Find $f(2)$ given the tangent line equation.

Question

I came across this question tutoring someone:

The tangent line to $y = f(x)$ at $x = 2$ has the equation $y = 3 - 7x$.

Find $f(2)$.

My student has only started limits and differentiation. How could you possibly solve this without integration?

Answer 1

Just you can use $$f(2)=3-7\cdot2=-11$$

Answer 2

First option

The tangent line to $y=f(x)$ at $x=a$ is

$$y-f(a)=f'(a)(x-a).$$ Thus we have at $x=2$,

$$y=f(2)+f'(2)(x-2)=f'(2)x+f(2)-2f'(2)=-7x+3.$$ That is,

$$\begin{cases}f'(2)&=-7,\\f(2)-2f'(2)&=3.\end{cases}$$

Solve the linear system and you'll get the answer.

Second option

At $x=a$ the tangent line to $f(x)$ and $f(x)$ have the common value $f(a)$. Thus,

$f(2)=3-7\cdot 2=-11.$

Answer 3

Hint:

the point $P=(2,f(2))$ is also a point of the tangent line at this point.