You are being asked to find an such that:Originally Posted by Mathn00b
Sketching/graphing the right hand side shows that it equals 1 near , but exactly, so:
1. If g(x)= 3 + x + e^x, find g^-1(4).
It seems as if I cannot isolate "x" by itself.
2. lim x / ( √(1+3x) - 1 )
I know the answer is 2/3 (due to graphing), but I cannot prove this answer utilizing the Limit Laws.
3. A Tibetan monk leaves the monastery at 7:00 A.M. and takes his usual path to the top of the mountain, arriving at 7:00 P.M. The following morning, he starts at 7:00 A.M. at the top and takes the same path back, arriving at the monastery at 7:00 P.M. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.
I am utterly stumped on this problem. I realize that I have to chart the path on the same graph, but then what?
4. Find lim x⇒->∞ f(x) if, for all x > 1,
5√(x) / (√(x-1)) < f(x) < (10e^x - 21) / (2e^x)
Let be the length of the path, and let beOriginally Posted by Mathn00b
the distance that the monk is from the bottom at time
on day 1, and , be the distance the monk is form the bottom
at time on day 2.
Now consider the function: .
At 7am ,
at 7pm .
So by the intermediate value theorem there must be some time t between
7am and 7pm where , as is between and .
( of course means that at time on both days the monk is at
the same disrance from the bottom, that is at the same point)
Raionalize the denominator; multiply top and bottom by
This is an application of the famous "Squeeze Theorem".
Divide top and bottom of the first fraction by
Divide top and bottom of the second fraction by
We have: .
. . .Then: . . . .
. . Hence: . . . . . . . .