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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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Round each number to the nearest ten. what is the best estimate for the answer 303.55
bixtya [17]
Answer=303.6

When rounding numbers to the nearest tenth, we need to look to the number in the hundredths place.

303.55

If the number is 5 or greater, we round up. If the number is 4 or less, we round down. The number IS 5, so we round up.

303.55 becomes 303.6
3 0
3 years ago
Use calculus to find the area of the triangle with the vertices (0, 5), (2, -2), and (5, 1).
guajiro [1.7K]

The area of the triangle with the vertices (0, 5), (2, -2), and (5, 1) by using the calculus is  21 square unit.

We need to find the equation among all possible pairs and then integrate the equations from one co-ordinate to another co-ordinate

Equation of line passing through (0,5) and (2,-2) is

y-5 = [(-2-5)/(2-0)](x-0)

=>y-5 = (-7) /2x

=>y= (-7/2x)+5 -------(eq1)

Equation of line passing through (0,5) and (5,1) is

y-5 = [(1-5) / (5-0)](x-0)

=>y-5 = (-4/5)x

=>y = (-4/5)x+5-------(eq2)

Equation of line passing from (2,-2) and (5,1) is

y-(-2) = [[1 - (-2)] / (5-2)] / (x-2)

=>y+2=(3/3)(x-2)

=>y=x-4--------(eq3)

Now, we use definite integration to find the area between the different equation of line.

So, area enclosed between the equations is given by the

area =\int\limits^5_2[(-4/5)x+5 - (-7/2)x + 5)dx  + \int\limits^5_1[(-4/5)x+5 -(x-4)]dx

=>area=\int\limits^5_2(7/2-4/5)x dx + \int\limits^5_1((-9/5)x+9)dx

Using properties of integration,\int\limits x\, dx=x^{2}/2

=>area=\int\limits^5_2(27/10)x dx + \int\limits^5_1(-9/5)x+9)dx

=>area=([27/10)×[5² - 2²])/2 + [ (-9/5)×(5²-1²) ]/2 +9×(5-1)

=>area=(27/20)×(25-4) + (-9/5)×24+9×4

=>area = (27×21)/20 + (-216)/5+ 36

=>area=(567/20) - (216/5) + 36

=>area= [(567-261×4)+(36×20)]/20

=>area=[(567-864)+720]/20

=>area=423/20

=>area=21 square unit.

Hence, area of triangle is 21 square unit.

To know more about area of triangle, visit here:

brainly.com/question/19305981

#SPJ4

4 0
1 year ago
How many different 5-digit whole
Lerok [7]

Answer:

120

Step-by-step explanation:

so

<h2 /><h2><em><u> ?????</u></em></h2>

this 5 interrogative marks are our 5 digit number

for the first place you have 5 choices(2, 3, 4, 5, 6)

for the second:4

for the third:3

for the fourth:2

and for the fifth:1

you multiply this choices

5×4×3×2=120

4 0
2 years ago
Mrs.Lorentz bought 12 pounds 8 ounces of flour. This is 1/4 of the flour she will use to make sugar cookies in her bakery this w
Lerok [7]

Answer:

153 batches

Step-by-step explanation:

4 0
3 years ago
What is 63 over 75 as a decimal?
marissa [1.9K]

Answer:

0.84

Step-by-step explanation:

Use a calculator.  That would be 0.84.

                               3*21

Note that 63/75 = ---------- = 21/25.  Multiplying top and bottom by 4 results

                                3/25                   in 0.84 (so this answer checks out ok)

3 0
3 years ago
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