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3 years ago

Differentiating Exponential functions In Exercise,find the derivative of the function. See Example 2 and 3.

Mathematics

1 answer:

nadya68 [22]3 years ago

4 0

Answer:

f'(x) $=2(e^4^x)+(x^2+1)(4e^4^x)$

Step-by-step explanation:

The derivative of the function:

$(x^2+1)e^4^x$

The rule for the product of two functions:

f'(x) $=g'(x)h(x)+g(x)h'(x)$

Therefore

g(x) $=x^2+1$

g'(x) $=2$

f(x) $=e^4^x$

f'(x) $=4e^4^x$

f'(x) $=2(e^4^x)+(x^2+1)(4e^4^x)$

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Suppose a rectangular pasture is to be constructed using 1 2 linear mile of fencing. The pasture will have one divider parallel

timama [110]

Answer:

$\displaystyle A=\frac{1}{192}$

Step-by-step explanation:

<u>Maximization With Derivatives</u>

Given a function of one variable A(x), we can find the maximum or minimum value of A by using the derivatives criterion. If A'(x)=0, then A has a probable maximum or minimum value.

We need to find a function for the area of the pasture. Let's assume the dimensions of the pasture are x and y, and one divider goes parallel to the sides named y, and two dividers go parallel to x.

The two divisions parallel to x have lengths y, thus the fencing will take 4x. The three dividers parallel to y have lengths x, thus the fencing will take 3y.

The amount of fence needed to enclose the external and the internal divisions is

$P=4x+3y$

We know the total fencing is 1/2 miles long, thus

$\displaystyle 4x+3y=\frac{1}{2}$

Solving for x

$\displaystyle x=\frac{\frac{1}{2}-3y}{4}$

The total area of the pasture is

$A=x.y$

Substituting x

$\displaystyle A=\frac{\frac{1}{2}-3y}{4}.y$

$\displaystyle A=\frac{\frac{1}{2}y-3y^2}{4}$

Differentiating with respect to y

$\displaystyle A'=\frac{\frac{1}{2}-6y}{4}$

Equate to 0

$\displaystyle \frac{\frac{1}{2}-6y}{4}=0$

Solving for y

$\displaystyle y=\frac{1}{12}$

And also

$\displaystyle x=\frac{\frac{1}{2}-3\cdot \frac{1}{12}}{4}=\frac{1}{16}$

Compute the second derivative

$\displaystyle A''=-\frac{3}{2}.$

Since it's always negative, the point is a maximum

Thus, the maximum area is

$\displaystyle A=\frac{1}{12}\cdot \frac{1}{16}=\frac{1}{192}$

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ValentinkaMS [17]

Answer:

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Step-by-step explanation:

If 7x = 7y, then...

7x = 7y

7x / 7 = 7y / 7

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Answer:

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Step-by-step explanation:

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3 years ago

Read 2 more answers

People enjoyed attending the concert in the park.

Luden [163]

Answer:

w = 238

Step-by-step explanation:

So create a system of equations if the number of m = 2w where the men's number is twice the women's number. and m + w = 714 so substitute 2w for m

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3 years ago