Class 12

Math

Calculus

Application of Derivatives

If the sum 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 $3π ˙$

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The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2cm/minute. When $x=10$cm and $y=6$cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.

Find the approximate change in the volume V of a cube of side x meters caused by increasing the side by 2%.

Find the equation of the normal to curve $x_{2}=4y$which passes through the point (1, 2).

Let f be a function defined on [a, b] such that $f_{prime}(x)>0$, for all $x∈(a,b)$. Then prove that f is an increasing function on (a, b).

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

Find the maximum and the minimum values, if any, of the function f given by$f(x)=x_{2},x∈R$.

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is $tan_{−1}2 $.

For all real values of x, the minimum value of $1+x+x_{2}1−x+x_{2} $is(A) 0 (B) 1 (C) 3 (D) $31 $