Find an equation of the tangent line to the graph of f at the given point

Question

(a) Find an equation of the tangent line to the graph of f at the given point, (b) Use a graphing utility to graph the function and its tangent line at the point, and (c) Use the derivative feature of the graphing utility to confirm your results.

Function                                    point

y = (4x3+ 3)2                             (-1, 1)

Answer 1

Step 1:

(a)

The function is .

Apply derivative on each side

Apply chain rule of derivative .

Step 2:

slope at the point , Substitute x = -1  in .

Find the tangent line equation using point-slope form .

The tangent line equation is .

 

Answer 2

Step 3:

(b)

Graph the function and tangent line  to verify the result.

Step 4:

(c)

Verify the derivative of the function using graphing utility .

So , the derivative of the function verified graphically.

The derivative of the function is .

 

Answer 3

(a) The tangent line equation is  .

(b)

Graph of the function and tangent line .

(c)

The derivative of the function is .