Application of Derivatives

Prove that is an increasing function of in . - Wired Faculty

Question

Prove that $straight y equals space fraction numerator 4 space sin space straight theta over denominator 2 plus cos space straight theta end fraction minus straight theta$ is an increasing function of $straight theta$ in $open square brackets 0 comma space straight pi over 2 close square brackets$ .

Solution

Here, $straight y equals space fraction numerator 4 space sin space straight theta over denominator 2 plus cos space straight theta end fraction minus straight theta$
$therefore space space space space space dy over dθ space equals space fraction numerator left parenthesis 2 plus cos space straight theta right parenthesis thin space left parenthesis 4 space cos space straight theta right parenthesis space minus space 4 space sin space straight theta space left parenthesis negative sin space straight theta right parenthesis over denominator left parenthesis 2 plus space cos space straight theta right parenthesis squared end fraction minus 1$
$equals space fraction numerator 8 space cos space straight theta space plus space 4 space cos squared space straight theta space plus space 4 space sin space squared straight theta over denominator left parenthesis 2 plus cos space straight theta right parenthesis squared end fraction minus 1 equals space space fraction numerator 8 space cos space straight theta space plus space 4 left parenthesis cos squared straight theta plus sin squared straight theta right parenthesis over denominator left parenthesis 2 plus cosθ right parenthesis squared end fraction minus 1 equals space fraction numerator 8 cosθ plus 4 over denominator left parenthesis 2 plus cos space straight theta right parenthesis squared end fraction minus 1 space equals space fraction numerator 8 space cos space straight theta space plus space 4 space minus left parenthesis 2 plus space cos space straight theta right parenthesis squared over denominator left parenthesis 2 plus space cos space straight theta right parenthesis squared end fraction space equals space fraction numerator 8 space cos space straight theta plus 4 minus 4 minus 4 space cos space straight theta minus space cos squared straight theta over denominator left parenthesis 2 plus space cos space straight theta right parenthesis squared end fraction space equals fraction numerator 4 space cos space straight theta space minus space cos squared straight theta over denominator left parenthesis 2 plus cos space straight theta right parenthesis squared end fraction space equals space fraction numerator cos space straight theta left parenthesis 4 minus cos space straight theta right parenthesis over denominator left parenthesis 2 plus cos space straight theta right parenthesis squared end fraction$
For y to be increasing, $dy over dθ greater than 0$
$rightwards double arrow space space space space space fraction numerator cos space straight theta space left parenthesis 4 minus cos space straight theta right parenthesis over denominator left parenthesis 2 plus space cos space straight theta right parenthesis squared end fraction greater than 0 space space space space rightwards double arrow space space space cos space straight theta thin space greater than 0 space space space space space left square bracket because space space left parenthesis 2 plus cos space straight theta right parenthesis squared greater than 0 comma space space 4 minus cos space straight theta space greater than 0 right square bracket which space is space so space space if space space straight theta space element of open parentheses 0 comma space straight pi over 2 close parentheses Hence space the space result. space$

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Application Of Derivatives

Prove that $straight y equals space fraction numerator 4 space sin space straight theta over denominator 2 plus cos space straight theta end fraction minus straight theta$ is an increasing function of $straight theta$ in $open square brackets 0 comma space straight pi over 2 close square brackets$ .

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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