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A farmer has 16 meters of fencing, and wants to use it to create an enclosure for his sheep to graze in. He wants the enclosure to have a rectangular section and a semi-circular section with radius r, as illustrated in the diagram below.
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Give an expression for the area of the enclosure in terms of r.
Find the maximum possible area of the enclosure.
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Consider the function f defined by f(x)=√x+m+n where n,m∈R.
Find f′(x).
Now consider the function g(x)=ex+2n. The graphs of f and g share a tangent at x=1.
Show that m=4e21−1.
Hence, find the value of n.
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Consider the function f defined by f(x)=3(x−h)2+k where h,k∈R.
Find f′(x).
Now consider the function g(x)=ex−1−k. The graphs of f and g share a tangent at x=2.
Show that h=612−e.
Hence, find the value of k.
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