Class 12

Math

Calculus

Application of Derivatives

The volume of a cube is increasing at a rate of 9 cubic centimetres per second. How fast is the surface area increasing when the length of an edge is 10 centimetres?

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Find the value of $a$ if $x_{3}−3x+a=0$ has three distinct real roots.

Find the approximate change in the volume $V$ of a cube of side $x$ meters caused by increasing side by $1%˙$

If $φ(x)$ is differentiable function $∀x∈R$ and $a∈R_{+}$ such that $φ(0)=φ(2a),φ(a)=φ(3a)andφ(0)=φ(a)$ then show that there is at least one root of equation $φ_{prime}(x+a)=φ_{prime}(x)∈(0,2a)$

Separate the intervals of monotonocity of the function: $f(x)=−sin_{3}x+3sin_{2}x+5,x∈[−2π ,2π ]˙$

If the tangent at $(1,1)$ on $y_{2}=x(2−x)_{2}$ meets the curve again at $P,$ then find coordinates of $P˙$

Let $f(x)$ be differentiable function and $g(x)$ be twice differentiable function. Zeros of $f(x),g_{prime}(x)$ be $a,b$ , respectively, $(a<b)˙$ Show that there exists at least one root of equation $f_{prime}(x)g_{prime}(x)+f(x)g_{x}=0$ on $(a,b)˙$

Let $x$ be the length of one of the equal sides of an isosceles triangle, and let $θ$ be the angle between them. If $x$ is increasing at the rate (1/12) m/h, and $θ$ is increasing at the rate of $180π $ radius/h, then find the rate in $m_{3}$ / $h$ at which the area of the triangle is increasing when $x=12mandthη=π/4.$

Let $f(x)=xa +x_{2}˙$ If it has a maximum at $x=−3,$ then find the value of $a˙$