kipriyanovichAlright, future engineers!
A **Derivative** is the instantaneous rate of change of a function, or the slope of its tangent line. Ex: For `f(x) = x^2`, `f'(x) = 2x`. Pro-Tip: It's how fast things *are changing* right now!
HoldMyTypeFixed Point Intersection or involution in #calculus ever wonder how #lambdacalculus handles such cases , plain vanilla lc doesn't butits fixed point combinator : a means to allow for recursive definitions.
untyped lambda calculus, the function to apply the fixed-point combinator to may be expressed using an encoding, like Church encoding. In this case particular lambda terms (which define functions) are considered as values. "Running" (beta reducing) the fixed-point combinator on the encoding gives a lambda term for the result, which may then be interpreted as fixed-point value.
In lambda calculus function (or term) is an implementation of a mathematical function. In the lambda calculus there are a number of combinators (implementations) that satisfy the mathematical definition of a fixed-point combinator.
A combinator is a closed lambda expression, meaning that it has no free variables. The combinators may be combined to direct values to their correct places in the expression without ever naming them as variable
untyped lambda calculus, the function to apply the fixed-point combinator to may be expressed using an encoding, like Church encoding. In this case particular lambda terms (which define functions) are considered as values. "Running" (beta reducing) the fixed-point combinator on the encoding gives a lambda term for the result, which may then be interpreted as fixed-point value.
In lambda calculus function (or term) is an implementation of a mathematical function. In the lambda calculus there are a number of combinators (implementations) that satisfy the mathematical definition of a fixed-point combinator.
A combinator is a closed lambda expression, meaning that it has no free variables. The combinators may be combined to direct values to their correct places in the expression without ever naming them as variable
Alright, future engineers! A **Limit** describes the value a function approaches as its input gets closer to some value. Ex: `lim (x->0) (sin x)/x = 1`. Pro-Tip: It's key for understanding continuity and the foundation of derivatives!
#Calculus #Limits #STEM #StudyNotes
#Calculus #Limits #STEM #StudyNotes
Alright, future engineers!
A **Definite Integral** calculates the *net area* under a function's curve between two points. Ex: Area = ∫[a,b] f(x) dx. Pro-Tip: Think of it as summing infinitely many tiny rectangles!
A **Derivative** measures a function's *instantaneous* rate of change. Ex: `d/dx(x^n) = nx^(n-1)`. Pro-Tip: Think slope of the tangent line! It's crucial for analyzing motion, rates, & optimization.
#Calculus #Derivatives #STEM #StudyNotes
#Calculus #Derivatives #STEM #StudyNotes
Adrian Riskin 
The other day I had a fairly popular post talking about how mathematicians easily and often admit that they don't know things or don't understand things. Today at work a real-life example came up!
Original post linked to in the next toot since apparently I can't post a link and have an image at the same time ... wtf?!
I was helping a student with Calc I in my office. The question gave a function and asked for values of x where the tangent line was horizontal. The function is the first in the image.
This requires taking the derivative with the product rule. The result of this is the second in the image. Since the second term has a denominator (other than 1 of course) we need to combine the two terms so we can set the numerator to 0 and solve.
The result of this operation is the third in the image. Fractions are 0 when their numerators are 0, so the fourth line shows the equation to be solved.
The student got this far without any help but was unable to solve the equation. This is commonplace. After all, the hardest part of calculus is algebra. But I couldn't see how to solve it either, so I told the student this.
At this moment two of my colleagues were talking in the hall outside my office so I told the student I'd ask them about it. Neither knew how to solve it and told me as much. So I told the student, who was actually thrilled that none of us could solve it either.
So I asked Wolfram Alpha, which gave a solution using the Lambert W, aka the productlog function. I'm a combinatorial topologist -- I do graph theory of various kinds. I've heard of this function but otherwise know nothing at all about it. And I'm happy to admit it! Anyway, that's how mathematicians roll.
ETA: Of course this problem shouldn't have appeared in an introductory calculus text since no undergraduate at that level would be able to solve it, so its inclusion was a mistake of the author or the editor.
#Calculus #LambertW #HorizontalTangent #DifferentialCalculus #Math #Mathematics #Mathematicians
Alright, future engineers! The **Mean Value Theorem (MVT)** says for a smooth f(x) on [a,b], there's a 'c' where `f'(c) = (f(b)-f(a))/(b-a)`. Pro-Tip: It links average rate of change to instantaneous! Essential for error bounds.
#Calculus #MVT #STEM #StudyNotes
#Calculus #MVT #STEM #StudyNotes
Alright, future engineers!
A **Limit** is the value a function *approaches* as its input gets infinitesimally close to a specific point. Ex: `lim(x->0) sin(x)/x = 1`. Pro-Tip: Evaluate from both sides to confirm the limit exists!
Your Autistic Life FR/EN/ES"This first odor differential equation does not pass the smell test."
The derivative measures a function's instantaneous rate of change (its slope!). Ex: `d/dx(x^3) = 3x^2`. Pro-Tip: Crucial for optimization – find max/min points when `f'(x)=0`.
#Calculus #Derivatives #STEM #StudyNotes
#Calculus #Derivatives #STEM #StudyNotes