OH, I forgot to post this one with my last two questions! Here it is:
3. f is a differentiable function for all x. Which of the following statements must be true?
I- the derivative with respect to x of the integral from 0 to 4 of f of x, dx is equal to f of x
II- d dx of integral from x to 4 of f of t d t equals negative f of x
III- the integral from 4 to x of f prime of x, dx is equal to negative f of x
A. II only
B. III only
C. I and II only
D. II and III only
THANK YOU!
#3 II only. Check the FTC
I is false because ∫[0,4] f(x) dx is just a number
III is false because ∫[4,x] f'(x) dx = f(x)-f(4)
To determine which statements must be true, let's analyze each statement one by one:
I- The derivative with respect to x of the integral from 0 to 4 of f(x) dx is equal to f(x).
To determine if this statement is true, we need to apply the Fundamental Theorem of Calculus. According to the theorem, if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a), where a and b are the limits of integration.
In this case, the definite integral is from 0 to 4 of f(x) dx. Let's call the antiderivative of f(x) as F(x). By the Fundamental Theorem of Calculus, the integral from 0 to 4 of f(x) dx is equal to F(4) - F(0).
Now, we need to find the derivative of this integral with respect to x:
d/dx [ integral from 0 to 4 of f(x) dx ] = d/dx [ F(4) - F(0) ]
Since F(4) and F(0) are constants (not dependent on x), their derivatives with respect to x will be zero.
Therefore, the derivative with respect to x of the integral from 0 to 4 of f(x) dx is equal to zero, not f(x). So Statement I is not necessarily true.
II- d/dx of the integral from x to 4 of f(t) dt equals -f(x).
To determine if this statement is true, we need to apply the Second Fundamental Theorem of Calculus. According to this theorem, if F(x) is an antiderivative of f(x), then the derivative of the integral from a to x of f(t) dt gives us f(x). Mathematically, it can be expressed as:
d/dx [ integral from a to x of f(t) dt ] = f(x)
In this case, we need to find the derivative of the integral from x to 4 of f(t) dt with respect to x:
d/dx [ integral from x to 4 of f(t) dt ] = f(x)
This matches the form of the Second Fundamental Theorem of Calculus, where f(x) is equal to -f(x) times -1. Therefore, Statement II is the correct one.
III- The integral from 4 to x of f'(x) dx is equal to -f(x).
To determine if this statement is true, we need to analyze the relationship between the integral of the derivative of a function and the original function.
By the Fundamental Theorem of Calculus, if F(x) is an antiderivative of f(x), then the derivative of the integral from a to x of f(t) dt gives us f(x). Mathematically, it can be expressed as:
d/dx [ integral from a to x of f(t) dt ] = f(x)
In this case, we are given the function f'(x), which represents the derivative of f(x). If we integrate f'(x) with respect to x from 4 to x, we obtain the integral of the derivative, which should correspond to the original function f(x).
However, in Statement III, the integral is from 4 to x of f'(x), not f(x). This means that the integral does not directly correspond to the original function f(x).
Hence, Statement III is not necessarily true.
To recap:
Statement I is not true.
Statement II is true.
Statement III is not true.
Therefore, the correct answer is:
B. III only