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astraxan [27]
3 years ago
6

Find gradient xe^y + 4 ln y = x² at (1, 1)​

Mathematics
1 answer:
cricket20 [7]3 years ago
4 0

xe^y+4\ln y=x^2

Differentiate both sides with respect to <em>x</em>, assuming <em>y</em> = <em>y</em>(<em>x</em>).

\dfrac{\mathrm d(xe^y+4\ln y)}{\mathrm dx}=\dfrac{\mathrm d(x^2)}{\mathrm dx}

\dfrac{\mathrm d(xe^y)}{\mathrm dx}+\dfrac{\mathrm d(4\ln y)}{\mathrm dx}=2x

\dfrac{\mathrm d(x)}{\mathrm dx}e^y+x\dfrac{\mathrm d(e^y)}{\mathrm dx}+\dfrac4y\dfrac{\mathrm dy}{\mathrm dx}=2x

e^y+xe^y\dfrac{\mathrm dy}{\mathrm dx}+\dfrac4y\dfrac{\mathrm dy}{\mathrm dx}=2x

Solve for d<em>y</em>/d<em>x</em> :

e^y+\left(xe^y+\dfrac4y\right)\dfrac{\mathrm dy}{\mathrm dx}=2x

\left(xe^y+\dfrac4y\right)\dfrac{\mathrm dy}{\mathrm dx}=2x-e^y

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2x-e^y}{xe^y+\frac4y}

If <em>y</em> ≠ 0, we can write

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2xy-ye^y}{xye^y+4}

At the point (1, 1), the derivative is

\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=1,y=1}=\boxed{\dfrac{2-e}{e+4}}

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The number y of deer in a county after x years can be modeled by y=0.75x^2 + 10.5x + 1300. When were there about 2000 deer? Roun
Lemur [1.5K]

Answer:

24 years

Step-by-step explanation:

We are told that the model of number of deers is;

y = 0.75x² + 10.5x + 1300

Where;

y represents number of deers

x represents number of years

Now, we want to find how many years it will take to get to 2000 deers.

Thus;

2000 = 0.75x² + 10.5x + 1300

0.75x² + 10.5x + 1300 - 2000 = 0

0.75x² + 10.5x - 700 = 0

Using quadratic formula, we have;

x = [-b ± √(b² - 4ac)]/2a

x = [-10.5 ± √(10.5²- (4*0.75*-700)]/(2*0.75)

x = [-10.5 ± √(110.25 + 2100)]/(1.5)

x = [-10.5 ± √2210.25]/(1.5)

x ≈ -38 or 24

Number of years can't be negative, so we will take the positive value.

So, x = 24 years

3 0
2 years ago
Andy takes an average of 12 breaths per minute About how many breaths did Andy take by the time he turned 13 years old?
Diano4ka-milaya [45]
6832800 breathes in 13 years exactly
6 0
3 years ago
I’ll give you the BRAINLYEST answer if you can help me on this question!
igomit [66]

Answer:

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Step-by-step explanation:

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6 0
3 years ago
Which is a factor of 144 - 49x²?<br> O 12-7x²<br> O 72-7x²<br> O 12-7x<br> O 72 + 7x
lisabon 2012 [21]

Answer: C: 12-7x

Step-by-step explanation:

This expression is a good example of the difference of two squares, as 144 is the square of 12, while 49x² is the square of 7x.

Any difference of two squares a^2-b^2 can be factored into the sum of their square roots times the difference of their square roots (a+b)(a-b).

Let's factor our expression.

144-49x^2=(12+7x)(12-7x)

Out of all the options given, only 12-7x is a factor. Hence, option C is correct.

4 0
2 years ago
How do you do this? Please give an example to #1 and explain thoroughly please. I have asked multiple times and gotten answers I
liberstina [14]

Answer:

y=2x

Step-by-step explanation:

y = kx is the format of the graph we are looking for (as told in the instructions) where k is the variable we are supposed to find (like y = x, y=2x, y=3x, etc)

Note that as x increases by 1, y increases by 2. We can see that by writing a table of the points of points that the line given intersects (picture below). We can see that by the values of the x-axis and the y-axis. Why does this happen? This is because y =2x for each value of x. We notice that in our table.

Therefore, the equation is y=2x. I hope this helps! If you still have questions, ask me in the comments!

4 0
3 years ago
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