Asked by Laura
Of the infinitely many lines that are tangent to the curve
y = −7 sin x
and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.
y = −7 sin x
and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.
Answers
Answered by
Steve
The slope at any point is
y' = -7cosx
The line through (0,0) and (x,-7sinx) has slope -7sinx/x. That means we need
-7sinx/x = -7cosx
x = tanx
Solutions are
x = 4.493409, 7.725252, 10.904122, ...
The slopes of the lines are
1.521, -0.899, 0.639
See the first couple of lines at
http://www.wolframalpha.com/input/?i=plot+y+%3D+-7sin%28x%29%2C+y%3D1.521x%2C+y%3D-0.899x%2C+from+0+to+15
y' = -7cosx
The line through (0,0) and (x,-7sinx) has slope -7sinx/x. That means we need
-7sinx/x = -7cosx
x = tanx
Solutions are
x = 4.493409, 7.725252, 10.904122, ...
The slopes of the lines are
1.521, -0.899, 0.639
See the first couple of lines at
http://www.wolframalpha.com/input/?i=plot+y+%3D+-7sin%28x%29%2C+y%3D1.521x%2C+y%3D-0.899x%2C+from+0+to+15
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