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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