Sketch the graph of y = x - ln x. By adding a suitable graph to the sketch, find the set of values of x that satisfies x^3 - x^2lnx - 5x^2 +1 < / = 0. How do I draw the graph y = x - ln x? And is the use of GC needed to find the values?
*I know how to draw -ln x graph but how does the y = x - ln x one looks like?
Hi,
You may use GC to sketch the graph of y = x - ln x, since it is a non-standard function.
Thanks.
Cheers,
Wen Shih
Afternote: One can sketch the graph of y = x - ln x without GC, with basic knowledge of curve sketching.
1. Find x-intercepts, i.e. let y = 0. Then x = ln x, but no such x exists since y = x and y = ln x do not intersect at all.
2. Find y-intercepts, i.e. let x = 0. This is not possible because ln x is not defined when x = 0. Therefore, x = 0 is a vertical asymptote. Note also, that x cannot take negative values, since ln x will not be defined.
3. Investigate the behaviour as x approaches infinity. When that happens, we observe that y approaches x, so y = x is an oblique asymptote.
4. Determine stationary points by differentiation, i.e. let dy/dx = 0. In this case, we have dy/dx = 1 - 1/x, from which x = 1 when dy/dx = 0. It is not difficult to ascertain that x = 1 gives a minimum turning point by the second derivative test.
With these facts, we can go ahead and sketch the graph of y = x - ln x.
Thanks again.
Originally posted by Audi:Sketch the graph of y = x - ln x. By adding a suitable graph to the sketch, find the set of values of x that satisfies x^3 - x^2lnx - 5x^2 +1 < / = 0. How do I draw the graph y = x - ln x? And is the use of GC needed to find the values?
*I know how to draw -ln x graph but how does the y = x - ln x one looks like?
I suggest you use the GC. It is technically possible to sketch the graph without the GC, but that requires you to have a very strong foundation/talent in curve-sketching, and few people can visualize graphs in their minds correctly. I don't have a GC, but my guess is y=x-lnx should look like a y=lnx graph, though the x-intercept and the curvature will be different. However, why risk trusting my mind when you can know the definite answer from your GC? It is a great aide for normal mortals like us, so do learn to use it to your advantage.
You would need to find x-value of the point of intersection between the two graphs, and I don't think that can be solved analytically. I believe your GC should have a function for that.