Question

Determine the slope of the curve at the given point.

x² + 2y² - 3x - 4y + 2 = 0 at (0, 1)
(x² + 2y)² = x + 10 (-1, 2)
(3x - y)² = 6x + 2y + 23 at (1, - 2)
(3x - y)² = 10 at (3, 2)
(3x - y)² = 16 at (4, 2)

Answer 1

The slope of the curve at the point (0, 1) is undefined.

Step-by-step explanation:

The equation given represents a curve, not a straight line. To find the slope of the curve at the given point (0, 1), we need to take the derivative of the equation with respect to x. The derivative represents the slope of the tangent line at any given point on the curve. So, the first step is to differentiate the given equation.

d/dx(x² + 2y² - 3x - 4y + 2) = d/dx(0)

2x + 4yy' - 3 - 4y' = 0

4yy' - 4y' = 3 - 2x

(4y - 4)y' = 3 - 2x

y' = (3 - 2x) / (4y - 4)

Now, substitute the values of x and y at the given point (0, 1) into the derivative to find the slope.

y' = (3 - 2(0)) / (4(1) - 4)

y' = 3 / 0

The slope is undefined at (0, 1) since the denominator is zero.