Use the definition of the slope of a tangent line to find equation for the slope of the tangent line to the graph of f(x) = 2x^2+1 at any point c

Question

Use the definition of the slope of a tangent line to find equation for the slope of the tangent line to the graph of f(x) = 2x^2+1 at any point c

Answer 1

The definition of the slope of tangent line is the same as definition of derivative.

 

 

lim           f(c + h) - f(c)

h->0     _______________

                        h

 

 

lim         2(c + h)² + 1 - (2c² + 1)

h-->0    ______________________

                           h

 

 

lim           2(c² + 2ch + h²) + 1 - 2c² - 1

h-->0       _________________________

                                  h

 

 

 

lim           2c² + 4ch + 2h² + 1 - 2c² - 1

h-->0     _________________________

                                   h

 

 

lim             4ch + 2h²

x-->0        __________

                        h

 

 

lim           2h(2c + h)

h-->0      ___________                 h  cancels out.

                        h

 

 

lim          2(2c + h)                      plug in h=0

h-->0

 

 

 

lim         4c

h-->0

 

 

 

The slope of the tangent line is 4c.   Now the point of the original function at any point c is   (c, 2c² + 1).  Use this point to find the equation of the tangent line.  We need to also find b using the slope-intercept form.

 

y = mx + b

 

2c² + 1 = 4c(c) + b

 

2c² + 1 = 4c² + b

 

 

-2c² + 1 = b

 

 

The equation of the tangent line is

 

y = 4cx + (-2c² + 1)

 

y = 4cx - 2c² + 1

 

 

 

 

Answer 2

The slope of the tangent line to the graph of f(x) at c is f'(c). 

 

f'(x)=4x, so f'(c)=4c.