Answer to Question #273607 in Calculus for Alunsina

Question

Given the function y=√x

a. Find the differential dy.

b. Evaluate dy and ∆y if x=1 and dx=∆x=1

c. Find the equation of the tangent line at x=1

d. Sketch the graph of the curve y=√x and the tangent line in the Cartesian Plane using a scale of 1 unit = 1cm. Show in your diagram the line segments dx, dy, and ∆y. (Note: the curve us an upper semi-parabola whose vertex is at the origin and concaving to the right. Use 0, 1, 4, and 9 as x-coordinates.)

Answer 1

a.

$dy=y'dx=\frac{dx}{2\sqrt x}$

b.

at x = 1:

$dy=dx/2$

$\Delta y=y(2)-y(1)=\sqrt 2-1$

c.

equation of the tangent line:

$y-y_0=f'(x_0)(x-x_0)$

$f'(x)=\frac{1}{2\sqrt x}$

$f'(1)=1/2$

$y(1)=1$

$y-1=(x-1)/2$

$2y=x+1$

d.