A guide wire follows the curve y = tan x, with x in radians, for 0 ≤ x < π/2…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A guide wire follows the curve , with in radians, for . Use the derivative (which you can derive from the definition, as in the lesson) to find the slope of the curve at and at . Then write the tangent at as and find where it meets the -axis, to four decimal places.
Need a reminder? The key ideaShow
With in radians: , and . A tangent is horizontal where the derivative is . The angle of a tangent with the -axis satisfies , when both axes have the same scale.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Given. and .
Plan. Evaluate at the two points, then write the tangent.
Step 1. At . , so .
Step 2. At . , so .
Step 3. The tangent at . The point is and the slope is . So .
Step 4. The intercept. .
Check. At the line gives , the height of the curve.
Answer: the slopes are and , and the tangent at meets the -axis at .
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