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katovenus [111]
3 years ago
14

Find the absolute extrema of f(x) = e^{x^2+2x}f ( x ) = e x 2 + 2 x on the interval [-2,2][ − 2 , 2 ] first and then use the com

parison property to find the lower and upper bounds for I = \displaystyle \int_{-2}^{2} f(x) \, dxI = ∫ − 2 2 f ( x ) d x.
Mathematics
1 answer:
fredd [130]3 years ago
5 0

f(x)=e^{x^2+2x}\implies f'(x)=2(x+1)e^{x^2+2x}

f has critical points where the derivative is 0:

2(x+1)e^{x^2+2x}=0\implies x+1=0\implies x=-1

The second derivative is

f''(x)=2e^{x^2+2x}+4(x+1)^2e^{x^2+2x}=2(2x^2+4x+3)e^{x^2+2x}

and f''(-1)=\frac2e>0, which indicates a local minimum at x=-1 with a value of f(-1)=\frac1e.

At the endpoints of [-2, 2], we have f(-2)=1 and f(2)=e^8, so that f has an absolute minimum of \frac1e and an absolute maximum of e^8 on [-2, 2].

So we have

\dfrac1e\le f(x)\le e^8

\implies\displaystyle\int_{-2}^2\frac{\mathrm dx}e\le\int_{-2}^2f(x)\,\mathrm dx\le\int_{-2}^2e^8\,\mathrm dx

\implies\boxed{\displaystyle\frac4e\le\int_{-2}^2f(x)\,\mathrm dx\le4e^8}

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alex41 [277]

Step-by-step explanation:

there 5^(n+1) + 5^(n+2) = 5^n x 5^1 + 5^n x 5^2 breaking them as x^(a+b) = x^a x x^b

then taking common 5^n from both terms

5 0
3 years ago
Slope of (0,1) and (3,0)
Marysya12 [62]

Answer:

\boxed{ \bold{ \huge{ \boxed{ \sf{  \frac{ - 1}{3}}}}}}

Step-by-step explanation:

Let the points be A and B

Let A ( 0 , 1 ) be ( x₁ , y₁ ) and B ( 3 , 0 ) be ( x₂ , y₂ )

<u>Finding</u><u> </u><u>the </u><u>slope</u><u> </u><u>of</u><u> </u><u>the </u><u>points</u>

\boxed{ \sf{slope =  \frac{y2 - y1}{x2 - x1} }}

\longrightarrow{ \sf{slope =  \frac{0 - 1}{ 3 - 0}}}

\longrightarrow{ \sf{slope =  \frac{ - 1}{3}}}

Hope I helped!

Best regards! :D

5 0
3 years ago
Read 2 more answers
What is the y-intercept of the graph below?
lutik1710 [3]

Answer:

C

Step-by-step explanation:

The y-intercept is where the line passes through the y-axis

On this graph, we can see this is at the point (0, 2)

6 0
3 years ago
Read 2 more answers
10x +1 12x-5 solve for x
dimulka [17.4K]

Answer:

x = 3

Step-by-step explanation:

10x + 1 = 12x - 5

Subtract 1 from each side

10x = 12x - 6

Subtract 12x from each side

-2x = -6

Divided by -2

x = 3

5 0
3 years ago
What is the percent of discount on a $44 video game on sale for $28.60
Agata [3.3K]

Answer:

35% discount.

Step-by-step explanation:

28.60 / 44 = 13/20

13/20 = 65/100 = 65%

28.60 is 65% of 44

100 - 65 = 35

Therefore, the discount is 35%.

8 0
3 years ago
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