Find the linearization L(x) of the function g(x)=xf(x2) at x=2 given the following information.

Question

Find the linearization L(x) of the function g(x)=xf(x2) at x=2 given the following information.

f(2)=1

f′(2)=10

f(4)=5

f′(4)=−2

Answer 1

When applying local linearization to a function you only need to know three things to find the formula for the line:

1. The x-coordinate
2. The y-coordinate
3. The slope (which in this context is the derivative at the x-coordinate)

Once you know these three things, you can plug in any x value in order to perform a linear approximation.

In this problem, the x-coordinate is given: x=2. From this, we can find the y-coordinate; that is, we can find g(2).

g(2) = 2*f(2^2) = 2*f(4). We are told in the problem that f(4) = 5, so g(2) = 10.

Now we have our point: (2,10). So, we must now find the slope by taking the derivative of g(x) which will involve both the chain rule and the product rule.

g ' (x) = f(x^2) + x*2x*f ' (x^2). More simply, g ' (x) = f(x^2) + 2x^2 * f ' (x^2). Since we are particularly looking for the slope at x = 2, we can go ahead and plug in 2 wherever we see an x in the equation:

g ' (2) = f(4) + 8 * f ' (4). From the problem, we know that f(4) = 5 and f'(4)= -2. Plugging these in shows us that g ' (2) = 5 + 8* -2. Therefore, the slope of the line would be 5 - 16 which is -11.

So, now we have a point and a slope. So, now we can use the point-slope form of a line:

-11(x - 2) = y - 10 or y = -11x + 32. This may then be written as L(x) = -11x + 32.