Maths: Applications and Interpretation · Calculus. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
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A stationary point is a point where the gradient of the curve is zero, so the tangent is horizontal. To find one, differentiate, set dy/dx = 0 and solve for x. Then put x back into the original equation to get y.
At a local maximum the gradient changes from positive to negative (up, then down). At a local minimum it changes from negative to positive. Test the sign of dy/dx just before and just after the point, or look at the graph on your calculator.
In optimisation you find the largest or smallest value of a quantity such as area, volume, cost or profit. Write the quantity as a function of one variable, then find its stationary point.
Students stop at the x-value when the question asks for the maximum or minimum VALUE. Substitute x back into the original function (not into the derivative, which just gives 0) to get the area, volume or profit.
Answer: Minimum surface area = 48 cm2, when x = 4 cm.
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