Prove that $y=\frac{4\sin \theta }{(2+\cos \theta )}-\theta$ is an increasing function of $\theta$ in $\left[0,\frac{\pi }{2}\right]$.

1. Rate of Change of Quantities:

(i) If $y=f\left(x\right),$ then dydx measures the rate of change of $y$ with respect to $x.$

(ii) dydxx=x0represents the rate of change of $y$ with respect to $x$ at $x=x_0.$

(iii) If the displacement of a particle moving in a straight line at time $t$ is given by $s=f\left(t\right),$ then

(a) Velocity $\left(v\right)$ at time t=dsdt, Acceleration $\left(a\right)$ at time $t=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}=v\frac{dv}{ds}$

(b) If a particle moving in a straight line comes to rest, then dsdt=0 and d2sdt2=0

(c) If a particle moving in a straight line comes at rest instantaneously, then dsdt=0 but d2sdt2≠0

2. Increasing and Decreasing Functions:

(i) A function $f\left(x\right)$ is said to be monotonic on $(a,b)$ if it is either strictly increasing or strictly decreasing on $(a,b)$

(ii) A function $f\left(x\right)$ is said to be a strictly increasing function on $(a,b)$ if $x_1<x_2\Rightarrow f\left(x_1\right)<f\left(x_2\right)$ for all $x_1,x_2\in \left(a,b\right)$  

(iii) The necessary and sufficient condition for a differentiable function defined on $(a,b)$ to be strictly increasing on $(a,b)$ is that $f\text{'}\left(x\right)>0$ for all $x\in \left(a,b\right)$.

(iv) The necessary and sufficient condition for a differentiable function defined on $(a,b)$ to be strictly decreasing on $(a,b)$ is that $f\text{'}\left(x\right)<0$ for all $x\in (a,b)$

3. Tangents and Normals:

(i) Slope of Tangent and Normal: 

(a) If $y=f\left(x\right),$ then ${\left(\frac{dy}{dx}\right)}_P=$Slope of the tangent to $y=f\left(x\right)$ at point $P$

(b) -1dydxP$=$Slope of the normal to $y=f\left(x\right)$ at point

(ii) Equations of Tangent and Normal:

(a) If $P\left(x_1,y_1\right)$ is a point on the curve $y=f\left(x\right),$ then y-y1=dydxPx-x1 is the equation of tangent at $P$

(b) y-y1=-1dydxPx-x1 is the equation of the normal at $P$

(iii) Angle between the two curves:

(a) The angle between the tangents to two given curves at their point of intersection is defined as the angle of intersection of two curves.

(b)  If $C_1$ and $C_2$ are two curves having equations $y=f\left(x\right)$ and $y=g\left(x\right)$ respectively such that they intersect at point $P.$ The angle $\theta$ of intersection of these two curves is given by tanθ=dydxC1-dydxC21+dydxC1dydxC2

(c) If the angle of intersection of two curves is at right angle, then the curves are said to intersect orthogonally. The condition for orthogonality of two curves $C_1$ and $C_2$ is dydxC1×dydxC2=-1

4. Approximations:

(i) Let $y=f\left(x\right)$ be a function of $x,$ and let $∆x$ be a small change in $x$ and $\Delta y$ be the corresponding change in $y.$ Then, Δy=dydxΔx approximately.

(ii) Let $y=f\left(x\right)$ be a given function of $x$. If $∆x$ is an error in $x,$ then the corresponding error $∆y$ in $y$ is given by $\Delta y=\frac{dy}{dx}\Delta x$

The error $\Delta x$ in $x$ and $\Delta y$ in $y$ are known as absolute errors

(iii) If $\Delta x$ is an error in $x,$ then Δxx is called relative error in $x$

(iv) If $\Delta x$ is an error in $x,$ then Δxx×100 is called the percentage error in $x$.

5. Maxima and Minima: 

(i) First derivative test for local maxima and minima: 

Let $f\left(x\right)$ be a function differentiable at $x=a$, then 

(a) $x=a$ is a point of local maximum of $f\left(x\right),$ if $f\text{'}\left(a\right)=0$ and $f\text{'}\left(x\right)$ changes sign from positive to negative as $x$ passes through $a$.

(b) $x=a$ is a point of local minimum of $f\left(x\right),$ if $f\text{'}\left(a\right)=0$ and $f\text{'}\left(x\right)$ changes sign from negative to positive as $x$ passes through $a$.

(c) If $f\text{'}\left(a\right)=0,$ but $f\text{'}\left(x\right)$ does not change sign, that is, $f\text{'}\left(a\right)$ has the same sign in the complete neighbourhood of $a$, then $a$ is neither a point of local maximum nor a point of local minimum.

(d) The points at which a function attains either the local maximum values or local minimum values are known as the extreme points or turning points and both local maximum and local minimum values are called the extreme values of a function.

(ii) The maximum and minimum values of a function: 

Defined on a closed interval may be obtained by using the following steps.

Let $y=f\left(x\right)$ be a function defined on $\left[a,b\right]$

STEP I: Find dydx=f'x

STEP II: Put $f\text{'}\left(x\right)=0$ and find values of x. Let $c_1,c_2,\dots ,c_n$ be the values of $x.$ The value of $x$ for which $f\text{'}\left(x\right)=0$  are called stationary values or critical values of $x$ and the corresponding values of $f\left(x\right)$ are called stationary or turning values of function.

STEP III: Take the maximum and minimum values out of the values $f\left(a\right),f\left(c_1\right),f\left(c_2\right),\dots ,f\left(c_n\right),f\left(b\right)$.

The maximum and minimum values obtained in step III are respectively the largest or absolute maximum and the smallest or absolute minimum values of the function.