A road is drawn on a map in kilometres (km) as the curve y = x³ - 3x + 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 road is drawn on a map in kilometres (km) as the curve . A straight cable is laid along the tangent to the road at the point where km. Find the slope of the cable. Find where the cable meets the -axis. Then find where it meets the -axis, to two decimal places.
Need a reminder? The key ideaShow
The derivative is the limit of the secant slopes: . Also , and . The tangent at is .
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. in km, tangent at km.
Plan. Find the point and the slope, write the tangent, and read the intercepts.
Step 1. The point. , so the point is .
Step 2. The slope. , so .
Step 3. The tangent. , which is .
Step 4. The intercepts. At the height is . At , .
Check. The point satisfies , since .
Answer: slope , -intercept km and -intercept km.
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