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once you get to the cos(x+dx)=cosx-dxsinx step, you can subtract cosx from both sides. Then since cos(x+dx)-cosx=dy, you get dy=-dxsinx, which matches definition of derivative of cosx

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once you get to the cos(x+dx)=cosx-dxsinx step, you can subtract cosx from both sides. Then since cos(x+dx)-cosx=dy, you get dy=-dxsinx, which matches definition of derivative of cosx

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0.1-(0.1^3)/6+(0.1^5)/120= 0.0998334167

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dy = dx * cos x
-> dy/dx = cos x

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Comment:

Question:

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Setting the stage by moving from polynomials to transcendental functions

Introducing sine and cosine

Euler's goal was to differentiate sin x

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dy -> change in sine *Task 8*

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To Euler, "vanishing terms" mean extremely small terms in a series that will head to 0 as they shrink. From Task 8 we saw that as angle x gets smaller (approaches 0) it will approach itself aka 0.

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start to diverge around pi/2, which is approximately 1.57

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As the distance gets smaller and smaller, the sin approaches the angle itself

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This was my approach for Task 8, which uses two points on a single graph. Does this work seem correct?

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Key is in differentiating omega=0 from the limit as omega approaches 0

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Key is in differentiating omega=0 from the limit as omega approaches 0

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Allows the ratio to remain 2x:1 despite "vanishing increments"

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(a) using work from task 3 without limits

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(a) using work from task 3 without limits

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grey=sin(x)
green=sin(x+dx)

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The beginning of the project uses h to represent the increase to x, as opposed to w. Both definitions give the ratio of the increase of x, h or w, to the change in the function of x (shown in the top of the fraction in the first definition). The definition in the beginning, however, takes into the account the limit of this ratio as the increase in x goes to 0.

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Compare:

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"ratio" -> fraction
"w gets smaller" -> limit as w heads to 0

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I think that Euler believed 'transcendentals' to mean any quantity that is subject to change or that could be determined

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Comments:

Questions: what does differential calculus have to do with the analysis of the infinite?

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A function f from a set D to a set Y is a rule that assigns a unique (single) element f(x) ∈ Y to each element x ∈ D.

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A function f defines from a set A to a set B is a rule that associates with each element of the set A one, and only one, element of the set B.

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To derive a definition for differential calculus

Giving context to the definition provided at the end of the paragraph

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In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and root extraction. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.

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A function maps each input x in a subset D of the real numbers to a unique real number y. The set D is the domain of the function, and the range is the set of all real numbers y that have at least one input x in D such that f(x)=y.

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"Finally, it was Hardy who, in 1908, defined a function as a relation between two variables 𝑥 and 𝑦 such that “to some values of 𝑥 at any rate correspond values of 𝑦.”

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"Finally, it was Hardy who, in 1908, defined a function as a relation between two variables 𝑥 and 𝑦 such that “to some values of 𝑥 at any rate correspond values of 𝑦.”