Find all point on the curve y= √x at which the slope of the tangent line...

Question

Find all point on the curve y= √x at which the slope of the tangent line is the same slope of the straight line y= x/6 + 8. Find equations of the tangent line at these points. Explain how you got your answer.

Answer 1

Find all point on the curve y= √x at which the slope of the tangent line...

Find all point on the curve y= √x at which the slope of the tangent line is the same slope of the straight line y= x/6 + 8. Find equations of the tangent line at these points. Explain how you got your answer.

Hello Sara,

The straight line y = x/6 + 8, has a slope of 1/6. So, we need to take the derivative and set it equal to 1/6:

d/dx [ y = x^1/2 ]

dy/dx = 1/2 x^-1/2

1/6 = 1/2 x^-1/2 ...* Solve for x. This finds the x-value on the curve that has a tangent line with slope1/6.

√x = 3

x = 9

Substitute x = 9 into the function y = √x to get the y-value of the point on the curve with a slope of 1/6.

y = √9

y = 3

The point (9,3) on the curve y = √x has a tangent line with a slope of 1/6

Lastly, we have to get the equation of the line that lies tangent to the curve at point (9,3).

Easiest method is to use the point-slope form: Y - Y₁ = m(X - X₁) with point (9,3) and slope 1/6.

Y - 3 = 1/6(X - 9)

y = (1/6)x + 3/2