Find the slope of the tangent line to the parabola

Question

(a) Find the slope of the tangent line to the parabola y =4x - x^2

at the point (1,3)

( i ) using Definition 1 ( ii ) using Equation 2

(b) Find an equation of the tangent line in part (a).

Answer 1

Step 1 :

The function is   and the point .

a-i)

Definition 1:

The tangent line to the curve  at the point  is the line through point P with slope

provided that this limit exists.

Apply the limit .

The slope of the tangent line is 2.

Step 2 :

a-ii)

Equation 2 :

Here a = 1. 

Evaluate :

  

Evaluate :

Substitute and  in above equation.

Apply the limit .

The slope of the tangent line is 2.

Step 3 :

b)  

Point-slope form .

Substitute slope and the point .
 

Solution :

a) The slope of the tangent line is 2.

b) The equation of tangent line