Answer :
Final answer:
In College-level Mathematics, you use implicit differentiation to differentiate both sides of the ellipse equation to get the derivative, set the derivative equal to the desired slope, and solve for x. The x-coordinate of the point in the first quadrant on the ellipse 10x²+4y²=36.9 where the slope of the tangent line is -0.25 is approximately 1.85.
Explanation:
To use implicit differentiation to find the point on the ellipse 10x²+4y²=36.9 where the slope of the tangent line is -0.25, we first take the derivative of both sides of the equation with respect to x. This gives us:
20x + 8yy' = 0
Then, we solve for y' to get the slope of the tangent line:
y' = - (20x) / (8y)
We set y' equal to -0.25 and solve for x, then discard the negative solution because we're asked for a point in the first quadrant. Here, x comes out to be approximately 1.85.
Learn more about Implicit Differentiation here:
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