Intuitive Calculus

Subject classification: this is a mathematics resource.

Book: Infinite Powers by Steven Strogatz (ISBN#: 1328879984)

Educational level: this is a tertiary (university) resource.

Type classification: this is a notes resource.

Completion status: this resource is a stub, which means that pretty much nothing has been done yet.

Attribution: User Atcovi created this resource and is actively using it. Please coordinate future development with this user if possible.

Notes

4/11/2026 (Archimedes and the method of exhaustion)

Archimedes and figuring out the quadratic (or computation of the area) of a parabolic segment. This is just basically spamming smaller triangles into a parabola to equal one big triangle ( ${\textstyle =1}$ ) in order to figure out the area.

Total area of a parabolic segment from Archimedes findings: ${\textstyle 1}$ + ${\textstyle 1/4}$ + ${\textstyle 1/16}$ + ${\textstyle 1/64}$ ← geometric series.

^each term is ${\textstyle 1/4}$ of the term preceding it as the daughter triangles always contribute a total of 1 quarter as much area as their parents do.

Archimedes proved that ${\textstyle a=4/3}$ through a double reductio ad absurdum^[1] using the method of exhaustion, an analytical way of finding a result^[2].

5/2/2026 (Johannes Kepler)

Johannes Kepler

Elliptical orbits
- Ellipse: Plane curve where the sum of distances from any point on the curve to two fixed points (foci) is constant. For example, a circle is a type of ellipse. A circle is a set of points where distance from a given point (aka its center) is constant. Kepler stated that all planets follow an elliptical orbit.
Equal Areas in Equal Times
- Formula: Time (P₁ → P₂) = Time (P₃ → P₄) [their sectors have equal areas]
Third Law and the Sacred Frenzy^[3]
- ${\textstyle T}$ ${\textstyle T}$ ² = ${\textstyle a}$ ${\textstyle a}$ ³
  - ${\textstyle T}$ = how long it takes for a planet to go around the sun just once.
  - ${\textstyle A}$ = avg. of the planet's nearest and farthest distance from the sun.

5/14/2026 (Calculus definitions, introduction to adequality)

Differential calculus: cuts complicated problems into infinitely many simpler pieces. Ex, derivatives.
Integral calculus: puts the pieces back together again to solve the original problem. Ex, integrals.

Analytical geometry: Also known as Cartesian geometry, is geometry using a coordinate system (pictured towards the left). Analytical geometry is used in physics, engineering, and aviation. "Analysis" in analytic geometry is meant to be understood as a way of figuring out the results rather than proving the results^[5].

Adequality

See pages 103 to 107, which provide a breakdown of Pierre de Fermat and his concept of adequality.

Pierre de Fermat's concept of adequality (meaning approximate equality^[6]) was a way of finding the maxima, minima, tangents, and other problems in calculus. For example, two nearly equal values, [let's say] a and b at the maximum of a parabola, are used to find the maxima of a parabola through a small 'nudge' in the variable^[7].

Fermat's ideas eventually led to the concept of derivatives (illustrated towards the right) in modern calculus.

5/16/2026 (continuation of Fermat's adequality)

What does b - (x₁ + x₂) = 0 represent?

b = x₁ + x₂

Reference the hill diagram in Figure 1 (you may have to open the file and zoom in). X₁ and X₂ represent two nearby points on both sides of the "hill" which both produce almost the same output.

For both of the values, adding both X₁ and X₂ would equal ${\textstyle b}$ (the total length). B = x₁ + x₂ would come out to B = 2x, with x = b/2 (where the maximum is). This is the value of ${\textstyle x}$ that would ideally give the highest value for ${\textstyle c}$ (see below).

Purpose of bx - x² = c?

What is the purpose of the equation (see https://youtube.com/AOKoo_nQSts?si=1RfOYMAHm-Ll5sVT&t [minute 4:17] for context/writing of this equation): ${\textstyle bx}$ - ${\textstyle x}$ ² = ${\textstyle c}$ ?

If we take a line (total = ${\textstyle b}$ ), and make a cut at some point in the line (and designate the cut 'mark' as ${\textstyle x}$ ), how could we figure out ${\textstyle c}$ (output produced by the equation, ${\textstyle bx}$ - ${\textstyle x}$ ² = ${\textstyle c}$ )?

${\textstyle x}$ represents a portion of the line, while ${\textstyle b-x}$ represents the remaining portion of the line. The product of both ${\textstyle x}$ and ${\textstyle b-x}$ is ${\textstyle bx}$ - ${\textstyle x}$ ². The goal is to find the value of ${\textstyle x}$ that would produce the highest ${\textstyle c}$ value.

5/20/2026 [Fermet's Theorem]

Pages 107 to 113 detail Fermat's concept of adequality and other mathematical findings led to the decompression of fingerprint files for the FBI in the 1990s. Read this for more about the FBI's decision to digitalize fingerprint files and the process behind it.
[expand upon Fermat's optimization? Use the PDF?]
Fermet's Theorem = If a real-valued function, $f(x)$ $f(x)$ , is differentiable^[8] in an interval $(a,b)$ $(a,b)$ and $f(x)$ $f(x)$ has a maximum OR minimum at $c$ $c$ ∈ $(a,b)$ $(a,b)$ , then ${\textstyle f'(c)}$ ${\textstyle f'(c)}$ = ${\textstyle 0}$ ${\textstyle 0}$ ^[9].
- Explanation of ∈: essentially "belongs to/inside/a member of." For example, $c$ ∈ $(a,b)$ → "the number c is inside the interval between $a$ and $b$ ".

5/23/2026 [Logarithms]

[insert logarithms introduction/lesson]

log(a x b) = log a + log b

Multiply two numbers together, take the log = answer is the SUM of their individual logs. Logarithms are like an "undo" tool. They "undo" the mathematical operations done by exponential functions, and the relationship between logarithms and exponential functions is reciprocal.

e = 2.71828... similar to π in circles^[10]. See e (mathematical constant) (simple-wiki) & w:Natural logarithm (wikipedia). The rate of change of e^x is e^x. The rate of exponential growth is proportional to the function's current level^[11]. An example to illustrate this is the following: as a microphone picks up a noise that increases in volume (perhaps the source of the sound is moving closer to the microphone), the loudspeaker amplifies the noise at a constant, exponential rate in proportional (NOT equal) to the noise it is picking up through the microphone.

5/27/2026 [Derivatives]

What is the definition of a derivative? Essentially the rate of change: dy/dx. An example of a derivative is acceleration. Another example of a derivative is the following question: how many calories will I consume per bite of a hot pocket (each bite being infinitesimally small)?

The question posed by the book is as follows: how do we define the slope when the slope keeps changing?^[12]

Shifting our mindset from algebra: In calculus, the rate of change is not constant, as the IV changes (and is therefore regarded as a function). We go from Δy/Δx [set rate of change] → dy/dx [infinitesimally tiny, varied changes].

So instead of thinking of the hourly rate for a cashier as a set number (let's say $16/hr), we should think of the $16/hr as a constant function. This is going to pay off in calculus as we deal with rates of changes that are not always 'set in stone', or constant. For example, measuring a horse's total speed in a horse race is not going to be a constant, set number - it will be a function with a constantly changing rate. For this specific example:

x = time
y = speed
dy/dx = rate of change of horse's speed with respect to time (think of it as: "rate of change of [y] in respect to [x]").

6/6/2026 [Definite Integrals & Area Function]

Note:
TO-DO [6/6/2026]: provide example problem of area function modeling of YT video, do the work, take a pic and upload.

background info...

A definite integral, in calculus, is the generalized area under the curved function^[13]^[14].

Overview (page 184):

"finding the rate of change/derivative of a known function" = differentiation.
"inferring an unknown function from its rate of change" = integration.

Make sure to apply these principles on your understanding of differential calculus and integration calculus.

Area function (calculus) - Accumulated area under a curved line on an xy graph from point a to point x [upper bound that can be moved as opposed to point b]. YT video.
- What is the area problem? "Predicting the relationship between anything that changes at a changing rate and how much that thing builds up over time"^[15].

Wikipedia/Study Links

Archimedes

approximations of pi
quadrature (computation of area) of a parabolic segment
Archimedes Palimpsest
Archimedes' Law of the Lever

Pierre de Fermat

Fermat’s Method for Finding Maxima and Minima- Kenneth M Monks (2023)

Other

Glossary of mathematical symbols

References/Sources

↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 36. ISBN 978-1-328-87998-1.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 102. ISBN 978-1-328-87998-1.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 84. ISBN 978-1-328-87998-1.
↑ "Derivative". Wikipedia. 2026-04-13. https://en.wikipedia.org/w/index.php?title=Derivative&oldid=1348562692.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 101. ISBN 978-1-328-87998-1.
↑ "Number Theory: An Approach Through History from Hammurapi to Legendre". Wikipedia. 2024-09-18. https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&oldid=1246411217.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 106. ISBN 978-1-328-87998-1.
↑ function has a well-defined, smooth slope at every single point
↑ "Fermat's Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students | Mathematical Association of America". old.maa.org. Retrieved 2026-05-21.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 136. ISBN 978-1-328-87998-1.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 137. ISBN 978-1-328-87998-1.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 143. ISBN 978-1-328-87998-1.
↑ Strang, Gilbert; Herman, Edwin “Jed” (2016-03-30). "5.2 The Definite Integral - Calculus Volume 1 | OpenStax". openstax.org. Retrieved 2026-06-06.
↑ "Khan Academy". www.khanacademy.org. Retrieved 2026-06-06.
↑ Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 184. ISBN 978-1-328-87998-1.

[1] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 36. ISBN 978-1-328-87998-1.

[2] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 102. ISBN 978-1-328-87998-1.

[3] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 84. ISBN 978-1-328-87998-1.

[4] "Derivative". Wikipedia. 2026-04-13. https://en.wikipedia.org/w/index.php?title=Derivative&oldid=1348562692.

[5] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 101. ISBN 978-1-328-87998-1.

[6] "Number Theory: An Approach Through History from Hammurapi to Legendre". Wikipedia. 2024-09-18. https://en.wikipedia.org/w/index.php?title=Number_Theory:_An_Approach_Through_History_from_Hammurapi_to_Legendre&oldid=1246411217.

[7] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 106. ISBN 978-1-328-87998-1.

[8] unction has a well-defined, smooth slope at every single point

[9] "Fermat's Method for Finding Maxima and Minima: A Mini-Primary Source Project for Calculus 1 Students | Mathematical Association of America". old.maa.org. Retrieved 2026-05-21.

[10] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 136. ISBN 978-1-328-87998-1.

[11] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 137. ISBN 978-1-328-87998-1.

[12] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 143. ISBN 978-1-328-87998-1.

[13] Strang, Gilbert; Herman, Edwin “Jed” (2016-03-30). "5.2 The Definite Integral - Calculus Volume 1 | OpenStax". openstax.org. Retrieved 2026-06-06.

[14] "Khan Academy". www.khanacademy.org. Retrieved 2026-06-06.

[15] Strogatz, Steven (2020). Infinite powers: how calculus reveals the secrets of the universe (First Mariner books edition ed.). Boston New York: Mariner Books ; Houghton Mifflin Harcourt. p. 184. ISBN 978-1-328-87998-1.

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