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Question

# Give an example of:Two different functions that have the same linear approximation near $x = 0$.

Solution

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Consider two functions

$f(x) =e^x \text{ and } g(x) = x^2+x+1$

The linearisation of $f(x)$ near $x=0$ is

$f(x) \approx e^0+e^0(x-0) = 1+x$

The linearisation of $g(x)$ near $x=0$ is

$g(x) \approx (0+0+1)+(2(0)+1 )(x-0) = 1+x$

Thus, the two different functions have the same linearisation.

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