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) and provide an answer as well, I would appreciate it. Thank you very much.

Answer 1

The idea is to let the x coordinate of point Q get closer and closer to 1 (from both sides).

So pick x coordinate values for point Q like 1.01, 1.001, 1.0001, .9, .99, .999. Determine the y-value(s) of point Q then calculate the slope PQ. See if those slopes are approaching a value. That value is the slope of the tangent line to the curve.

See the following graphs to get the idea:

desmos.com/calculator/wpiaogjz2n

desmos.com/calculator/u8dvdt8fyi