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The turning point of a curve occurs when the gradient of the line = 0
The differential equation (dy/dx) equals the gradient of a line. Therefore in this case the differential equation will equal 0.
dy/dx = 0
Let's work through an example. If the equation of a line = y =x2 +2x
Therefore the differential equation will equal
dy/dx = 2x +2
therefore because dy/dx = 0 at the turning point then
2x+2 = 0
2x+2 = 0
This is the x- coordinate of the turning point
You can then sub this into the main equation (y=x2+2x) to find the y-coordinate. So if x = -1:
y = (-1)2+2(-1)
y = (1) +( - 2)
y = 3
This is the y-coordinate of the turning point
Therefore the coordinates of the turning point are x=-1, y =3
= (-1,3)see more
Eutrophication is the loss of oxygen in a body of water (e.g. pond or lake). This is usually due to algal bloom caused by artificial fertilisers and other chemicals being washed into the water by rain.
The 5 stages:
1.Chemicals/ artificial fertilisers: Chemicals and artificial fertilisers used on the land (e.g. for farms/crops) are washed into a pond/ lake by rainwater.
2. Algal bloom: The excess nutrients from the chemicals cause the algae in the water to grow at a fast rate and bloom over the surface of a pond/lake.
3. Plants die: The algal bloom on the surface of the water blocks sunlight to the plants at the bottom of the pond/lake. As plants require light for photosynthesis and this light is blocked, the plants die.
4. Decomposition of the dead plants and algae: The algae eventually die and bacteria decompose both the dead plants and the dead algae, further using up the oxygen in the pond/lake.
5. Fish die: As there are no longer plants to supply oxygen to the water and bacteria are decomposing dead plants and algae further using up the oxygen, the oxygen in the lake depletes. As there is not enough oxygen in the pond/lake to survive, the fish suffocate and die.
At ths point the pond/lake is anoxic (no oxygen in the water) and only bacteria/aglae which do not need oxygen to survive are left.see more