Question

If an equation of the tangent line to the curve `y = f(x)` at the point where `a = 2` is `y = 4x-5`, find `f(2)` and `f'(2)`? I know f(2) is 3 but how do I find `f'(2)`?

The textbook says that `f'(2) = 4` but how exactly?

Answer 1

I presume the statement "where `a=2`" should read "where `x=2`"

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point.

We are given that the equation of the tangent when `x=2` is:

`y=4x-5`

Comparing with the standard form equation of a straight line:

`y=mx+c`

We see that:

`m=4`

And as indicated above this is the same as the value of the derivative when `x=2`, thus:

`f'(2) = 4` QED

Without additional information, we cannot infer any information regarding `f(2)`.