Consider the curve given by x^2 + sin(xy) + 3y^2 = C, where C is a constant. The point (1, 1) lies on this curve. Use the tangent line approximation to approximate the y-coordinate when x = 1.01.

0.996

1

1.004

Cannot be determined

1.388

Question

Consider the curve given by x^2 + sin(xy) + 3y^2 = C, where C is a constant. The point (1, 1) lies on this curve. Use the tangent line approximation to approximate the y-coordinate when x = 1.01.

0.996

1

1.004

Cannot be determined

1.388

anonymous · Answer

1.004

dumbcow · Answer

$$f(x) \approx f'(a) (x-a) + f(a)$$
a = 1
$$f'(1) = -\frac{2x+y \cos(xy)}{x \cos(xy) +6y} = -\frac{2+\cos(1)}{\cos(1) + 6} \approx -.388407$$
$$f(1.01) \approx -.388407 (.01) + 1 = .996$$