I don't think it is possible, because IVT requires that for some d (in this case 3) between g(a) and g(b), there is a point c where g(c) = d. but the function g(x) gives the value of 1/2 and 2 on [1,2]
Did I miss something? Maybe it could work and I forgot something important
I don't think Rolle's theorem would work though, then g(1) would need to be equal to g(3), but they are not...
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if x is not 0, and 0 at 0. Thus f'(x) exists for all x but since the limit of f'(x) as x goes to 0 is not 0, f'(x) is not continuous at x= 0.
I know it does not...
One more question, we did not learn that property in class. Is it possible to prove that f' is a continuous function? so I can use the intermediate value theorem to say that f'(c) = 1/2 exists
Prove that if f is a differentiable function on R such that f(1) = 1, f(2) = 3, f(3) = 3. There is a c \in (1 , 3) such that f'(c) = 0.5
I think the mean value theorem should be used, but I can't figure out how to prove such value exists
I wanted to prove that the Fibonacci sequence is a divisibility sequence, but I don't even know how to prove it.
all I know is that gcd\left({F}_{m},{F}_{n}\right)={F}_{gcd\left(m,n\right)} and I should somehow use the Euclidean algorithm?
What would be the standard partition? P= \left\{[{x}_{0},{x}_{1}],...,[{x}_{i-1},{x}_{i}],...,[{x}_{n-1},{x}_{n}]\right\} ?
I think 1-x where x is irrational would be the lower sum, and it should be bounded above by 1, since 0 is not irrational. And 1+x would be the upper sum bounded by 1 as...
Prove that the function
f(x) = 1+x, 0 \le x \le 1, x rational
f(x) = 1-x, 0 \le x \le 1, x irrational (they are one function, I just don't know how to use the LATEX code properly)
is not integrable on [0,1]
I don't know where to start, I tried to evalute the lower and upper Riemann sum but it...
Ooooo, that is clever! and just so happen that U(f,{P}_{n}) is almost the same since {M}_{i} is \frac{i}{n}! which further proves that this function is integrable
I ran into some issues when trying to calculate the lower Riemann sum of f\left(x\right)={x}^{3}, x\in[0,1]
I am asked to use the standard partition {P}_{n} of [0,1] with n equal subintervals and evaluate L(f,{P}_{n}) and U(f,{P}_{n})
What I did:
L(f,{P}_{n}) =...