The equation x^2+4y^2=100 describes an ellipse.
First, use implicit differentiation to find its slope at the points (8,3) and (8, -3).
Then, solve the equation for y, obtaining two functions and differentiate both to find the slopes at x = 8.
Do your answers agree? Should they agree? Why or why not?
Explain your reasoning
clearly they should not agree, since one point is on the bottom of the ellipse, and the other is just above it.
To find the slope at the points (8,3) and (8, -3) on the ellipse x^2 + 4y^2 = 100 using implicit differentiation, we differentiate both sides of the equation with respect to x.
Differentiating x^2 + 4y^2 = 100 implicitly, we get:
2x + 8y * dy/dx = 0
Now, let's solve for dy/dx. Rearranging the equation, we have:
dy/dx = -2x / (8y)
To find the slope at x = 8, we substitute this value into the equation:
dy/dx = -2(8) / (8y)
At the point (8,3), y = 3, and at the point (8, -3), y = -3. Let's calculate the slopes for both points:
For (8,3):
dy/dx = (-2 * 8) / (8 * 3) = -16/24 = -2/3
For (8,-3):
dy/dx = (-2 * 8) / (8 * -3) = -16/-24 = 2/3
The slopes at the points (8,3) and (8, -3) are -2/3 and 2/3, respectively.
Next, let's solve the equation x^2 + 4y^2 = 100 for y. Rearranging the equation, we have:
4y^2 = 100 - x^2
y^2 = (100 - x^2) / 4
y = ± √((100 - x^2) / 4)
Differentiating both functions, we get the slopes at x = 8:
For y = √((100 - x^2) / 4):
dy/dx = (1/2) * (100 - x^2)^(-1/2) * (-2x)
Substitute x = 8: dy/dx = -8 / √(100 - 64) = -8/√36 = -8/6 = -4/3
For y = -√((100 - x^2) / 4):
dy/dx = (1/2) * (100 - x^2)^(-1/2) * (-2x)
Substitute x = 8: dy/dx = -8 / √(100 - 64) = -8/√36 = -8/6 = -4/3
The slopes at x = 8 for both functions are -4/3.
Therefore, the answers obtained from implicit differentiation and solving for y and differentiating both functions agree. In this case, they should agree because the equation is for an ellipse, which is symmetric in the y-axis. Hence, the slopes at corresponding positive and negative y-values, such as (3, x) and (-3, x), should be equal.