if f(1)=12 and f' is continuous, what is the value of f(4)?

integral from 1 to 4 of f'(x)dx = 17

IF the integral of f'(x) dx from 1 to 4 is 17, as you say, then the function f(x), which is the integral with an arbitrary constant, changes by 17 from 1 to 4. Then the value of f(x) at x=4 must be 12 + 17 = 29.

Oh, I see what you did there! You integrated my answer and got 17! But unfortunately, that's not quite how it works. The integral of f'(x) dx from 1 to 4 represents the net change of the function f(x) from 1 to 4, but it doesn't tell us the actual value of f(4). We need some more information to determine the value of f(4). Keep those math questions coming!

To find the value of f(4), we can use the given information that f(1) = 12. This implies that the function f(x) passes through the point (1, 12).

We are also given that f'(x) is continuous. This means that f(x) is differentiable for all values of x.

Given that the integral from 1 to 4 of f'(x)dx = 17, we can use the Fundamental Theorem of Calculus to relate the integral of a function to the original function itself.

According to the Fundamental Theorem of Calculus, if F(x) is an antiderivative of f(x) on the interval [a, b], then the definite integral of f(x) from a to b is equal to the difference between F(b) and F(a).

In this case, we can say that F(x) - F(1) = 17. Since F(1) is an arbitrary constant, we can represent it as C.

So, F(x) - C = 17.

Since we know that f(x) is the derivative of F(x), we can differentiate both sides of the equation with respect to x:

f(x) - 0 = d/dx (17)
f(x) = 0

This means that f(x) is a constant function, and its derivative is 0. However, this contradicts the given information that f(1) = 12. Therefore, it seems that there might be some inconsistency in the information provided.

Without further information, it is not possible to determine the exact value of f(4).

To find the value of f(4), we need to determine the function f(x) based on the given information.

We know that f'(x) is continuous and that f(1) = 12.

Since f'(x) is continuous, we can apply the Fundamental Theorem of Calculus, which states that the derivative of the integral of a function is equal to the original function.

Therefore, if the integral from 1 to 4 of f'(x) dx is equal to 17, we can conclude that the change in f(x) from 1 to 4 is 17.

Since f(1) = 12, the initial value of f(x) is 12 at x = 1.

To find f(4), we need to add the change in f(x) (17) to the initial value of f(x) (12):

f(4) = f(1) + integral from 1 to 4 of f'(x) dx
= 12 + 17
= 29

Therefore, the value of f(4) is 29.