Derivatives and parallel lines

In summary, the conversation discusses finding the slope of a curve at any point, determining if lines tangent to the curve at the x-intercepts are parallel, and finding points on the curve where the tangent lines are vertical. The solution involves using implicit differentiation to find the slope at any point, finding the x-intercepts by setting y=0 in the derivative formula, and finding the points where the derivative is infinity (which indicates a vertical tangent line).
  • #1
chris40256
5
0

Homework Statement



Consider the curve defined by x^2+xy +y^2=27

a. Write an expression for the slope of the curve at any point (x,y)
b. Determine whether the lines tangent to the curve at the x-intercepts of the curve are parallel. Show the analysis that leads to your conclusion.
c. Find the points on the curve where the lines tangent to the curve are vertical.


The Attempt at a Solution



a. I used implicit differentiation
x^2 +xy +y^2 =27
2x +y + xy' +2yy' = 0
y'(x+2y)= - (2x+y)
y' = -(2x+y)/(x+2y). I am pretty sure this is right.

b. i know that the lines must have the same slope for them to be parallel, but I am not actually sure what else to do with this. how do u find the x-intercepts and show that their slopes are equal?

c. the vertical line would be like x=k, then you plug k into the orginal curve and then solve for k? i got 6 and -6 is that right? can someone please help me with this
 
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  • #2
a. Correct

b. Find x-intercepts of the curve by putting y=0, and find the x values. Then put these x and y in dy/dx.

c. Find where dx/dy is zero.
 
  • #3
(a) is almost right, but you neglect the possibility that x+2y=0. You would either have to find an alternative expression for that case, or show it never happens on the curve (e.g. by substituting x=-2y into the original equation)

For (b), I suggest being really slick and not calculating where the x-intercepts are located if you can (try plugging y=0 into your dy/dx expression. Where can this go wrong?). Note a line that's tangent to a curve has a slope equal to dy/dx

For (c), if the curve is parallel to a vertical line, then dy/dx is +/- infinity. Taking a page from part (a), when does this occur?
 
  • #4
Office_Shredder said:
For (b), I suggest being really slick and not calculating where the x-intercepts are located if you can (try plugging y=0 into your dy/dx expression. Where can this go wrong?). Note a line that's tangent to a curve has a slope equal to dy/dx

I didn't quite get what exactly you meant.
 
Last edited by a moderator:
  • #5
He is saying, rather than set y= 0 in the original equation, to find the x-intercepts, set y= 0 in the formula for the derivative. It happens, for this particular problem, that you don't need to know x!
 

1. What are derivatives?

Derivatives are mathematical tools used to measure the rate of change of a function. They tell us how much a function is changing at a specific point.

2. How are derivatives calculated?

Derivatives are calculated using the process of differentiation, which involves finding the slope of a tangent line at a specific point on a function.

3. What is the relationship between derivatives and parallel lines?

Derivatives and parallel lines are related because they both involve the concept of slope. Just as the slope of a tangent line at a specific point on a curve is equal to the slope of the curve at that point, the slopes of any two parallel lines are equal.

4. Can derivatives be negative?

Yes, derivatives can be negative. A negative derivative indicates that the function is decreasing at a specific point, while a positive derivative indicates that the function is increasing at that point.

5. How are derivatives used in real life?

Derivatives have many real-life applications, such as in physics to calculate velocity and acceleration, in economics to determine marginal cost and revenue, and in engineering to optimize designs. They are also used in financial markets to analyze risk and make predictions.

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