f is a differentiable function on the interval [0, 1] and g(x) = f(2x). The table below gives values of f '(x). What is the value of g '(0.1)
x| .1 .2 .3 .4 .5
f'(x)| 1 2 3 -4 5
So I know f(x) would be the integral of f'(x) which you can get with a Riemann sum, right? But then what do you do when you get that?
g(x)=f(2x)
g'(x)=f'(2x)*2
It's just the chain rule
Yes, you are correct that the integral of f'(x) gives you the function f(x). However, in this question, you are asked to find g'(0.1), the derivative of the function g(x). To find g'(0.1), we need to use the chain rule.
The chain rule states that if you have a composition of functions, such as g(x) = f(2x), then the derivative of g(x) with respect to x is equal to the derivative of the outer function (f(2x)) times the derivative of the inner function (2x) with respect to x.
So, to find g'(0.1), we need to find the derivative of f(2x) and then evaluate it at x = 0.1.
Let's apply the chain rule:
g'(x) = f'(2x) * (2)
Using the given values of f'(x):
For x = 0.1, we have 2x = 2 * 0.1 = 0.2
Using the table, f'(0.2) = 2
Now, substitute the values back into the chain rule formula:
g'(0.1) = f'(0.2) * 2 = 2 * 2 = 4
Therefore, g'(0.1) is equal to 4.
Yes, you are correct that to find the function f(x) from the derivative f'(x), you can integrate f'(x) using a Riemann sum or any other integration method.
To find g'(x), we need to use the chain rule. Since g(x) = f(2x), we can write g'(x) = f'(2x) * (d/dx)(2x).
Now, let's find g'(0.1). Since we have values of f'(x) for specific values of x, we can substitute x = 0.1 into f'(x) to find f'(0.1). From the given table, we see that f'(0.1) = 1.
Now, let's use the chain rule to find g'(x) = f'(2x) * (d/dx)(2x) = f'(2x) * 2.
Since we want to find g'(0.1), we can substitute x = 0.1 into g'(x) = f'(2x) * 2 to get g'(0.1) = f'(2 * 0.1) * 2 = f'(0.2) * 2.
From the given table, we see that f'(0.2) = 2. Therefore, g'(0.1) = f'(0.2) * 2 = 2 * 2 = 4.
So, the value of g'(0.1) is 4.