Write as an equivalent decimal 17 1/3

Hi, help with question 18 please. thanks

Vadim26 [7]

Answer:

See Below.

Step-by-step explanation:

We are given the equation:

$\displaystyle y^2 = 1 + \sin x$

And we want to prove that:

$\displaystyle 2y\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right) ^2 + y^2 = 1$

Find the first derivative by taking the derivative of both sides with respect to x:

$\displaystyle 2y \frac{dy}{dx} = \cos x$

Divide both sides by 2y:

<h3> $\displaystyle \frac{dy}{dx} = \frac{\cos x}{2y}$ </h3>

Find the second derivative using the quotient rule:

$\displaystyle \begin{aligned} \frac{d^2y}{dx^2} &= \frac{(\cos x)'(2y) - (\cos x)(2y)'}{(2y)^2}\\ \\ &= \frac{-2y\sin x-2\cos x \dfrac{dy}{dx}}{4y^2} \\ \\ &= -\frac{y\sin x + \cos x\left(\dfrac{\cos x}{2y}\right)}{2y^2} \\ \\ &= -\frac{2y^2\sin x+\cos ^2 x}{4y^3}\end{aligned}$

Substitute:

$\displaystyle 2y\left(-\frac{2y^2\sin x+\cos ^2 x}{4y^3}\right) + 2\left(\frac{\cos x}{2y}\right)^2 +y^2 = 1$

Simplify:

$\displaystyle \frac{-2y^2\sin x-\cos ^2x}{2y^2} + \frac{\cos ^2 x}{2y^2} + y^2 = 1$

Combine fractions:

$\displaystyle \frac{\left(-2y^2\sin x -\cos^2 x\right)+\left(\cos ^2 x\right)}{2y^2} + y^2 = 1$

Simplify:

$\displaystyle \frac{-2y^2\sin x }{2y^2} + y^2 = 1$

Cancel:

$\displaystyle -\sin x + y^2 = 1$

Substitute:

$-\sin x + \left( 1 + \sin x\right) =1$

Simplify. Hence:

$1\stackrel{\checkmark}{=}1$

Q.E.D.

8 0

3 years ago

What is the answer for 4m - t = m Solve for m?

blagie [28]

solve for m by simplifying both sides of the equation, then isolating the variable.

m = t/3

4 0

3 years ago

This visual representation shows the sets of real numbers. Classify the number −85. Select all terms that are correct. Venn diag

Olenka [21]

♡ The Question/Task ♡

Classify the number -85.

Rational, Whole, Natural, Integer, Irrational.

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♡ The Answer ♡

-85 is an integer, and a rational number!

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♡ The Explanation/Step-By-Step ♡

Natural numbers are positive integers from 1 till infinity, though, nautral numbers don't include zero. Since -85 is a negative number, this wouldn't be a natural number.

A whole number is a set of numbers including all positive integers and 0. Since -85 isn't a positive number, this wouldn't be a whole number.

An irrational number is where any real number that cannot be expressed as the quotient of two integers. Irrational numbers are commonly square roots, too. Although, -85 isn't a square root.

A rational number is where any number that can be written as a ratio, or a fraction, of two integers can be classified as a rational number. Rational numbers are commonly fractions, decimals, negative decimals, etc!

An integer is a whole number that can be positive, negative, or zero. 85 is a whole number, and can be negative, which in this case, is -85. The number we're using. Therefore, -85 is an integer.

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♡ Tips ♡

-No Tips Provided!