Mathematics, 13.07.2020 22:01 rmcarde4432

Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.) f(θ)=6cosθ+3sin2θ g

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The critical value of $f(\theta) = 6\cdot \cos \theta + 3\cdot \sin 2\theta$ are given by $\theta \approx 0.091\pi \pm 2\pi\cdot n$ or $\theta \approx 0.909\pi \pm 2\pi \cdot n$ , $\forall \,n \in \mathbb{N}$

Step-by-step explanation:

The function to be evaluated is $f(\theta) = 6\cdot \cos \theta + 3\cdot \sin 2\theta$ , the first derivative of the function must be taken in order to determine the set of critical numbers. Each derivative are found by using the differentiation rule for a sum of functions and rule of chain and subsequently simplified by trigonometric and algebraic means:

First derivative

$f'(\theta) = - 6 \cdot \sin \theta +6\cdot \cos 2\theta$

$f'(\theta) = -6\cdot \sin \theta + 2\cdot (\cos^{2}\theta-\sin^{2}\theta)$

$f'(\theta) = -6\cdot \sin \theta + 2\cdot [(1-\sin^{2}\theta-\sin^{2}\theta)]$

$f'(\theta) = -6\cdot \sin \theta + 2\cdot (1-2\cdot \sin^{2}\theta)$

$f'(\theta) = -6\cdot \sin \theta + 2 - 4\cdot \sin^{2}\theta$

$f'(\theta) = -4\cdot \sin^{2}\theta - 6\cdot \sin \theta +2$

The procedure to determine the critical number of the given function are described briefly:

1) First derivative is equalised to zero.

2) The resultant equation is solved.

Then,

$-4\cdot \sin^{2}\theta - 6\cdot \sin \theta +2 = 0$

Whose roots are:

$\sin \theta_{1} \approx 0.281$ and $\sin \theta_{2} \approx -1.781$

The sine function is a continuous function with a range between 1 and -1, so, only the first root offers a realistic solution. In addition, such function is positive at first and second quadrants and has a periodicity of $2\pi$ radians, the family of critical values are determined by the unse of inverse trigonometric functions:

$\theta \approx \sin^{-1} 0.281$

There are two subsets of solutions:

$\theta \approx 0.091\pi \pm 2\pi\cdot n$ or $\theta \approx 0.909\pi \pm 2\pi \cdot n$ , $\forall \,n \in \mathbb{N}$

Answer from: Quest

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1

Answer from: Quest

the rocket cannot penetrate the ground, in a contextual sense, therefore the roots of this equation should be the domain of this function. i'll let you find the roots of this quadratic equation (hint: use the quadratic formula:

1- c

2-a

step-by-step explanation:

mark brainliest!

The graph below represents the height in meters of a toy rocket t seconds after it was launched upwa

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Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to d...