University:
Rice University
Course:
MATH 212 | Multivariable Calculus
Academic year:

2023
Views:

454

Pages:

2
Author:

Deskov

Implicit Differentiation. Example Problem 3 For what values of x does the implicit equation x cubed plus y cubed = 3xy have a horizontal tangent? Again, we're taking the derivative. Specifically, a horizontal tangent indicates that dy dx = 0, whereas a vertical tangent indicates that dx dy = 0. Let's begin by finding the derivative of x cubed.The derivative of x cubed is 3x squared. For y cubed, we get the same result. The derivative of y is dy dx plus the derivative of 3x is 3.Keeping only the y term, we find that dy dx equals 3y. We will isolate dy dx. We are going to move all of the terms with dy dx to the left side of the equation, leaving a 3y minus 3x squared term on the right side. We can then factor out a dy dx from this expression. 3y minus 3x squared. We are striving to find the value of dy/dx that produces a horizontal tangent, so we need a numerator of 0. We also need the denominator not to be 0 because if you get 0 over 0, it's not necessarily a horizontal tangent or a vertical tangent. The graph is indeterminate in the first quadrant. As long as the numerator of a fraction is 0 and the denominator is nonzero, there will be no solution to that equation when x = y = 0. If x = 1 or y = 1, then there will not be a solution. Let y equal x squared divided by 3, so that 3y squared equals 3x squared. Letting y equal x squared gives us a value for y that we can substitute into the equation to find x. So, as long as y is equal to x squared, we will have horizontal tangents. However, we must make sure that our solution is a point on the original curve; therefore, we will substitute back into the original equation and solve for y. x cubed plus y cubed is equal to 3xy. We were told that y equals x squared, so everywhere we see y in the equation, we can replace it with x squared. So the equation becomes x cubed plus x to the sixth is equal to 3x cubed. To solve this, subtract x cubed from both sides of the equation. To solve the equation x to the sixth minus 2x cubed is equal to 0, we can factor out an x cubed and obtain x cubed minus 2 equals 0. So we find that x equals 0 and y equals 0. Therefore, if x equals 0, then y equals 0. Since we already said that no solution exists for this equation when y = 0, then we should ignore this value of x. We can then say that x cubed minus 2 equals 0. Therefore, x = the cube root of 2.