kipriyanovichAlright, future engineers!
The **Product Rule** differentiates a product of two functions.
Formula: `(fg)' = f'g + fg'`
Pro-Tip: 'First D Second + Second D First' is a classic mnemonic!
Alright, future engineers!
The **Fundamental Theorem of Calculus (FTC)** links differentiation & integration. It states: If `F(x)=∫_a^x f(t)dt`, then `F'(x)=f(x)`.
Ex: If `F(x)=∫_0^x cos(t)dt`, `F'(x)=cos(x)`.
Pro-Tip: It's the ultimate 'undo' button between derivatives & integrals!
#Calculus #FTC #STEM #StudyNotes
The **Fundamental Theorem of Calculus (FTC)** links differentiation & integration. It states: If `F(x)=∫_a^x f(t)dt`, then `F'(x)=f(x)`.
Ex: If `F(x)=∫_0^x cos(t)dt`, `F'(x)=cos(x)`.
Pro-Tip: It's the ultimate 'undo' button between derivatives & integrals!
#Calculus #FTC #STEM #StudyNotes
Sci-books.comThe Calculi of Lambda-Conversion (AM-6), Volume 6 (Annals of Mathematics Studies) by Alonzo Church (PDF)
Author: Alonzo Church
File Type: PDF
Download at https://sci-books.com/the-calculi-of-lambda-conversion-am-6-volume-6-annals-of-mathematics-studies-b088pk9pc7/
#Calculus, #AlonzoChurch
Author: Alonzo Church
File Type: PDF
Download at https://sci-books.com/the-calculi-of-lambda-conversion-am-6-volume-6-annals-of-mathematics-studies-b088pk9pc7/
#Calculus, #AlonzoChurch
Your Autistic Life FR/EN/ES
Alright, future engineers!
The **Mean Value Theorem (MVT)** guarantees that a function's instantaneous rate of change equals its average rate over an interval.
Ex: `f'(c) = (f(b)-f(a))/(b-a)` for some `c` in `(a,b)`.
Pro-Tip: It's all about guaranteeing a *specific point* where slopes match!
#Calculus #MVT #STEM #StudyNotes
The **Mean Value Theorem (MVT)** guarantees that a function's instantaneous rate of change equals its average rate over an interval.
Ex: `f'(c) = (f(b)-f(a))/(b-a)` for some `c` in `(a,b)`.
Pro-Tip: It's all about guaranteeing a *specific point* where slopes match!
#Calculus #MVT #STEM #StudyNotes
**Critical Points**: Where `f'(x)=0` or `f'(x)` is undefined. These are spots where local max/min *might* occur.
Ex: `f(x)=x^2`, `f'(x)=2x`. `2x=0` means `x=0` is a crit. pt.
Pro-Tip: Always test critical pts (and endpoints!) for absolute extrema!
#Calculus #Optimization #STEM #StudyNotes
Ex: `f(x)=x^2`, `f'(x)=2x`. `2x=0` means `x=0` is a crit. pt.
Pro-Tip: Always test critical pts (and endpoints!) for absolute extrema!
#Calculus #Optimization #STEM #StudyNotes
Alright, future engineers!
A **Limit** is the value a function approaches as its input gets arbitrarily close to a specific point.
Ex: `lim(x->2) (x^2 - 4)/(x - 2) = 4`.
Pro-Tip: Always simplify first if you hit 0/0!
Alright, future engineers!
**U-Substitution:** Simplifies integrals by changing variables.
Ex: For `∫2x cos(x^2) dx`, let `u=x^2`, so `du=2x dx`.
Pro-Tip: Find a function & its derivative in the integrand!
Alright, future engineers!
A **Critical Point** is where a function's derivative is zero or undefined. Ex: For `f(x)=x^3-3x`, `f'(x)=3x^2-3=0` at `x=±1`. Pro-Tip: They're candidates for local max/min. Use 1st/2nd Derivative Tests!
Alright, future engineers!
**Optimization** is finding the maximum or minimum value of a function to solve a real-world problem.
Ex: To maximize profit P(x), set P'(x)=0 & check critical points.
Pro-Tip: It’s how we design for peak performance or minimal cost!
#Calculus #Optimization #STEM #StudyNotes
**Optimization** is finding the maximum or minimum value of a function to solve a real-world problem.
Ex: To maximize profit P(x), set P'(x)=0 & check critical points.
Pro-Tip: It’s how we design for peak performance or minimal cost!
#Calculus #Optimization #STEM #StudyNotes