# Linear Approximation

Let's say want to computer the value of a function $$f(x)$$ at $$x = 1$$ but all we know about this function is this:

$f(x) = f'(x)$

and

$f(0) = 1$

One way to do this to use linear approximation repeatedly with a defined step size (say h). The lesser will be the value of step, the better approximation we can have.

Let's take an example,

Let's use the step size of $$0.5$$

Using, a step size of $$0.5$$ or $$h = 0.5$$, we can first compute the value at $$f(0.5)$$ and then at f(1) which we ultimately require.

Generally, linear approximation states:

$f(h) \approx f(0) + h.f'(0)$

$f(2h) \approx f(h) + h.f'(h)$

$f(3h) \approx f(2h) + h.f'(2h)$

and this can go forever.

To approximate the value of f(1) using a step size of $$0.5$$ that is for $$h = 0.5$$, we can do it like this:

First, we compute value of $$f$$ at $$0.5$$ like this:

$$f(0.5)\approx f(0) + 0.5 \times f'( 0) = 1 + 0.5 \times 1 = 1.5$$

Secondly, we can use the value of $$f$$ at $$0.5$$ to computer $$f(1)$$ like this:

$$f(1) \approx f(0.5) + 0.5 \times f'(0.5) = 1.5 + 0.5 \times 1.5 = 2.25$$

Now, write a program to find the value of $$f(1)$$ using a step size of $$0.05$$ or $$h = 0.05$$ with repeated linear approximations.

The approximation of $$f(1)$$ using this program is:

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