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3 years ago

Find an equation of the tangent line to the curve x^2+2xy+4y^2=12 at the point (2,1)

Mathematics

1 answer:

ohaa [14]3 years ago

4 0

We have that
x²+2xy+4y²<span>=12

we know that
</span>A better way to do this problem is using what is called “Implicit Differentiation
so
step 1
Differentiate both sides of the equation using the Chain Rule
x²+2xy+4y²=12-------> 2x+2y+2xy1+8yy1=0
2xy1+8yy1=-2x-2y--------> y1*[2x+8y]=-2x-2y
y1=(-2x-2y)/(2x+8y)
for the point (2,1)
y1=(-2*2-2*1)/(2*2+8*1)-------> y1=-6/12------> y1=-1/2

step 2
find the equation <span>of the tangent line with m=-1/2 and the point (2,1)
y-y0=m*(x-x0)------> </span>y-1=(-1/2)*(x-2)-----> y-1=(-1/2)x+1
y=(-1/2)x+2

the answer is
y=(-1/2)x+2

using a graph tool
see the attached figure

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snow_tiger [21]

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The question tells us that all of the digits in the price, except for the thousand digit, are even numbers, and that the scooter is priced between $1, 000 and $2, 000.

Effectively, this means that:

a) it must cost more than $1, 000 but less than $2, 000

b) the second, third and fourth digits are even (0, 2, 4, 6 or 8)

We also know that the price is a multiple of 10, thus we can modify b) from above and say that:

b) the second and third digits are even (0, 2, 4, 6 or 8)

c) the fourth digit is 0 (since the price must be divisible by 10)

Furthermore, we are told that the value of the hundreds digit is 30 times the value of the tens digit.

Now, this is where we need to remember that the second (ie. hundreds) and third (ie. tens) digits must be even and multiples of 10. Thus, we must do a little trial and error to find two values that fit these constraints, with x representing the tens value and y representing the hundreds value. We will start with x = 20 (we cannot start with x = 0 since the price would then be $1, 000, and we cannot start with x = 10, since the digit in the tens place would be a 1, which is odd, not even):

(1) if x = 20: y = 30*20 = 600

6 is an even number, which fits the constraint of the problem.

Thus, the tens digit is 2 and the hundreds digit is 6.

Given this information and that which we defined at the very beginning, we can now see that the price of the scooter is $1, 620.

I hope this does help you, however if you have any further questions or feel that I may have misinterpreted something (I personally found 'The value of the hundreds digit is 30 times the value of the tens digit' to be a little poorly defined), please feel free to comment below :)

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