Final answer:
To find y' (2), the derivative of y with respect to x at x = 2, we use implicit differentiation on the equation x²/16 + y²/36 = 1. After differentiating and substituting x = 2 and the given y(2) = 5.2, we solve for y' and find it to be approximately -4.35.
Step-by-step explanation:
The student has provided the equation of an ellipse, x²/16 + y²/36 = 1, and the value of y when x is 2, which is y(2) = 5.2. To find y' (2), which represents the derivative of y with respect to x at x = 2, we must use implicit differentiation. Differentiating both sides of the given ellipse equation with respect to x gives:
2x/16 + (2yy')/36 = 0
Since x = 2, we substitute x and y into the differentiated equation:
(2· 2)/16 + (2· 5.2 y')/36 = 0
1/8 + 10.4y'/36 = 0
After simplifying, we solve for y':
10.4y' = -36/8
y' = -36/8 × 1/10.4
y' = -4.35 (approx)
So, y' (2) is approximately -4.35.