Question

The point \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\) lies on the curve \(y = {\bf{cos}}\pi x\).

(a) If Q is the point \(\left( {x,{\bf{cos}}\pi x} \right)\), find the slope of the secant line PQ (correct to six decimal places) for the following values of x:

(i) 0 (ii) 0.4 (iii) 0.49 (iv) 0.499 (v) 1 (vi) 0.6 (vii) 0.51 (viii) 0.501

(b) Using the results of part (a), guess the value of the slope of tangent line to the curve at \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\).

(c) Using the slope from part (b), find an equation of the tangent line to the curve at \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\).

(d) Sketch the curve, two of the secant lines, and the tangent line.

Short answer

(a)

	\(x\)	\(Q\left( {x,\cos \pi x} \right)\)	\({m_{PQ}}\)
(i)	0	\(\left( {0,1} \right)\)	\( - 2\)
(ii)	0.4	\(\left( {0.4,0.309017} \right)\)	\( - 3.090170\)
(iii)	0.49	\(\left( {0.49,0.031411} \right)\)	\( - 3.141076\)
(iv)	0.499	\(\left( {0.499,0.003142} \right)\)	\( - 3.141587\)
(v)	1	\(\left( {1, - 1} \right)\)	\( - 2\)
(vi)	0.6	\(\left( {0.6, - 0.309017} \right)\)	\( - 3.090170\)
(vii)	0.51	\(\left( {0.51, - 0.031411} \right)\)	\( - 3.141076\)
(viii)	0.501	\(\left( {0.501, - 0.003142} \right)\)	\( - 3.141587\)

(b) The slope of tangent line is \( - \pi \).

(c) The equation of tangent line is \(y = - \pi x + \frac{\pi }{2}\).

Step-by-step solution

Step 1:Find answer for part (a)

Find the coordinate of point \(P\)for \(x = 0\).

\(\begin{aligned}\left( {x,\cos \pi x} \right) &= \left( {0,\cos \pi \left( 0 \right)} \right)\\ &= \left( {0,1} \right)\end{aligned}\)

Find the slope os secant line for \(x = 0\).

\(\begin{aligned}{c}m &= \frac{{1 - 0}}{{0 - 0.5}}\\ &= - 2\end{aligned}\)

The table below represents the slope of secant lines for different values of x.

	\(x\)	\(Q\left( {x,\cos \pi x} \right)\)	\({m_{PQ}}\)
(i)	0	\(\left( {0,1} \right)\)	\( - 2\)
(ii)	0.4	\(\left( {0.4,0.309017} \right)\)	\( - 3.090170\)
(iii)	0.49	\(\left( {0.49,0.031411} \right)\)	\( - 3.141076\)
(iv)	0.499	\(\left( {0.499,0.003142} \right)\)	\( - 3.141587\)
(v)	1	\(\left( {1, - 1} \right)\)	\( - 2\)
(vi)	0.6	\(\left( {0.6, - 0.309017} \right)\)	\( - 3.090170\)
(vii)	0.51	\(\left( {0.51, - 0.031411} \right)\)	\( - 3.141076\)
(viii)	0.501	\(\left( {0.501, - 0.003142} \right)\)	\( - 3.141587\)

Find the answer for part (b)

From the table in step-1 it can be observed that the slope of tangent line is close to \( - \pi \).

Find the answer for part (c)

For the slope 1, the equation of the tangent line at \(P\left( {0,0.5} \right)\) is:

\(\begin{aligned}{c}y - 0 &= - \pi \left( {x - 0.5} \right)\\y &= - \pi x + \frac{\pi }{2}\end{aligned}\)

Thusm the equation of tangent line is \(y = - \pi x + \frac{\pi }{2}\).

Find the answer for part (d)

The curve below shows the curve of \(y = \cos \pi x\) with secant line and tangent line.