Question

X2 + y2 = 49 (a) Find two explicit functions by solving the equation for y in terms of x.

y1 = (positive function) y2 = (negative function)

(b) Sketch the graph of the equation and label the parts given by the corresponding explicit functions.

(c) Differentiate the explicit functions. dy/dx = ±

(d) Find dy/dx through implicit differentiation. dy/dx = Is the result equivalent to that of part

e)? Yes N

Answer 1

To find two explicit functions, we solve the equation x2 + y2 = 49 for y in terms of x. The positive function is y1 = √(49 - x2) and the negative function is y2 = -√(49 - x2). When the graph of the equation is sketched, it forms a circle centered at the origin. The derivatives of y1 and y2 with respect to x are dy1/dx = -x/√(49 - x2) and dy2/dx = x/√(49 - x2). Through implicit differentiation, the derivative dy/dx is found to be -x/y, which is equivalent to the derivatives of the explicit functions.

Step-by-step explanation:

To find two explicit functions, we solve the equation for y in terms of x.

Given the equation x2 + y2 = 49, we can rewrite it as y2 = 49 - x2. Taking the square root of both sides gives us y = ±√(49 - x2).

(a) The positive function is y1 = √(49 - x2) and the negative function is y2 = -√(49 - x2).

(b) To sketch the graph, plot points for the positive function, y1, and the negative function, y2. Connect the points to form a circle centered at the origin with a radius of 7 units.

(c) To differentiate the explicit functions, we find the derivatives of y1 and y2 with respect to x. dy1/dx = -x/√(49 - x2) and dy2/dx = x/√(49 - x2).

(d) To find dy/dx through implicit differentiation, differentiate x2 + y2 = 49 with respect to x. We get 2x + 2y(dy/dx) = 0, which simplifies to dy/dx = -x/y.

Yes, the result is equivalent to that of part (c).