Applications of derivatives
Increasing and decreasing, maxima and minima, tangents and rates.
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Where the slope is zero, the graph is at the top of a hill or bottom of a valley.
Look: Flat at the top of the hill (x = 1) and the bottom of the valley (x = 3).
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1Smallest value of x² − 6x + 10?
2Two numbers add to 20. Biggest product?
3Where is x² going up?
! Careful!
✗Assuming every flat point is a top.
✓It could be a bottom. Check either side.
✗Stopping at x = 50.
✓Answer the question: the sizes and the area.
See the whole lesson on one page
Where the slope is zero, the graph is at the top of a hill or bottom of a valley.
Slope above 0: going up. Below 0: going down. Find where the derivative is 0 to find the biggest or smallest value.
Think of it like thisAt the very top of a roller coaster, for a moment, you are neither going up nor down. The slope there is zero.
Solved together
You have 200 m of fence for a field by a river (no fence on the river side). Which shape gives the biggest field?
Two short sides x and one long side 200 − 2x.
Area = x(200 − 2x) = 200x − 2x²Set the derivative to 0.
200 − 4x = 0x = 50Long side = 200 − 100 = 100.
50 × 100 = 5,000 m²
Answer50 m × 100 m, area 5,000 m².
Now practise it
Kanakku makes new questions on applications of derivatives every time. It shows the working and gives a hint when you are stuck.