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Wednesday, 28 October 2015

GROVE HSC 2U EX 2.6 Q11 STATIONARY POINT - TRICKY FUNCTION

QUESTION: Find any stationary points on the curve y=(x-3)\sqrt{4-x}

This is a product of two functions, so we must use the Product rule to differentiate it.

If y=uv, y'=vu'+uv'

u=x-3         u'=1

v=\sqrt{4-x}
v=(4-x)^{1\over 2}
v'={1\over 2}(4-x)^{-{1\over 2}}(-1)
v'=\displaystyle {-1\over 2\sqrt{4-x}}

[RECALL chain rule. If y=f(x)^n, then y'=n\,f(x)^{n-1}\times f'(x)]

Therefore,
y'=vu'+uv'
y'=\sqrt{4-x} .1 + (x-3)\left( \displaystyle {-1\over 2\sqrt{4-x}}\right)

y'=\displaystyle {\sqrt{4-x} \over 1} + \displaystyle{-(x-3) \over 2\sqrt{4-x}}


y'=\displaystyle {\sqrt{4-x} \over 1} + \displaystyle{3-x \over 2\sqrt{4-x}}


y'=\displaystyle {\sqrt{4-x} \over 1}\times {2\sqrt{4-x} \over 2\sqrt{4-x}} + \displaystyle{3-x \over 2\sqrt{4-x}}

[RECALL: \sqrt{a}\times \sqrt{a}=a]

y'=\displaystyle{2(4-x)+3-x \over 2\sqrt{4-x}}

y'=\displaystyle{8-2x+3-x \over 2\sqrt{4-x}}

y'=\displaystyle{11-3x\over 2\sqrt{4-x}}



FOR STAT POINTS: SOLVE y'=0.
\displaystyle{11-3x\over 2\sqrt{4-x}}=0
\therefore 11-3x=0
3x=11
x=\displaystyle {11\over 3}

When
x=\displaystyle {11\over 3},
y=\displaystyle\left({11\over 3}-3\right)\sqrt{4-{11\over 3}}
y=\displaystyle{2\over 3}\sqrt{{1\over 3}}
y=\displaystyle{2\over 3\sqrt{3}}

y=\displaystyle{2\sqrt{3}\over 9}

Stationary point is \displaystyle\left({11\over 3},{2\sqrt{3}\over 9}\right).

Now determine the nature of this stat point.

So it's a maximum turning point.

Correct to 2 decimal places it's (3.67, 0.38). (Grove gives the answer to 2 d.p. even though the question didn't ask for it)

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