Use the limit process to find the slope of the line tangent to the graph

Question

Let F(x) = 3x^4+4

 

(a) Use the limit process to find the slope of the line tangent to the graph of f at x = 2.

 

 

 

(b) Find an equation of the line tangent to the graph of f at x = 2.

Answer 1

Using the limit definition of the derivative is certainly a pain. It helps if you know that (x+h)⁴=x⁴+4x³h+6x²h²+4xh³+h⁴.

I used the binomial theorem for that but you could take the time to actuallly multiply it out.

 

So lim     f(x+h) - f(x)

    h->0          h

 

Hopefully this looks familiar to you.

so we've got 3(x+h)⁴ - 3x⁴  for the numerator. Expanding it as above and multiplying by 3 you get

lim    3x⁴+12x³h + 18x²h² + 12xh³+3h⁴ +4- (3x⁴+4)

h->0                      h

 

The first and last term of the numerator cancel out and then the h divides into each of the terms in the numerator to leave you with

lim     12x³+18x²h +12xh²+3h³

h->0                                             which means f'(x)= 12x³ This is the derivative function.

So f'(2) = 96 which is the slope of the tangent line at x=2 and f(2) =52.

 

So you can use the point slope form of a line to write y-52 = 96(x-2) to get the equation of the tqngent line.

 

Hope this helped some. There is another method to find the slope of 96, but it comes out to the same thing.

 

OK, I won't be lazy.

 

lim   f(x)-f(2)      = 3x⁴+4  -   52     = 3x⁴-48  =    3(x⁴-16)  =  3(x-2)(x³+2x²+4x+8)

x->2   x-2         

 

The x-2 cancels out in the numerator and denominator to leave 

lim 3(x³+2x²+4x+8

x->2                                     which when substituting 2 in, you get 96.