Hi, I am not sure how to go about doing these problems and was wondering if someone could show/tell me how to start them? Thanks!
1) Determine whether the Mean Value Theorem can be applied to f on the closed interval [a,b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a,b) such that f'(c) = f(b)-f(a)/b-a.
f(x) = x log[sub:lnmqh1ob]2[/sub:lnmqh1ob]x, [1,2]
2) Use the Intermediate Value Theorem and Rolle's Theorem to prove that the equation has exactly one real solution.
x[sup:lnmqh1ob]5[/sup:lnmqh1ob]+x[sup:lnmqh1ob]3[/sup:lnmqh1ob]+x+1=0
3. Find a function f that has the derivative f'(x) and whose graph passes through the given point.
f'(x) = 2x, (1,0)
On this one, I did not know if it was possible to do without a calculator or not and was uncertain how to determine the critical points.
4. Identify the open intervals on which the function is increasing or decreasing.
y = x-2 cos x, 0 < x < 2 TT
My work:
y'=1+2 sin x
y'(0) = sin x = -1/2
Critical points: 7 TT/6, 11 TT/6
First question: Why would you pick these points as the critical points? And why 2 TT/3 and 4 TT/3 wouldn't work?
Second question: When determining which intervals are increasing and decreasing, is it possible to figure out which is which without the use of a calculator?
One point I used for the interval of 7 TT/6 < x < 11 TT/6 was 4 TT/3. I calculated each to compare each value in decimals and found 7 TT/6 = 3.67, 11 TT/6 = 5.76 and 4 TT/3 = 4.19, which is how I figured I could use 4 TT/3 to plug back into the derived equation to find a decreasing slope with the use of a calculator. But is there another way to do figure it out without a calculator? Thanks!!
1) Determine whether the Mean Value Theorem can be applied to f on the closed interval [a,b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a,b) such that f'(c) = f(b)-f(a)/b-a.
f(x) = x log[sub:lnmqh1ob]2[/sub:lnmqh1ob]x, [1,2]
2) Use the Intermediate Value Theorem and Rolle's Theorem to prove that the equation has exactly one real solution.
x[sup:lnmqh1ob]5[/sup:lnmqh1ob]+x[sup:lnmqh1ob]3[/sup:lnmqh1ob]+x+1=0
3. Find a function f that has the derivative f'(x) and whose graph passes through the given point.
f'(x) = 2x, (1,0)
On this one, I did not know if it was possible to do without a calculator or not and was uncertain how to determine the critical points.
4. Identify the open intervals on which the function is increasing or decreasing.
y = x-2 cos x, 0 < x < 2 TT
My work:
y'=1+2 sin x
y'(0) = sin x = -1/2
Critical points: 7 TT/6, 11 TT/6
First question: Why would you pick these points as the critical points? And why 2 TT/3 and 4 TT/3 wouldn't work?
Second question: When determining which intervals are increasing and decreasing, is it possible to figure out which is which without the use of a calculator?
One point I used for the interval of 7 TT/6 < x < 11 TT/6 was 4 TT/3. I calculated each to compare each value in decimals and found 7 TT/6 = 3.67, 11 TT/6 = 5.76 and 4 TT/3 = 4.19, which is how I figured I could use 4 TT/3 to plug back into the derived equation to find a decreasing slope with the use of a calculator. But is there another way to do figure it out without a calculator? Thanks!!
