The function
- Cubic to examine
Maths · sample calcsheet · 2 pages
Turning points with symbolic maths
Symbolic maths works on formulae, not just numbers. Here a cubic is differentiated symbolically, and that derivative is solved for zero to find both of its turning points.
What this calculation covers
Differentiate
With x undefined there is no number to return, so the answer is the derivative itself. Keywords chain, and expand puts the result in standard order.
- The derivative, as a formula
- The same derivative, as a function
- Slope at x = 2
Turning points
A turning point is where the slope is zero. solve() returns every root at once, as a column to read elements from.
- Where the slope is zero
- First turning point
- Second turning point
- Height of the first
- Height of the second
Maximum or minimum
- Second derivative, from the first
- Negative, so a maximum
- Positive, so a minimum
- Slope zero at the maximum
- Slope zero at the minimum
Results summary
- The cubic turns at x = 1, a maximum of 4, and at x = 3, a minimum of 0. Nothing was differentiated by hand: change the function and every row follows.
Try changing…
- Set the cubic to x³ − 12x — the turning points move to −2 and 2, and every row below follows.
- Right-click the derivative row and pick Show as Function Call — the spelling changes, the answer does not.
- Drop the → expand from the derivative row — the same formula returns, just not in standard order.