How do I know the slope of the tangent in this problem? (Regarding tangent problem)

Question

Q. The point P(1,0) lies on the curve y=sin(17π/x).

If Q is the point (x, sin(17π/x)), find the slope of the secant line PQ (correct to four decimal places) for the following values of x.

I got,

x=2 -> 1

1.5. -1.7321

1.4. 1.0847

1.3 -0.7977

1.2 2.5

1.1 -9.8982

0.5 0.0000

0.6 -2.1651

0.7 -2.6061

0.8 3.5355

0.9 -3.4202

Throught this calculation, I knew that as x approaches 1, the slopes do not appear to be approaching any particular value.

SO, my question is

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

How do I get the slope of the tangent line at P? I have to submit this homework due tonight. So, I hope to receive the answer as soon as possible. I want to know the exact answer of this question and also the process of solving this problem.

Answer 1

I checked your answers for the slope of the secant line and agree with them. You'll notice they jump all over the place, which is due to the oscillatory nature of the function.

To find the slope of the tangent line at x = 1, you need to choose values VERY close to 1:

x y

0.999 53.4351

0.9999 53.4122

0.99999 53.4076

0.999999 53.4071

0.9999999 53.4071

0.99999999 53.4071

0.999999999 53.4074

0.9999999999 53.4140

Now switch to values above x = 1

x y

1.001 53.3284

1.0001 53.4015

1.00001 53.4065

1.000001 53.4070

1.0000001 53.4070

1.00000001 53.4071

If I was going to guess, I'd guess 53.41

The exact answer is found by taking the derivative which you probably haven't learned how to do yet. When you do, you will find that the exact value of the slope of the tangent line at x = 1 is "17π"

Answer 2

Pick two x values from the table around the x value where they want you to find the tangent.

If you calculate the secant slope using the slope formula y2-y1/x2-x1,

that secant slope is going to be the best estimation of the tangent slope at the x value in question!

Hope that helps!