Question

Which of the following describes the y-coordinates of the points on the curve e^x = sin y in the xy-plane where the curve has a vertical tangent line?

Options:
Option 1: Always negative
Option 2: Always positive
Option 3: Undefined
Option 4: Equal to zero

Answer 1

The y-coordinates of the points on the curve e^x = sin y where the curve has a vertical tangent line can be both positive and negative, occurring at y = (2n+1)π/2, where n is an integer.

Step-by-step explanation:

The question is about finding the y-coordinates of the points on the curve e^x = sin y in the xy-plane where the curve has a vertical tangent line. A curve has a vertical tangent line when the slope of the tangent is undefined, which happens when the derivative of the function with respect to x goes to infinity or does not exist. To find when e^x = sin y has a vertical tangent, we need to differentiate both sides with respect to x, treating y as an implicit function of x. The derivative of e^x is e^x, and by the chain rule, the derivative of sin y with respect to x is cos y * dy/dx. Setting these equal gives us e^x = cos y * dy/dx. For the tangent to be vertical, dy/dx must be undefined, which occurs when cos y = 0. This happens when y = (2n+1)π/2, where n is an integer. The values of y-coordinate for these points are multiples of π/2 (e.g., π/2, 3π/2, etc.), and they are neither always positive, negative, equal to zero, nor undefined but can be both positive and negative depending on the value of n.

Answer 2

The y-coordinates of the points on the curve e^x = sin y where it has a vertical tangent line are undefined. The answer is Option 3: Undefined.

Step-by-step explanation:

The equation e^x = sin y represents the relationship between the x-coordinates and the y-coordinates of the points on the curve.

To find where the curve has a vertical tangent line, we need to determine the values of y that make the derivative of the equation equal to infinity. Differentiating both sides of the equation, we get d/dx(e^x) = d/dx(sin y), which simplifies to e^x = cos y.

For the curve to have a vertical tangent line, the derivative of y with respect to x, dy/dx, must be undefined. Therefore, the y-coordinates must make cos y equal to 0, which occurs at y = (2n + 1)π/2, where n is an integer. At these values of y, the curve has a vertical tangent line. So, the answer is Option 3: Undefined.