INTUITION MATTERS! Applied Calculus in Engineering: Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

Created byMark Misin Engineering

Last updated 11/2025

English

What you'll learn

You will develop very strong intuition & understanding in Calculus
You will also receive very strong training in problem solving in Calculus
You will learn how to apply Calculus in real life to the level not seen in other courses
You will see how to apply Calculus in control systems by designing a PID controller

Course content

15 sections • 230 lectures • 33h 31m total length

Functions - what are they - high school review 19:14
Functions - what are they - high school review 28:58
SIM: Straight line functions in motion3:18
Practice functions - airplane homework7:37
Abstract World VS Real World - 15:13
Abstract World VS Real World - 25:10
Straight line functions - How to manipulate them9:41
Inverse functions - what are they and how to find them7:35
Find inverse functions - example5:00
Parabolic functions - How to manipulate them11:16
Cubic functions - How to manipulate them6:17
SIM: Car race exercise - who will win the race?1:01
SIM: Car race exercise - who will win the race - SOLUTION2:38
What is a function, what is not - 18:14
What is a function, what is not - 26:06
Inverse function of a parabola - how to find it5:53
Inverse function of a cubic function- how to find it6:26
Functions in general form - how to manipulate them9:51
SIM: General functions in motion5:35
Piecewise defined functions - how to form them9:24
How to shift functions7:41
How to do function operations11:05
Practice function operations7:06
Quadratic function roots - How to get them 19:52
Quadratic function roots - How to get them 26:51
SIM: 3 airplane race - who wins, and when?1:19
Follow-up2:06

Pi number and circumference of a circle - what are they and how to get them12:38
Radians VS Degrees & how to get the arc length of a circle10:03
Sine function - where does it come from11:09
Cosine and tangent function - where do they come from7:01
Inverse of trigonometric functions - how to get them9:26
Practice trigonometric functions - exercises4:11
Abstract VS Real - Aerospace Engineering example11:53
SIM: Avoid the crash - exercise3:39
SIM: Avoid the crash - solution 11:46
SIM: Avoid the crash - solution 24:14
SIM: Avoid the crash - solution 32:52
SIM: Avoid the crash - solution 46:30
SIM: Avoid the crash - solution 55:13
SIM: Avoid the crash - solution 61:55
SIM: Avoid the crash - solution 73:42
SIM: Avoid the crash - solution 81:01
SIM: Avoid the crash - solution 94:41
SIM: Avoid the crash - solution 101:01

Variable equaling VS approaching something - what's the difference11:10
Variable approaching infinity - what happens7:07
Smart financial decision using limits - how to do it9:44
Manipulate trigonometric functions in different ways - how10:05
Model a wave with a trigonometric function - how9:53
Get from a sine to a cosine function - exercise2:15
Practice limits on a trigonometric function - does the limit exist?3:58
SIM: Limit simulation2:55
SIM: sin_limit simulation1:13

Derivatives - what are they - falling sphere example 111:28
Where does the the equation Hi-5*t^2 come from?0:24
Derivatives - what are they - falling sphere example 29:52
Derivatives - what are they - falling sphere example 311:29
SIM: Earth, Mars, Moon - exercise3:21
SIM: Earth, Mars, Moon - solution3:37
Practice your derivative intuition - exercises9:02
Prove that d(sin(theta))/d(theta)=cos(theta)
chain rule - how to apply it to a multi link problem - water pipe example 19:18
chain rule - how to apply it to a multi link problem - water pipe example 213:18
Differential operators - What are they and how to apply them8:33
Multilevel derivatives - double, triple, quadruple6:11
Derivative rules - shortcuts 15:43
Derivative rules - shortcuts 210:33
Implicit VS Explicit functions - chain rule - water pipe example11:55
Derivative rules - BIG exercise11:07
Trigonometric function derivatives - how to take them 17:56
Trigonometric function derivatives - how to take them 27:17
Minimize the material used to produce a can - how to apply derivatives 111:09
Minimize the material used to produce a can - how to apply derivatives 26:42
Minimize the material used to produce a can - how to apply derivatives 312:49
Exponential functions - what are they - how to manipulate them8:16
Logarithmic functions & log scale - what are they & derivatives11:42
Maximize your profit - how to apply derivatives for that11:12
Motion of a Projectile (The Crazy Motorcyclist)
Practice trigonometry with this free online game0:53

Vectors in 2 dimensions - what are they9:17
Vectors in multiple dimensions - what are they14:02
Vector terminology and disclaimers5:51
Vector magnitude - what is it10:19
Vector components - exercise5:25
Vector operations - how do they work4:17
Apply the dot product - Forces on a box example5:33
Cross product intuition- what is it5:57
Cross product - how to calculate it10:55
Apply the cross product - electron and magnetic field example8:55
MEGA Application - Water tank example 16:01
MEGA Application - Water tank example 28:36
MEGA Application - Water tank example 310:46
CORRECTION: MEGA Application - Water tank example 34:31
MEGA Application - Water tank example 411:30
CORRECTION: MEGA Application - Water tank example 41:36
MEGA Application - Water tank example 54:53
SIM: MEGA Application - Water tank example - Complete1:30
MEGA Application - Water tank example - exercise 19:52
MEGA Application - Water tank example - exercise 29:36
SIM: MEGA Application - Water tank example - exercise0:46
MEGA Application - Water tank mass flow- use derivatives to get it12:57
Distance VS Position vectors - what's the difference13:36
Displacement VS Velocity VS Acceleration vectors - what's the difference12:37
Average displacement, velocity, and acceleration vectors - clarification1:57
Graph Displacement, Velocity, & Acceleration vectors7:18
Acceleration vector - intuition5:41
Vector derivatives - how do they work2:06

Intro into Parametric Equations - Intuition7:32
MEGA Application - Landing Airplane in 2D7:03
SIM: MEGA Application: Landing simulation3:22
MEGA Application - Airplane flying in an oscillatory way (in 2D)11:25
MEGA Application - Airplane's velocity and acceleration in 2D18:09
SIM: MEGA Application - Sin trajectory simulation5:31
MEGA Application - Airplane doing circles in 2D - Cartesian & Polar coordinates16:29
MEGA Application - Airplane doing circles in 2D - Velocity & Acceleration17:41
SIM: MEGA Application - Circle simulation7:28
MEGA Application - Airplane flying in a spiral in 3D14:47
SIM: MEGA Application - 3D trajectories - simulation 12:44
SIM: MEGA Application - 3D trajectories - simulation 21:35
SIM: MEGA Application - 3D trajectories - simulation 33:06
SIM: MEGA Application - 3D trajectories - simulation 42:38
SIM: MEGA Application - 3D trajectories - simulation 51:14
MEGA Application - 2 joint robot - Angular position & velocity4:44
MEGA Application - 2 joint robot - 2 joints rotating simultaneously10:03
MEGA Application - 2 joint robot - assigning multiple reference frames14:32
MEGA Application - 2 joint robot - describing a point in two different frames8:43
MEGA Application - 2 joint robot - deriving a 2D rotational matrix12:58
MEGA Application - 2 joint robot - describing rotating and translating joints18:45
MEGA Application - 2 joint robot - joints rotating at different frequencies7:48
SIM: MEGA Application - 2 joint robot - simulation 14:27
SIM: MEGA Application - 2 joint robot - simulation 23:24
3D Rotational matrix - how to derive it 118:19
3D Rotational matrix - how to derive it 211:05
3D Rotational matrix - how to derive it 312:50
Spiral - Final shape

Implicit functions - what are they and how to graph them13:22
Implicit function derivatives - how to take them11:42
Expanding function dimensions 16:40
Expanding function dimensions 26:28
Machine Learning example - how to apply multidimensionality16:50
Partial derivatives - intro 110:36
Partial derivatives - intro 214:51
Partial derivatives - intro 313:47
Higher dimensional partial derivatives & its graph12:57
Airplane example - Chain rule in parametric equations - partial derivatives 16:37
Airplane example - Chain rule in parametric equations - partial derivatives 29:46
Multilevel partial derivatives - how to take them14:48
Gradient - what is it and how to find it7:33
Neural Networks + Calculus (Article recommendation)0:20
Taylor Series - what is it and how to formulate it 116:57
Taylor Series - what is it and how to formulate it 219:42
Linearization and Quadratic Approximation - what is it and how to do it28:24
Multidimensional linearization - practical exercise22:44
Multidimensional quadratic approximation - Hessian Matrix16:46
Representing a system mathematically, in vector-matrix form, and block diagrams18:20

Discrete VS Continuous functions - what's the difference10:44
Intro to Integrals - The area of the wall calculation 110:30
Intro to Integrals - The area of the wall calculation 214:36
Integrals - what's the meaning of it 111:56
Integrals - what's the meaning of it 213:10
Integrals - what's the meaning of it 313:42
Integrals - what's the meaning of it 417:47
Indefinite Integrals VS Definite Integrals - what's the difference15:02
Definite Integrals - Area of the wall calculation 122:04
Definite Integrals - Area of the wall calculation 212:40
Improper Integrals - made intuitive20:42
Definite Integrals - formal definition20:41
Area of a circle & initial conditions - integrals applied13:29
Area of a circle - 2nd approach - integrals applied13:58
Volume of a solid - integrals applied15:58
Integration by parts - where it comes from and how to apply it14:03
Curve length calculation - integrals applied 116:19
Curve length calculation - integrals applied 210:26
Spiral length calculation in 3D - integrals applied12:31
Spiral trajectory length

Requirements

Addition, Subtraction, Multiplication, Division
very basic algebra

Description

I have NEVER believed in simply memorizing things. Your greatest asset in life is time. If you study something, you invest your time in it. If you memorize it, soon you will forget it - that is not a great investment. But if you truly understand something, that stays with you for good - it can serve you for a long time.

This course will give you strong intuition and understanding in Calculus, and you will be trained to apply it in real life. It will stay in your head for good - this is a GREAT investment! Watch some of my free preview videos, enroll in my course, and I will make Calculus second nature to you - that is a promise!

These are the topics that the course covers:

Single variable functions
Trigonometric functions
Multidimensional functions
Limits
Derivatives - single variable
Vectors
Transformations
Parametric Equations
Multivariable Calculus
Single variable integrals
Multivariable integrals
Introduction to Differential Equations & waves in time and frequency domain
Calculus in Control Engineering (PID controller & magnetic train simulation)

Most sections contain a small number of free preview videos for you to check out.

The great thing about this course is that it prepares you for pretty much all the other Engineering and Science courses, because Calculus is the basis of everything in this world.

Good luck!

Who this course is for:

Science and Engineering students
Working Scientists and Engineers

INTUITION MATTERS! Applied Calculus in Engineering: Complete

What you'll learn

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Course content

Single variable functions27 lectures • 3hr

Trigonometric functions18 lectures • 1hr 43min

Multidimensional functions3 lectures • 30min

Limits9 lectures • 58min

Derivatives - single variable24 lectures • 3hr 24min

Vectors28 lectures • 3hr 26min

Transformations6 lectures • 39min

Parametric Equations27 lectures • 4hr 8min

Multivariable Calculus20 lectures • 4hr 29min

Single variable integrals19 lectures • 4hr 40min

Requirements

Description

Who this course is for: