How to determine the value of $f(x)$ if we know $f'(x)$? (Concrete example in the text)

Question

Let

$f(x) = 21/x$

$f'(x) = -21/x^2$

So $f(4) = 5.25$ and $f'(4)= -\frac{21}{16}$

Is there a way, and if yes, how to calculate $f(3)$ if we only know $f(4)$ and $f'(x)$ without calculating with the original $f(x)$ function ?

Answer 1

The answer is no.

We can integrate $f'(x)$ to find $f(x)$ up to a constant and we use $f(4)$ to find the constant.

For example if $f'(x) = x^3+1$, then $f(x)= x^4/4+x+c$

If we know that $f(4) = 65$, we have $f(4)= 4^4/4+4+c=65$ which implies, $c=-3$ and $f(x)=x^4/4+x-3$

Now you can find $f(3)=81/4$

Answer 2

If we know $f'(x)$ and $f(4)$ we have that

$$f(x)=f(4)+\int_4^x f'(t) dt$$

Answer 3

What you have is called a differential equation, i.e. you have an equation that involves a function and its derivative.

Technically, the solution is obtained using an integral,

$$f(3)=f(4)+\int_4^3 f'(x)\,dx$$

but this is essentially a way to retrieve the function $f(x)$ so I would tend to answer no to your question.

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Anyway, there are so-called numerical methods that allow you to estimate the integral without building an analytical expression for the function and can return an approximation of $f(3)$ as the output of an algorithm.