Question

Write the equation of the line (in point-slope form) tangent to w(x)= p(x)/√x−3 at x=4.

A. y−y_1=m(x−x_1),m=w′(4), (x_1,y_1)=(4,w(4))
B. y−y_1=m(x−x_1),m=w′′(4), (x_1,y_1)=(4,w(4))
C. y−y_1=m(x−x_1),m=w ′(4)(x_1,y_1)=(4,w ′(4))
D. y−y_1=m(x−x 1), m=w′′(4),(x_1,y_1)=(4,w′(4))

Answer 1

To find the equation of the line tangent to the function w(x) = p(x)/√(x-3) at x = 4 in point-slope form, we need to find the slope of the tangent line and the point of contact.

Step-by-step explanation:

Equation of the Tangent Line in Point-Slope Form

To find the equation of the line tangent to the function w(x) = p(x)/√(x-3) at x = 4, we need to find the slope of the tangent line and the point at which it touches the function.

Step 1: Find the slope of the tangent line

The slope of the tangent line is equal to the derivative of the function evaluated at x = 4. Let's find the derivative of w(x) first:

w'(x) = p'(x)√(x-3) - p(x)/(2√(x-3)(x-3))

Now, substitute x = 4 in this derivative to find w'(4). This value represents the slope of the tangent line.

Step 2: Find the point of contact

The point of contact is (4, w(4)). Substitute x = 4 in the original function to find the y-coordinate.

Step 3: Write the equation in point-slope form

Now, we have the slope and the point of contact. Plug these values into the point-slope form:
y - y1 = m(x - x1)

Therefore, the correct option is A. y - y1 = w'(4)(x - 4), where (x1, y1) = (4, w(4)).