Precalculus Course

Learn Precalculus in this full college course. These concepts are often used in programming.

⭐️ Lecture Notes ⭐️

⭐️ Course Contents ⭐️
⌨️ (0:00:00) Functions
⌨️ (0:12:06) Increasing and Decreasing Functions
⌨️ (0:17:35) Maximums and minimums on graphs
⌨️ (0:26:38) Even and Odd Functions
⌨️ (0:36:12) Toolkit Functions
⌨️ (0:43:18) Transformations of Functions
⌨️ (0:55:48) Piecewise Functions
⌨️ (1:00:19) Inverse Functions
⌨️ (1:14:34) Angles and Their Measures
⌨️ (1:22:47) Arclength and Areas of Sectors
⌨️ (1:28:39) Linear and Radial Speed
⌨️ (1:33:02) Right Angle Trigonometry
⌨️ (1:40:38) Sine and Cosine of Special Angles
⌨️ (1:48:41) Unit Circle Definition of Sine and Cosine
⌨️ (1:54:11) Properties of Trig Functions
⌨️ (1:04:50) Graphs of Sine and Cosine
⌨️ (2:11:23) Graphs of Sinusoidal Functions
⌨️ (2:21:36) Graphs of Tan, Sec, Cot, Csc
⌨️ (2:30:29) Graphs of Transformations of Tan, Sec, Cot, Csc
⌨️ (2:39:02) Inverse Trig Functions
⌨️ (2:48:49) Solving Basic Trig Equations
⌨️ (2:55:49) Solving Trig Equations that Require a Calculator
⌨️ (3:05:44) Trig Identities
⌨️ (3:13:16) Pythagorean Identities
⌨️ (3:18:37) Angle Sum and Difference Formulas
⌨️ (3:26:33) Proof of the Angle Sum Formulas
⌨️ (3:31:09) Double Angle Formulas
⌨️ (3:38:39) Half Angle Formulas
⌨️ (3:44:50) Solving Right Triangles
⌨️ (3:51:24) Law of Cosines
⌨️ (4:01:24) Law of Cosines - old version
⌨️ (4:09:44) Law of Sines
⌨️ (4:17:34) Parabolas - Vertex, Focus, Directrix
⌨️ (4:29:24) Ellipses
⌨️ (4:40:33) Hyperbolas
⌨️ (4:54:23) Polar Coordinates
⌨️ (5:01:55) Parametric Equations
⌨️ (5:13:22) Difference Quotient

--

[coldandafraid]

No one:
Youtube at 3 am: "hey buddy, got 5 and a half hours to spare?"

[josephwhite]

if you watch this and the 12 hour calc after it... your basically set for the semester. This stuff is gold!

[ErikMrazBR]

After some days watching 30 minutes per day I finished this course and i'm going to calculus 1 course. Thank you for this incredible job, Dr. Linda

[slenderdluex]

🎯 Key Takeaways for quick navigation:

00:18 📊 A function relates input (usually x) to output (usually y), with each input having exactly one output.
02:34 🔄 Function notation (f(x)) represents the output value (y) when a value (x) is input into the function.
12:10 ↗️ A function is increasing if y values go up as x values increase; it's decreasing if y values go down as x values increase.
19:52 📉 Decreasing intervals: Describe where the function is decreasing in terms of x values.
22:44 📉 Local minima: Points where y values are lower than nearby values in an open interval around them.
24:39 🔁 Symmetry: A graph is symmetric with respect to the x-axis if (x, y) implies (x, -y) is on the graph.
27:41 ↔️ Symmetry: A graph is symmetric with respect to the y-axis if (x, y) implies (-x, y) is on the graph.
28:58 🔄 Graph is symmetric with respect to y-axis if (-x, y) is on the graph when (x, y) is.
29:26 🔄 Graph is symmetric with respect to origin if rotating it by 180 degrees makes it line up with itself.
32:33 🔄 Even functions: f(-x) = f(x) for all x. Odd functions: f(-x) = -f(x) for all x.
35:08 📈 Even function example: f(x) = x² + 3. Even both algebraically and symmetrically.
35:30 📉 Odd function example: f(x) = 5x - 1 / x. Odd both algebraically and symmetrically.
43:49 🔄 Transformations of functions: Outside changes affect y-values (vertical motions), inside changes affect x-values (horizontal motions).
50:24 🔄 Translations (adding/subtracting) affect y-values, horizontal shifts in opposite direction from sign.
54:35 📐 The rules for transformations of functions: Outside numbers affect vertical motions, inside numbers affect horizontal motions. Adding and subtracting correspond to shifts, multiplying and dividing correspond to stretches and shrinks. Negative sign corresponds to reflection.
55:48 📈 Piecewise functions are defined in pieces by different rules for different x values. Calculating values for piecewise functions involves applying the appropriate rule based on the given x.
59:21 🔀 Continuity of piecewise functions: Functions with transitions between different rules often have discontinuities at the transition points.
01:01:44 🔁 The graph of an inverse function is obtained by reflecting the graph of the original function over the line y = x.
01:05:37 🔄 Systematic method for finding inverses: Reverse roles of y and x, solve for y, and express the result as the inverse function.
01:07:24 ❌ Not all functions have inverses. A function must be one-to-one (satisfy horizontal line test) to have an inverse.
01:14:49 📐 Conversion between degrees, minutes, and seconds. 1 degree = 60 minutes, 1 minute = 60 seconds.
01:19:01 🔁 Conversion between decimal degrees and degrees, minutes, and seconds involves unit conversions based on ratios.
01:23:54 📐 Arc length is determined by the ratio of the angle to the total angle of the circle, multiplied by the circumference.
01:25:04 📐 Arc length formula: length = angle (in radians) × radius. Angle must be in radians.
01:27:28 📐 Area of sector formula: area = (angle / 2) × radius^2. Angle must be in radians.
01:29:12 📏 Linear speed = angular speed × radius for a rotating circle.
01:33:17 📐 Trig functions (sine, cosine, tangent) relate sides of a right triangle to its angles.
01:35:51 📐 Reciprocal trig functions (secant, cosecant, cotangent) are inverses of sine, cosine, tangent.
01:40:58 📐 Sine and cosine values for 30°, 45°, 60° angles: 1/2, √2/2, √3/2.
01:51:12 📐 The definitions of sine, cosine, and tangent are based on right triangle ratios: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.
01:52:32 🌀 Unit circle definition: cosine(theta) is the x-coordinate, sine(theta) is the y-coordinate, and tangent(theta) is y-coordinate/x-coordinate of a point on the unit circle at angle theta.
01:54:22 🔄 Periodic property: Sine and cosine values repeat with a period of 2π (360 degrees), meaning sine(theta + 2π) = sine(theta) and cosine(theta + 2π) = cosine(theta).
01:55:43 📐 Pythagorean property: cosine(theta)^2 + sine(theta)^2 = 1, which originates from the Pythagorean Theorem applied to points on the unit circle.
02:02:18 🔎 Given sine(t) = -2/7 and t in quadrant III, cosine(t) is calculated as the negative square root of 45/7.
02:11:58 ⚙️ Shifting the graph: Adding or subtracting constants inside the function shifts the graph vertically.
02:17:30 📈 Understanding sinusoidal functions: Horizontal shifts, period adjustments, and amplitude changes in graphs of functions like y = a sin(bx - c) + d.
02:21:47 📉 Intuition for graphing tangent (tan) function: Relating slope of a line on the unit circle to the tangent graph, vertical asymptotes, undefined slopes, and periodicity of pi.
02:26:20 🔄 Graphing secant (sec) and cosecant (csc) functions: Converting from cosine and sine, vertical asymptotes, period, range, and domain.
02:37:57 🔄 Graphing transformations of tangent, secant, cotangent, and cosecant: Effects of amplitude changes, period adjustments, vertical and horizontal shifts.
02:39:11 ↩️ Understanding inverse trigonometric functions: Relationship between original trig functions and their inverses, domain and range restrictions, notation (arc, inverse), and graphical interpretation.
02:46:17 📐 The restricted domain of tangent (tan) function is from -π/2 to π/2, with vertical asymptotes at these endpoints. The range of restricted tan is from negative infinity to infinity.
02:48:41 🧪 Sine inverse (arc sine), cosine inverse (arc cosine), and tangent inverse (arc tangent) functions were introduced as inverse trig functions.
02:53:29 🕰️ Solving trig equations involves isolating the trig function, finding principal solutions in a given interval, then adding multiples of the function's period for all solutions.
03:07:49 ❌ Not all equations are identities. An equation that holds for some values of the variable is not an identity.
03:15:34 🔁 Pythagorean identity holds across different quadrants through symmetry.
03:41:22 📐 We derived the half angle formulas for cosine and sine: cos(θ/2) = ±√((cosθ + 1)/2) and sin(θ/2) = ±√((1 - cosθ)/2).
03:42:41 🔍 Given sine θ = 4/5 and θ in the second quadrant (π/2 to π), we found exact values: cos(θ/2) = 1/√5 and sin(θ/2) = 2/√5.
03:48:56 📏 Given one side and one angle in a right triangle, we used tangent to find the other side, and Pythagorean Theorem to find the hypotenuse.
03:51:11 📐 Solved triangles using side-side-side (SSS) and side-angle-side (SAS) using Law of Cosines and trigonometric functions.
04:05:20 📏 Applied Law of Cosines to find side length and angles in a triangle with known side lengths.
04:17:44 📐 Equations of a parabola: The equations y - k squared = 4px - h and x - h squared = 4py - k define parabolas. They're shifted by (h, k), and the distance |p| represents focus-vertex and vertex-directrix distances.
04:34:57 🥇 Ellipse equation and relationships: Ellipses elongated horizontally: x²/a² + y²/b² = 1. Relationship: a² = b² + c² (major, minor axis, focus-vertex distance).
04:37:12 📌 Equations for ellipses elongated vertically have x and y roles reversed, involving shifts in both directions.
04:37:41 🌟 Key points like vertices and foci can be labeled and calculated for ellipses with shifted centers.
04:39:24 🔵 The distance definition of a hyperbola involves the constant difference of distances to its foci.
04:42:12 🌀 Hyperbolas can have horizontal or vertical transverse axes, affecting their orientation like lying down or standing up.
04:43:34 📏 Hyperbolas' equations differ based on their orientation, involving parameters A, B, and C representing distances.
04:45:15 ⚙️ Formulas for distances to foci and vertices help derive hyperbolic equations, with A, B, and C being related quantities.
04:46:37 📊 Hyperbolas' slopes and equations of asymptotes depend on A and B, creating guidelines for drawing the curves.
04:49:27 ✒️ The conversion between polar and Cartesian coordinates involves trigonometric equations and understanding angles and radii.
05:00:16 🔄 Polar coordinates allow points to be represented with radius and angle, with adjustments for negative radii and angles.
05:02:27 📈 Parametric equations enable the description of curves using separate functions for x and y in terms of a third variable, often time.
05:02:54 📊 Parametric equations relate x and y to a third variable called the parameter, and graphing them involves finding corresponding x and y values for a given parameter value.
05:04:41 🔵 Graphs like circles on the xy-plane can be described by parametric equations (e.g., x = cos(t), y = sin(t)), and their Cartesian equation is often derived using trigonometric identities.
05:05:50 ⭕ Unit circles with radius 1 are described by x^2 + y^2 = 1 in Cartesian coordinates, and this equation can be parameterized using x = cos(t) and y = sin(t).
05:08:57 🔢 Parametric equations can also be created from given Cartesian equations, like parameterizing curves using x = t or other equations based on trigonometric functions.
05:11:07 📍 The equation of a circle centered at (h, k) with radius r is (x - h)^2 + (y - k)^2 = r^2, and parametric equations for circles are derived by scaling and shifting unit circle parametric equations.
05:12:49 ⚖️ Difference quotient and average rate of change concepts relate to the derivative and calculus, where average rate of change represents the slope of secant lines between two points on a function graph.

Made with HARPA AI

[luizhenriquebinharaarantes]

2020 conronavirus pandemic, day 240.: today i've decided to learn some calculus

[miriosdick]

I just wanted to say, thank you so much for this free course and I'm really grateful for all the work you put in these videos.
I'm 16 years old and my school system sucks, our math is really bad and without you I don't know if I was going to continue learning math aside from the stupid school system that doesn't help at all.
Books are also a great source of information, I'm very thankful that people like you exist.

[ltcolumbo]

Great refresher before taking college calc finished in about a week, would recommend for any person to take the summer before they start a STEM major in college. Next up the calculus courses :)

[johnsmith]

Errors:
1. 1:24:47
10m should be 20m
2. 1:53:32
The answer to tan φ should be -2.7477 The answer shown was calculated by (cos φ) / (sin φ)
when in fact tan φ should be calculated as (sin φ) / (cos φ)

This is a great course and I am very thankful to have access to it for free. I am only listing these for the purpose of helping others, in case they were confused at these parts :)

[michaeldavis]

I’m 25 years old, have a full time job and not even in college, yet decided that I want to learn some math today! I’m only 7 minutes in and I got to say that the way the teacher is teaching this course is very relaxing and soothing!

[Your_Boy_Suraj]

⌨️ (0:00:00) Functions
⌨️ (0:12:06) Increasing and Decreasing Functions
⌨️ (0:17:35) Maximums and minimums on graphs
⌨️ (0:26:38) Even and Odd Functions
⌨️ (0:36:12) Toolkit Functions
⌨️ (0:43:18) Transformations of Functions
⌨️ (0:55:48) Piecewise Functions
⌨️ (1:00:19) Inverse Functions
⌨️ (1:14:34) Angles and Their Measures
⌨️ (1:22:47) Arclength and Areas of Sectors
⌨️ (1:28:39) Linear and Radial Speed
⌨️ (1:33:02) Right Angle Trigonometry
⌨️ (1:40:38) Sine and Cosine of Special Angles
⌨️ (1:48:41) Unit Circle Definition of Sine and Cosine
⌨️ (1:54:11) Properties of Trig Functions
⌨️ (1:04:50) Graphs of Sine and Cosine
⌨️ (2:11:23) Graphs of Sinusoidal Functions
⌨️ (2:21:36) Graphs of Tan, Sec, Cot, Csc
⌨️ (2:30:29) Graphs of Transformations of Tan, Sec, Cot, Csc
⌨️ (2:39:02) Inverse Trig Functions
⌨️ (2:48:49) Solving Basic Trig Equations
⌨️ (2:55:49) Solving Trig Equations that Require a Calculator
⌨️ (3:05:44) Trig Identities
⌨️ (3:13:16) Pythagorean Identities
⌨️ (3:18:37) Angle Sum and Difference Formulas
⌨️ (3:26:33) Proof of the Angle Sum Formulas
⌨️ (3:31:09) Double Angle Formulas
⌨️ (3:38:39) Half Angle Formulas
⌨️ (3:44:50) Solving Right Triangles
⌨️ (3:51:24) Law of Cosines
⌨️ (4:01:24) Law of Cosines - old version
⌨️ (4:09:44) Law of Sines
⌨️ (4:17:34) Parabolas - Vertex, Focus, Directrix
⌨️ (4:29:24) Ellipses
⌨️ (4:40:33) Hyperbolas
⌨️ (4:54:23) Polar Coordinates
⌨️ (5:01:55) Parametric Equations
⌨️ (5:13:22) Difference Quotient

[aidenramirez]

For anyone else using this, at 1:25:09 the arclength should be 20m instead of 10m. Otherwise the video is tremendously useful, thank you for your work!!!

[dulxbe]

took me a month, but I finally finished the video :))

[zionification]

You guys are really doing a awesome job to make this courses accessible to others for free, this is the best thing anyone can do for the whole mankind

[jameswoodford]

Watches on 2x speed to be confused twice as fast😎

[rickennmakesgames]

going to be putting this on while i sleep

[wonsoongi]

i took a gap year to study for my uni entrance exams and i watch these videos almost daily, as well as some of the programming tutorials, and it's like i've signed up for an amazing college boot camp for free! i truly appreciate the effort that goes into producing these videos <3

[misuzuflv]

i found this channel like a minute ago when this was posted wow

[ADA]

No Joke this is one of the best teachers i have ever seen. She explains to you things in a away that refreshes your memory and makes things that didn't make sense before finally make sense. probably because we got taught poorly. This is truly a gift. Thank you!!!

[bigmike]

I have a precalc final in about 12 hours this is gonna be so helpful!

[gdahlm]

I really do appreciate this video for my review after decades.

That said there is an error at 1:25:10 2.5 * 8 is not 10, it is 20.

It would be nice if there was an annotation added to call that out.