(a) Find the slope, m, of the tangent to the curve y = 2√x at the point where x = a.

Question

(b) Find equations of the tangent lines at the following points.

(4, 4)

y = ?

 

(25, 10)

y = ?

Answer 1

Tangent to the curve is the linear line tangent to the curve at a given x value. You are asked to first find the slope of the tangent line which is the first derivative of the function. Next, use the point slope formula to find the equation of the tangent line for 2 different points which will be two separate and different equations.

Part a) If y=2(x)^1/2 Then f'(x)=y'=dy/dx=x^-1/2 and f'(a)=a^-1/2 , therefore m=a^-1/2

Part b) point slope y-y₁=m(x-x₁) for point (x₁, y₁)

x=a

y-4=4^-1/2(x-4)

y=1/2(x-4)+4

y=1/2x-2+4

y=1/2x+2

Now for point (25,10)

y-10=25^-1/2(x-25)

y=1/5(x-25)+10

y=1/5x-5+10

y=1/5x+5