Science:Math Exam Resources/Courses/MATH110/April 2011/Question 01 (a)/Solution 1

The statement is false, so a linear approximation $L$ to a function $f$ may satisfy $L(x)=f(x)$ for more than one value of $x$. To prove this, we need an example of such a function. In particular, we would like to find a function with a tangent line (the linear approxmiation) that intersects the function at another point.

One example would be the function $f(x)=x^{3}$, with the linear approximation at the point, say, $(-1,1)$. Drawing that function and linear approximation looks like this:

So we see that $L(x)=f(x)$ both at $x=-1$ and at $x=2$. So the original statement is false.

If you want to make it even simpler, choose ƒ to be linear, e.g. f(x) = 17x+42. Then the linear approximation L coincides with ƒ at all points.