How do you calculate $$dy$$ using differentials?

$$dy = f$$ , where $$f$$ is the derivative of the function at point $$x$$ and $$dx$$ is the small change in $$x$$ .

What is the value of $$f(4)$$ for the function $$f(x) = x^2 - 2x + 3$$ ?

$$f(4) = 4^2 - 2(4) + 3 = 3$$ .

If $$\text{radius} = 20 \text{ cm}$$ and $$dr = 0.2 \text{ cm}$$ , what is the approximate change in area?

Using differentials, $$dA = 2\text{pi} \times 20 \times 0.2 = 8\text{pi} \text{ cm}^2$$ .

What is the relationship between $$\text{dx}$$ and $$\text{dy}$$ ?

$$dy = f$$ shows how a small change in $$x$$ (dx) affects the change in $$y$$ (dy) based on the slope of the function.

How do you find $$\text{dy}$$ for a given function at a specific point?

Calculate the derivative at that point and multiply by the change in $$x$$ (dx).

What is the value of $$\text{dx}$$ if $$\text{dx} = \frac{1}{2}$$ ?

$$dx = \frac{1}{2}$$ represents a small change in the input value of the function.

What is the significance of the tangent line in Euler's Method?

The tangent line at a point provides a linear approximation of the function's behavior near that point.

How do you apply Euler's Method to approximate $$f(10)$$ from $$f(8)$$ ?

Use the formula $$f(10) \text{ approx } f(8) + f$$ .

What is the value of $$\text{dy}$$ for the function $$f(x) = x^2 - 2x + 3$$ at $$x = 4$$ if $$dx = \frac{1}{2}$$ ?

$$dy = 3$$ , calculated using the derivative at that point.

What is the purpose of using differentials in calculus?

Differentials help in approximating changes in functions and are foundational in understanding derivatives and integrals.

Differentials & Euler's Method Flashcards

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1.

FLASHCARD QUESTION

Front

What is a differential?

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2.

FLASHCARD QUESTION

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What is Euler's Method?

Back

Euler's Method is a numerical technique used to approximate solutions of ordinary differential equations (ODEs) using tangent line approximations.

3.

FLASHCARD QUESTION

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4.

FLASHCARD QUESTION

Front

What is the formula for the area of a circle?

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