❌⇒ differentiable

  • NOT differentiable at:

    • Corners

    • Cusps

    • Vertical tangents

    • Discontinuities





Derivative Rules (Memorize These)



  • Constant → 0

  • Power Rule: x^n \to nx^{n-1}

  • Constant multiple: 3x^2 \to 6x

  • Sum/Difference → take separately

  • Product Rule

  • Quotient Rule

  • Chain Rule (outside · inside’)

  • Trig, exponential, logarithmic derivatives




Tangent Lines



  • Need slope (derivative) + point

  • Vertical tangent → undefined slope

  • Horizontal tangent → slope = 0




Motion Problems



  • Position → Velocity → Acceleration

  • Speed = |v|

  • Speeding up: velocity & acceleration same sign

  • Slowing down: opposite signs




Implicit Differentiation



  • Differentiate both sides

  • Chain rule when differentiating y

  • Solve for \dfrac{dy}{dx}






📙 CC3: Curve Sketching & Graph Behavior (SIGN CHARTS)




Key Relationships



  • f' > 0 → increasing

  • f' < 0 → decreasing

  • f'' > 0 → concave up

  • f'' < 0 → concave down




Critical Points



  • Where f' = 0 or undefined




First Derivative Test



  • + to - → local max

  • - to + → local min




Second Derivative Test



  • f'(a)=0 and f''(a)>0 → min

  • f'(a)=0 and f''(a)<0 → max




Inflection Points



  • Where concavity changes

  • f''=0 or undefined AND sign changes