I have no idea what to do with this problem, I don't even know what of the information is relevant.
"To specify a function we must give a procedure for obtaining an output (usually numerical and often denoted y) for each input (usually numerical and often denoted x). The definite integral gives us a new way to specify a function. Begin with an integrable function y=f(x) defined on some interval [a, b]. (Remember, all continuous functions are integrable andmany functions that are not continuous are also integreable.) Choose a number c in the interval [a, b]. for each number x between a and b, define F(x) to be the definite integral of f over the interval from c to x.
"For example, suppose f(x)=√(4-x²) so the interval on which f is defineed is [a, b]= [-2, 2]. the graph of y=f(x) is the top half of a circle with radius r=2 centered at (0,0). Choose c=0 and define the function F(x) for each x between -2 and 2 by
F(x)= (integral from 0 to x) f(t) dt = (integral from 0 to x) √(4-t²) dt."
Whew. Now for the questions (can you see why I'm having trouble with this?):
1. Exlpain why the variable of integration in the definition of the function F(x) is t rather than x.
I said that it's a different function than f(x), because if you take the integral of f(t) you should end up with f(x) but I don't know if that's right because he hasn't brought that up yet.
2. There are some values of x for which F(x) is negative. For what values of x is F(x) negative?
I don't even know where to start.
3. Explain how F(x) can have negative values when the graph of the integrand lies entirely above the x-axis.
Again, I have no idea where to start with this.
Hopefully that made a lot more sense to you than it did to me. Can you help me?
"To specify a function we must give a procedure for obtaining an output (usually numerical and often denoted y) for each input (usually numerical and often denoted x). The definite integral gives us a new way to specify a function. Begin with an integrable function y=f(x) defined on some interval [a, b]. (Remember, all continuous functions are integrable andmany functions that are not continuous are also integreable.) Choose a number c in the interval [a, b]. for each number x between a and b, define F(x) to be the definite integral of f over the interval from c to x.
"For example, suppose f(x)=√(4-x²) so the interval on which f is defineed is [a, b]= [-2, 2]. the graph of y=f(x) is the top half of a circle with radius r=2 centered at (0,0). Choose c=0 and define the function F(x) for each x between -2 and 2 by
F(x)= (integral from 0 to x) f(t) dt = (integral from 0 to x) √(4-t²) dt."
Whew. Now for the questions (can you see why I'm having trouble with this?):
1. Exlpain why the variable of integration in the definition of the function F(x) is t rather than x.
I said that it's a different function than f(x), because if you take the integral of f(t) you should end up with f(x) but I don't know if that's right because he hasn't brought that up yet.
2. There are some values of x for which F(x) is negative. For what values of x is F(x) negative?
I don't even know where to start.
3. Explain how F(x) can have negative values when the graph of the integrand lies entirely above the x-axis.
Again, I have no idea where to start with this.
Hopefully that made a lot more sense to you than it did to me. Can you help me?
