PROBLEM 51 (Page 113): How many tangent lines to the curve ...
PROBLEM 51 (Page 113): How many tangent lines to the curve [pic] pass through the point [pic]? At which points do these tangent lines touch the curve?
First, check to see if the point [pic]is on the curve [pic]:
[pic]. Thus, the point [pic] is not on the curve [pic].
In general, we have the following picture:
NOTE: The point [pic] is the tangent point.
Since the two points of [pic] and [pic] are two points on the tangent line, we can find the slope of the tangent line using algebra:
[pic]
Of course, we can find the slope of the tangent line to the graph of [pic] at the point [pic] using calculus: [pic]
Thus, we have that [pic]. For this problem, [pic].
Thus, [pic] = [pic] = [pic] and
[pic] = [pic] = [pic] = [pic] =
[pic] = [pic] = [pic]
Thus, [pic] [pic]
[pic]
[pic]
[pic]
[pic] [pic]
[pic] or [pic]
[pic]. Since the domain of the function f is all real numbers except [pic], then we do not have a tangent point at[pic].
[pic]
[pic] = [pic] = [pic] = [pic]
These are the x-coordinates of the tangent points to the graph of [pic] of the tangent lines that pass through the point [pic]. Thus, there are two tangent lines that pass through the point [pic].
If [pic], then [pic] = [pic] =
[pic] = [pic] = [pic]. Thus, one tangent point is [pic].
If [pic], then [pic] = [pic] =
[pic] = [pic] = [pic] = [pic] =
[pic]. Thus, the other tangent point is [pic].
Maple commands to solve this problem and draw the graph.
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