Application of Derivatives · Mathematics · TS EAMCET

For a given function $y=f(x), \delta y$ denote the actual error in $y$ corresponding to actual error $\delta x$ in $x$ and $d y$ denotes the approximately value of $\delta y$. If $y=f(x)=2 x^{2}-3 x+4$ and $\delta x=0.02$, then the value of $\delta y-d y$ when $x=5$ is

The length of the normal drawn at $t=\frac{\pi}{4}$ on the curve $x=2(\cos 2 t+t \sin 2 t), y=4(\sin 2 t+t \cos 2 t)$ is

If Water is poured into a cylindrical tank of radius 3.5 ft at the rate of $1 \mathrm{cu} \mathrm{ft} / \mathrm{min}$, then the rate at which the level of the water in the tank increases (in $\mathrm{ft} / \mathrm{min}$ ) is

$y=2 x^{3}-8 x^{2}+10 x-4$ is a function defined on [1,2]. If the tangent drawn at a point $(a, b)$ on the graph of this function is parallel to X-axis $a \in(1,2)$, then $a=$

If $m$ and $M$ are respectively the absolute minimum and absolute maximum values of a function $f(x)=2 x^{3}+9 x^{2}+12 x+1$ defined on $[-3,0]$, then $m+M=$

The maximum interval in which the slopes of the tangents drawn to the curve $y=x^{4}+5 x^{3}+9 x^{2}+6 x+2$ increase is

If $A=\{P(\alpha, \beta) /$ the tangent drawn at $P$ to the curve $y^{3}-3 x y+2=0$ is horizontal line $\}$ and $B=\{Q(a, b) /$ the tangent drawn at $Q$ to the curve $y^{3}-3 x y+2=0$ is a vertical line $\}$, then $n(A)+n(B)=$

$y=f(x)$ and $x=g(y)$ are two curves and $P(x, y)$ is a common point of the two curves. If at $P$ on the curve $y=f(x), \frac{d y}{d x}=Q(x)$ and at the same point $P$ on the curve $x=g(y), \frac{d x}{d y}=-Q(x)$, then

If the expression $7+6 x-3 x^2$ attains its extreme value $\beta$ at $x=\alpha$, then the sum of the squares of the roots of the equation $x^2+\alpha x-\beta=0$ is

The equation of the normal drawn to the curve $y^3=4 x^5$ at the point $(4,16)$ is

A point $P$ is moving on the curve $x^3 y^4=2^7$. The $x$-coordinate of $P$ is decreasing at the rate of 8 units per second. When the point $P$ is at $(2,2)$, the $y$-coordinate of $P$

If the function $f(x)=x^3+a x^2+b x+40$ satisfies the conditions of Rolle's theorem on the interval $[-5,4]$ and $-5,4$ are two roots of the equation $f(x)=0$, then one of the values of $c$ as stated in that theorem is

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$

If $4+3 x-7 x^2$ attains its maximum value $M$ at $x=\alpha$ and $5 x^2-2 x+1$ attains its minimum value $m$ at $x=\beta$, then $\frac{28(M-a)}{5(m+\beta)}=$

If $x=\cos 2 t+\log (\tan t)$ and $y=2 t+\cot 2 t$, then $\frac{d y}{d x}=$

The approximate value of $\sqrt[3]{730}$ obtained by the application of derivatives is

If $\theta$ is the acute angle between the curves $y^2=x$ and $x^2+y^2=2$, then $\tan \theta=$

The vertical angle of a right circular cone is $60^{\circ}$. If water is being poured in to the cone at the rate of $\frac{1}{\sqrt{3}} \mathrm{~m}^3 / \mathrm{min}$, then the rate ( $\mathrm{m} / \mathrm{min}$ ) at which the radius of the water level is increasing when the height of the water level is 3 m is

A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then semi-vertical angle of the cone is

If $f(x)=k x^3-3 x^2-12 x+8$ is strictly decreasing for all $x \in R$, then

The radius of a sphere is 7 cm . If an error of 0.08 sq cm is made in measuring it, then the approximate error (in cubic cm ) found in its volume is

The curve $y=x^3-2 x^2+3 x-4$ intersects the horizontal line $y=-2$ at the point $P(h, k)$. If the tangent drawn to this curve at $P$ meets the $X$-axis at $\left(x_1, y_1\right)$, then $x_1=$

If $f(x)=(2 x-1)(3 x+2)(4 x-3)$ is a real valued function defined on $\left[\frac{1}{2}, \frac{3}{4}\right]$, then the value(s) of $c$ as defined in the statement of Rolle's theorem

If the interval in which the real valued function $f(x)=\log \left(\frac{1+x}{1-x}\right)-2 x-\frac{x^3}{1-x^2}$ is decreasing in $(a, b)$, where $|b-a|$ is maximum, then $\frac{a}{b}=$

If the slope of the tangent drawn at any point $(x, y)$ on the curve $y=f(x)$ is $\left(6 x^2+10 x-9\right)$ and $f(2)=0$, then $f(-2)=$