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Hyperbolic Functions to the Rescue – My New Lecture on the Blog

16 August 2024 / Krystian Karczyński

I wrote a new Lecture entirely dedicated to hyperbolic functions and their inverses. In it, I explain what hyperbolic functions are, when - for example - they are useful, and why they sometimes appear in Wolfram results, generally causing panic among students. However, the hyperbolic sine isn't that scary - I invite you to the Lecture.

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Forbidden Formulas for Indefinite Integrals – Derivation of Formulas

08 August 2024 / Krystian Karczyński

Professors at universities have their own requirements. Many of them, for the good of their students, of course, will not hesitate to specify the rules for solving problems in great detail. Some do not recognize ready-made formulas for some indefinite integrals. See where they came from.

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Limit of the Function: x to the x to the x … What to Do? (Example with Morals)

07 August 2024 / Krystian Karczyński

As a curiosity, I'll calculate the limit of the function: (x^x^x - cosx). More interesting than the calculations themselves are the two morals that can be drawn from them. But the morals are at the end (has anyone ever seen a story with a moral at the beginning)?

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The Volume of an Ellipsoid (But Not a Rotational One, Just a Wild Type) Calculated by a Definite Integral

07 August 2024 / Krystian Karczyński

Let's say we need to calculate the volume of an ellipsoid: {x^2}/4+{y^2}/5+{z^2}/9=1. This is an ellipsoid that intersects the x, y, z axes at coordinates 2, \sqrt{5}, and 3, respectively. This is not a rotational ellipsoid, it is not formed by rotating any curve around any axis, so we can't use the standard formula for the volume of a rotational solid. We need to figure out another way.

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What to Do When Exponentiation Just Doesn’t Work (Complex Numbers)

06 August 2024 / Krystian Karczyński

There are days when nothing seems to work. And there are examples with complex numbers where nothing goes right. The known and memorized methods don't help. Take, for instance, this innocent-looking exponentiation: (1+2i)^8. Following the well-trodden path you've used in many examples, you want to write the number 1+2i in trigonometric form and then raise it to the eighth power using the appropriate formula. But you encounter complications along the way... See the trick I use.

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Complex Polynomial Equations Reducible to Quadratic Equations

05 August 2024 / Krystian Karczyński

Many fourth-degree polynomial equations can be transformed into quadratic equations using a well-known high school trick of substitution. This also works perfectly for polynomials in complex numbers. See how to turn these equations into lower-degree equations.

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Limits with Cube Roots (Is it Multiplication by Conjugate?)

05 August 2024 / Krystian Karczyński

When given a limit to calculate that involves some subtraction with a root (and which obviously couldn't be calculated more simply), such as: "Something - square root of something", "square root of something - Something", or "square root of something - square root of something", we used a trick I call - "multiplication by the conjugate." We simply multiplied this expression by its counterpart with a plus sign, or rather by a fraction where this counterpart was in the numerator and the denominator. What to do when the roots are cube roots?

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Limits of a Sequence with the Sum of Squares or the Sum of Cubes

05 August 2024 / Krystian Karczyński

In the limits of sequences, sometimes the expression contains the sum of squares or cubes of consecutive natural numbers. What then? The answer is simple: formulas for the sum of squares and the sum of cubes of consecutive natural numbers. They go like this...

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Cyclometric Functions (arcus) – New Lecture on the Blog

04 August 2024 / Krystian Karczyński

I've added a new article with a video to the "Lectures" section (on the right sidebar): Cyclometric Functions. It's dedicated to cyclometric functions (arcsinx, arccosx, arctgx, arcctgx) and is divided into two parts. In the first part, I quickly show how to calculate them, and in the second part, I delve deeper into the topic (definitions, graphs).

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