Question

if the tangent line to y=f(x) at(4,3) passes through the point (0,2) find f(4) and f'(4)

Answer 1

Well, the fact that we know that \((4,3)\) exists on the graph on \(f(x)\) means that \(f(4)=3\). And also, knowing that the line of \(y=mx+b\) is tangent to \(f(x)\) at \(x=4\) means that \(f'(4)\) would be the slope of this line.

We know that the line passes through the points of \((4,3)\phantom{..}and\phantom{..}(0,2)\)

So we can use the formula:
\[m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\]
And sub in the co-ordinates to find the slope of the line. In this case \(f'(4=m)\)

Answer 2

Er sorry edit on the last line:
\[f'(4)=m\]