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14) Find the slope of tangent to
a) Y =
[tex]4 \div x + 2 \sqrt{x \: } [/tex]
x=4​


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Réponse : Hi,

[tex]Y'=-\frac{4}{x^{2}}+2 \times \frac{1}{2\sqrt{x}}=-\frac{4}{x^{2}}+\frac{1}{\sqrt{x}}[/tex].

Equation of tangent to x=4 is:

[tex]y=Y'(4)(x-4)+Y(4)[/tex].

[tex]Y'(4)=-\frac{4}{4^{2}}+\frac{1}{\sqrt{4}}=-\frac{1}{4}+\frac{1}{2}=\frac{-1+2}{4}=\frac{1}{4}\\Y(4)=\frac{4}{4}+2\sqrt{4}=1+2 \times 2=1+4=5[/tex].

So the equation of tangent to x=4:

[tex]y=\frac{1}{4}(x-4)+5=\frac{1}{4}x-1+5=\frac{1}{4}x+4[/tex]

The slope of the tangent to x=4 is [tex]\frac{1}{4}[/tex].