Porisma
Back to lesson

Chapter exploration

Applications of differentiation

Explore

Plot f(x)=x3−3xf(x)=x^3-3x, g(x)=f(x)+5g(x)=f(x)+5 and their derivatives. Before inspecting the plots, predict the stationary inputs, the increasing/decreasing intervals and which coordinates move under the translation. Compare your predictions with exact derivatives. Explain why identical derivative graphs do not determine a unique function graph without a height condition.

Related interactive example

Explore this related example, then return to the prediction task above.

How fast does the level rise at each depth?

dh/dt = 1920/(9πh²) falls steeply as the water gets deeper. Slide the line to find the depth for a given rate.

Height of the horizontal linek = 5.65
Try an example
Number of distinct depths
1

Each rate corresponds to one depth. At 5.65 cm/s the depth is h≈3.47h\approx3.47 cm; the deeper the water, the slower the rise.

All listed intersections are within the original drawing window. A touching point counts once; hollow endpoints are excluded.

Coordinates are rounded to 6 significant figures. Inspect a row to highlight that intersection and bring it into view.
Pointxy
3.466815.65
  • Dh/dt against the depth h
  • Chosen rate
  • Depth with that rate
  • Intersection selected in the coordinate table
Key idea

Constant inflow in a widening container means a slowing rise.

Ideas to try
At what depth is the level rising at 1 cm/s?
Open the full interactive problem →
Back to lesson