Sample QuestionsAPPLICATION OF DERIVATIVES questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The slope of tangent to the curve $x = t^2 + 3t - 8, y = 2t^2 - 2t - 5$ at the point $(2, -1)$ is:
View full solution →he two curves $x^3 - 3xy^2 + 2 = 0$ and $3x^2y - y^3 - 2 = 0$ intersect at an angle of:
View full solution →Maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is:
View full solution →The interval on which the function $f(x) = 2x^3 + 9x^2 + 12x - 1$ is decreasing is:
View full solution →The equation of normal to the curve $3x^2 - y^2 = 8$ which is parallel to the line $x + 3y = 8$ is:
View full solution →Find the approximate value of $(1.999)^5.$
View full solution →Find an angle $\theta,0<\theta<\frac{\pi}{2},$ which increases twice as fast as its sine.
View full solution →Prove that $\text{f(x)}=\sin\text{x}+\sqrt{3}\cos\text{x}$ has maximum value at $\text{x}=\frac{\pi}{6}.$
View full solution →At what point, the slope of the curve $y = -x^3 + 3x^2 + 9x - 27$ is maximum? Also find the maximum slope.
View full solution →Show that $\text{f(x)}=2\text{x}+\cot^{-1}\text{x}+\log\Big(\sqrt{1+\text{x}^1}-\text{x}\Big)$ is increasing in R.
View full solution →Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are $3\ cm$ and $3.0005\ cm,$ respectively.
View full solution →If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is $\frac{\pi}{3}.$
View full solution →Prove that the curves $xy = 4$ and $x^2 + y^2 = 8$ touch each other.
View full solution →At what points on the curve $x^2 + y^2 - 2x - 4y + 1 = 0,$ the tangents are parallel to the $y-$axis$?$
View full solution →A man, 2m tall, walks at the rate of $1\frac{2}{3}\text{m/s}$ towards a street light which is $5\frac{1}{3}\text{m}$ above the ground. At what rate is the tip of his shadow moving? at what rate is the length of the shadow changing when he is $3\frac{1}{3}\text{m}$ from the base of the light?
View full solution →An open box with square base is to be made of a given quantity of card board of area $c^2.$ Show that the maximum volume of the box is $\frac{\text{c}^2}{6\sqrt{3}}$ cubic units.
View full solution →$x$ and $y$ are the sides of two squares such that $y = x - x^2.$ Find the rate of change of the area of second square with respect to the area of first square.
View full solution →A telephone company in a town has $500$ subscribers on its list and collects fixed charges of $Rs. 300/-$ per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of $Rs. 1/-$ one subscriber will discontinue the service. Find what increase will bring maximum profit?
View full solution →Show that $\text{f(x)}=\tan^{-1}(\sin\text{x}+\cos\text{x})$ is an increasing function in $\Big(0,\frac{\pi}{4}\Big).$
View full solution →Show that for $\text{a}\geq1,\text{ f(x)}=\sqrt{3}\sin\text{x}-\cos\text{x}-2\text{ax}+\text{b}$ is decreasing in R.
View full solution →A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is propotional to the surface. Prove that the radius is decreasing at a constant rate.
The equation of normal to the curve y = tan x at (0, 0) is ________.
The values of a for which the function f(x) = sinx - ax + b increases on R are ______.
The function $\text{f(x)}=\frac{2\text{x}^2-1}{\text{x}^4},\text{ x}>0,$ decreases in the interval _______.
View full solution →The least value of the function $\text{f(x)}=\text{ax}+\frac{\text{b}}{\text{a}}(\text{a}>0,\text{b}>0,\text{x}>0)$ is ______.
View full solution →The curves $y = 4x^2 + 2x - 8$ and $y = x^3 - x + 13$ touch each other at the point $..........$
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