By choosing appropriate secant lines, estimate the slope of the tangent line at P. (Round your answer to two decimal places.)

Question

The point P(1,0) lies on the curve y=sin((14pi𝜋)/(x)).

(a) If Q is the point (x,sin((14𝜋)/(x)), find the slope of the secant line PQ (correct to four decimal places) for the following values of x.

(i) 2 = 0

(ii) 1.5 = -1.7321

(iii) 1.4 = 0

(iv) 1.3 = 2.2104

(v) 1.2 = -4.3301

(vi) 1.1 = 7.5575

(vii) 0.5 = 0

(viii) 0.6 = 2.1651

(ix) 0.7 = 0

(x) 0.8 = 5

(xi) 0.9 = 9.8481

(b)By choosing appropriate secant lines, estimate the slope of the tangent line at P. (Round your answer to two decimal places.)

Note: I answered part (a) and provided answers for it as they were correct. I am just a bit confused on how to answer part B of this question. If you can kindly explain what to do for part (b), I would appreciate it. Thank you very much.

Answer 1

Notice that points (i), ..., (vi) are approaching x=1 from the right and points

(vii), ..., (xi) are approaching from the left.

The closer is a point Q to x=1, the closer is the slope of the line PQ to the slope of the tangent at x=1.

For this particular function you can treat a pair like (xi),(vi) as an upper and lower

bound on the slope of the tangent.

If you can find a pair (something like Q1=0.999, Q2=1.001) so close to x=1 that the two secant lines PQ1 and PQ2 have identical slopes within two decimal places, then you have your estimate.

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The exact value of the slope at x=1 is -14π.

The thing about this function is that it alternates between up and down extremely quickly.

If I could attach a graph I would; I would recommend that you look at it in some graphing tool.

Just imagine what is happening around x=1:

The argument of sin is 14π, making the value of the function 0.

If you keep increasing x beyond 1, the argument of sin gets smaller;

when the argument of sin gets to 13.5π the value of the function reaches -1

and the direction of the slope reverses.

That means that to get any meaningful estimate of the slope you need x

such that 14π/x is in the interval (13.5π, 14.5π).

That means x must be in the interval (0.97, 1.03).

And you probably need to try values even closer to 1 to get accuracy to 2 decimal digits.