find the equation of the perpendicular line to the curve

Question

 Use your answer from (a) to find the equation of the perpendicular line to the curve 𝑦=𝑓(𝑥)y=f(x) at 𝑥=4x=4.

part A answer: 7/2

Answer 1

Since the slope of the perpendicular line is the negative reciprocal of the slope of the curve at x=4, and the slope of the curve at x=4 is 7/2, then the slope of the perpendicular line is -2/7.

Since the perpendicular line passes through (4, f(4)), we can use point-slope form to write the equation of the line:

y - f(4) = m * (x - 4)

where m is the slope of the line and f(4) is the y-coordinate of the point on the curve.

Plugging in the values we know:

y - 7/2 = -2/7 * (x - 4)

Simplifying:

y = -2/7 * (x - 4) + 7/2

So the equation of the perpendicular line to the curve at x=4 is y = -2/7 * (x - 4) + 7/2.