Alright, future engineers!
**Chain Rule:** Differentiates composite functions (function within a function).
Ex: `d/dx sin(x^2) = cos(x^2) * 2x`.
Pro-Tip: Always differentiate outside in!
#Calculus #Differentiation #STEM #StudyNotes

Alright, future engineers!
The **Intermediate Value Theorem (IVT)** says a continuous function `f` on `[a,b]` takes every value between `f(a)` & `f(b)`.
Ex: If `f(0)=1, f(2)=5`, then some `x` in `[0,2]` has `f(x)=3`.
Pro-Tip: Use it to prove a root *exists*!
#Calculus #Theorem #STEM #StudyNotes

Alright, future engineers!
**Optimization:** Using derivatives to find the max or min value of a function (e.g., max volume, min cost).
Ex: Set `f'(x) = 0` to find critical points.
Pro-Tip: Don't forget to check *endpoints* of your interval too, not just critical points!
#Calculus #Optimization #STEM #StudyNotes

Alright, future engineers!
A **Limit** describes the behavior a function approaches as its input gets closer & closer to a certain value.
Ex: `lim (x->2) x^2 = 4`.
Pro-Tip: Think of it as predicting where the function is *going*, not necessarily where it *is*.
#Calculus #Limits #STEM #StudyNotes

Alright, future engineers!
**Derivative:** The instantaneous rate of change of a function, or the slope of its tangent line.
Ex: Power Rule: `d/dx (x^n) = nx^(n-1)`
Pro-Tip: It tells you how fast is this changing *right now*?
#Calculus #Derivatives #STEM #StudyNotes

Grazie ad Andrea proviamo questo programma scritto con il BASIC v7 del Commodore 128 che traccia grafici ed esegue calcoli numerici su funzioni reali di variabile reale. #commodore128 #basic #calculus https://www.youtube.com/watch?v=053DeWHXdUA

Alright, future engineers!
**Optimization** uses derivatives to find the max or min values of a function (e.g., max profit, min cost).
Ex: Set `f'(x)=0` to find critical points.
Pro-Tip: Always check endpoints and domain boundaries!
#Calculus #Optimization #STEM #StudyNotes

Alright, future engineers!
The **Chain Rule** differentiates a function inside another (a composite function).
Ex: `d/dx f(g(x)) = f'(g(x)) * g'(x)`
Pro-Tip: Differentiate the *outside*, then multiply by the derivative of the *inside*! Think 'onion layers.'
#Calculus #Differentiation #STEM #StudyNotes

Alright, future engineers!
**Concavity** describes how a curve bends – up (like a cup) or down (like a frown).
Ex: `f''(x) > 0` for concave up.
Pro-Tip: Inflection points occur where concavity changes, i.e., `f''(x)=0` or undefined!
#Calculus #CurveSketching #STEM #StudyNotes