Class 12

Math

Calculus

Application of Derivatives

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

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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 $(a,b)˙$

The graph $y=2x_{3}−4x+2andy=x_{3}+2x−1$ intersect in exactly 3 distinct points. Then find the slope of the line passing through two of these points.

Displacement $s$ of a particle at time $t$ is expressed as $s=21 t_{3}−6t˙$ Find the acceleration at the time when the velocity vanishes (i.e., velocity tends to zero).

If $f:[−5,5]R$ is differentiable function and if$f_{prime}(x)$ does not vanish anywhere, then prove that $f(−5)=f(5)˙$

Discuss the extremum of $f(x)=31 (x+x1 )$

Let $f(x)=2x_{3}=9x_{2}+12x+6.$ Discuss the global maxima and minima of $f(x)∈[0,2]and(1,3)$ and, hence, find the range of $f(x)$ for corresponding intervals.

If the tangent to the curve $xy+ax+by=0$ at $(1,1)$ is inclined at an angle $tan_{−1}2$ with x-axis, then find $aandb?$

Two cyclists start from the junction of two perpendicular roads, there velocities being $3um/m∈$ and $4um/m∈$ , respectively. Find the rate at which the two cyclists separate.