Mathematics, 21.12.2019 23:31 rosehayden21

Here we will study the function f (x) = e ^ x sin (x), where x ∈ [0, 2π]. a) determine where the function is decreasing and increasing. b) find all local maximam and minimam. does the absolute (global) maximam / minimam have? c) determine where f (x) curves up and down. also find any turning points.

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Answer from: whyidkmyself

we are given

f(x)=e^x sin(x)

(a)

Firstly, we will find critical numbers

so, we will find derivative

f'(x)=e^x sin(x)+e^x cos(x)

now, we can set it to 0

and then we can solve for x

we get

$x=\frac{3\pi }{4} ,x=\frac{7\pi }{4}$

now, we can draw a number line and then locate these values

and then we can find sign of derivative on each intervals

increasing intervals:

$[0,\frac{3\pi}{4} )U(\frac{7\pi}{4} , 2\pi]$

Decreasing interval:

$(\frac{3\pi}{4} ,\frac{7\pi}{4} )$

(b)

Local maxima:

It is the value of x where function changes from increasing to decreasing

so, local maxima is at

$x=\frac{3\pi}{4}$

Local minima:

It is the value of x where function changes from decreasing to increasing

so, local minima is at

$x=\frac{7\pi}{4}$

now, we will plug critical numbers and end values into original function

and we get

At x=0:

f(0)=e^0 sin(0)

f(0)=0

At $x=\frac{3\pi}{4}$ :

$f(\frac{3\pi}{4})=e^{\frac{3\pi}{4}} sin(\frac{3\pi}{4})$

$f(\frac{3\pi}{4})=7.46049$

At $x=\frac{7\pi}{4}$ :

$f(\frac{7\pi}{4})=e^{\frac{7\pi}{4}} sin(\frac{7\pi}{4})$

$f(\frac{7\pi}{4})=-172.640$

At $x=2\pi$ :

$f(2\pi)=e^{2\pi} sin(2\pi )$

$f(2\pi )=0$

Global maxima:

It is the largest value among them

so, we get

$f(\frac{3\pi}{4})=7.46049$

Global minima:

It is the largest value among them

so, we get

$f(\frac{7\pi}{4})=-172.640$

(c)

now, we can find second derivative

f'(x)=e^x sin(x)+e^x cos(x)

$f''(x)=\frac{d}{dx}\left(e^x\sin \left(x\right)+e^x\cos \left(x\right)\right)$

$=\frac{d}{dx}\left(e^x\sin \left(x\right)\right)+\frac{d}{dx}\left(e^x\cos \left(x\right)\right)$

$=e^x\sin \left(x\right)+\cos \left(x\right)e^x+e^x\cos \left(x\right)-e^x\sin \left(x\right)$

$f''(x)=2e^x\cos \left(x\right)$

now, we can set it to 0

and then we can solve for x

$f''(x)=2e^x\cos \left(x\right)=0$

so, we get

$x=\frac{\pi}{2} ,x=\frac{3\pi}{2}$

now, we can draw number line and locate these values

and then we can find sign of second derivative on each intervals

concave up intervals:

$[0,\frac{\pi}{2})U(\frac{3\pi}{2}, 2\pi]$

Concave down intervals:

$(\frac{\pi}{2} ,\frac{3\pi}{2})$

Turning points:

All values of x for which concavity changes

so, we get turning points at

$x=\frac{\pi}{2} ,x=\frac{3\pi}{2}$

Answer from: Quest

i dont know if you cant find a way its best to actually add all of them start from 105 tho

step-by-step explanation:

I’m not sure how to solve these type of problems

Answer from: Quest

d, x=8

step-by-step explanation:

simplifying

4x + -5 = 2x + 11

reorder the terms:

-5 + 4x = 2x + 11

reorder the terms:

-5 + 4x = 11 + 2x

solving

-5 + 4x = 11 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-5 + 4x + -2x = 11 + 2x + -2x

combine like terms: 4x + -2x = 2x

-5 + 2x = 11 + 2x + -2x

combine like terms: 2x + -2x = 0

-5 + 2x = 11 + 0

-5 + 2x = 11

add '5' to each side of the equation.

-5 + 5 + 2x = 11 + 5

combine like terms: -5 + 5 = 0

0 + 2x = 11 + 5

2x = 11 + 5

combine like terms: 11 + 5 = 16

2x = 16

divide each side by '2'.

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Here we will study the function f (x) = e ^ x sin (x), where x ∈ [0, 2π]. a) determine where the fun...

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