You are being asked to find an such that:Originally Posted by Mathn00b
,
or:
Sketching/graphing the right hand side shows that it equals 1 near , but exactly, so:
hence
RonL
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 )
x->0
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)
RonL
Hello, Mathn00b!
Raionalize the denominator; multiply top and bottom by
. .
Therefore: .
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: . . . . . . . .
Therefore: .