Question

find an equation of the tangent line to the graph of f at the given point
f(x)=x^3, (2,8) using the formula Mtan= lim delta x to 0 of f(x+delta x)/x or Mtan=lim h to 0 of f(x+h)/h) to find the answer, can you explain it using these?

Answer 1

here is general equation for tangent line:
\[y = f'(x_1) (x - x_1) + f(x_1)\]
where given point is (x1,y1)

Answer 2

so all you have to do is find derivative, then plug in x=2

Answer 3

we are actually using the formula Mtan= lim delta x to 0 of f(x+delta x)/x or Mtan=lim h to 0 of f(x+h)/h) to find the answer, can you explain it using these?

Answer 4

\[\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} = \frac{(x+h)^3 - x^3}{h}\]
\[= \frac{(x^3 +3x^2h +3xh^2+h^3) -x^3}{h}\]
\[= \frac{3x^2h +3xh^2 +h^3}{h}\]
\[=\frac{h(3x^2 +3xh+h^2)}{h} = 3x^2 +3xh +h^2\]
\[\lim_{h \rightarrow 0} 3x^2 +xh +h^2 = 3x^2\]