Part II: Slope of the Sine curve at (0,0) The general equation of a line is y- m

Question

Part II: Slope of the Sine curve at (0,0) The general equation of a line is y- mz+ b, where m is the slope of the line and (0, b) is the y-intercept. The slope of a line is constant (the same everywhere) and measures the "steepness" of the line. A curve, like y- sin does not have a constant slope, since the graph rises and falls at different rates. We can approximate slopes of the curve using slopes of secant lines, lines that cross a curve at two or more points. In the image below, there are four secant lines (dashed red) to the curve y sin z (solid black) that all pass through the origin To calculate the slope of a line, use the slope formula m-22 , GA. 0.7071) // (R/6, 0.5) (t/3. ? 866) 05 (o, o) 176 n/2 1. Calculate the slopes of each of the secant lines shown, where the first point of each is (0,0) and the each second point has an coordinate getting closer and closer to 0: ?/2, ?/3, ?/1, and ?/6. Give both an exact value of the slopes, and an approxinate value rounded to three decimal places. [Hint The coordinates of any point on the curve is (r, sin c))

Explanation / Answer

0,pi/2 :
= (sin(pi/2 - sin(0)) / (pi/2 - 0)
= (1 - 0) / (pi/2)
= 2/pi

0,pi/3 :
= (sin(pi/3) - sin(0)) / (pi/3 - 0)
= (sqrt3/2) / (pi/3)
= sqrt(3)/2 * 3/pi
= 3sqrt(3)/(2pi)

0,pi/4 :
= (sqrt2/2)/(pi/4)
= sqrt(2)/2 * 4/pi
= 2sqrt(2)/pi

0,pi/6 :
= (1/2)/(pi/6)
= 1/2 * 6/pi
= 3/pi

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2)
0,0.01 :
(sin(0.01) - sin(0)) / (0.01 - 0)
= 0.9999833334

-0.1,0 :
(sin(0) - sin(-0.1)) / (0 - (-0.1))
= 0.99833416647

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3)
Yes, they appear to get closer and closer to 1
This means, the instantaneous rate of change of y = sinx
at x = 0 is equal to 1

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