Pls help me with ROOT TANGENTS

[minhthai2312]

y= sqrt (x)
a. Find an equation of the tangent to the graph of f(x)=sqrt (x) at the point (9,3). Calculate the coordinates of the point where the tangent cuts the x-asix
Verify that it is a tangent by graphing both the square root function and your equation in Graphmatica ( on the same axes)
b. Find the coordinate of the point where the tangent to the graph of f(x)=sqrt (x) at ( 5, sqrt (5) ) crosses the x-axis. Can you see any pattern forming? Test with another point on the curve.
c. Let x(0) be any positive number, and let P be the point on the graph of y=sqrt (x) where x=x(0). Find an expression for the point where the tangent at P crosses the x-axis.
d. Describe a method for drawing tangents to the square root graph as it relate to the x-value. Illustrate this method with a diagram.

 

[BigGlenntheHeavy]

\(\displaystyle a.) \ Given: \ y \ = \ f(x) \ = \ x^{1/2}, \ find \ line \ tangent \ to \ f(x) \ at \ (9,3) \ and \ calculate \ the \ coordinates\)

\(\displaystyle \ when \ the \ tangent \ line \ cuts \ the \ x-axis.\)

\(\displaystyle f(x) \ = \ x^{1/2}, \ f \ ' \ (x) \ = \ \frac{1}{2x^{1/2}}, \ f \ ' \ (9) \ = \ \frac{1}{6} \ = \ m\)

\(\displaystyle Ergo, \ y-3 \ = \ \frac{1}{6}(x-9), \ y \ = \ \frac{x+9}{6}, \ when \ the \ tangent \ line \ cuts \ the \ x=axis, \ y \ = \ 0, \ hence\)

\(\displaystyle the \ coordinates \ are \ (-9,0).\)

\(\displaystyle See \ graph.\)

[attachment=0:2my3efnv]hero.jpg[/attachment:2my3efnv]

 

[minhthai2312]

how about question b and c.