Class 12

Math

Calculus

Application of Derivatives

Find the equation of tangent to the curve given by$x=asin_{3}t,y=bcos_{3}t$ ... (1)at a point where $t=2π $.

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Find the values of $p$ if $f(x)=cosx−2px$ is invertible.

If the function $f(x)=x_{3}−6x_{2}+ax+b$ defined on [1,3] satisfies Rolles theorem for $c=3 23 +1 $ then find the value of $aandb$

Let $f$ be continuous on $[a,b],a>0,and$ differentiable on $(a,b)˙$ Prove that there exists $c∈(a,b)$ such that $b−abf(a)−af(b) =f(c)−cf_{prime}(c)$

In the curve $x_{m+n}=a_{m−n}y_{2n}$ , prove that the $mth$ power of the sub-tangent varies as the $nth$ power of the sub-normal.

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

If the curves $ay+x_{2}=7andx_{3}=y$ cut orthogonally at $(1,1)$ , then find the value $a˙$

Discuss the extremum of $f(x)=asecx+bcosecx,0<a<b$

The acute angle between the curves $y=∣∣ x_{2}−1∣∣ $and $y=∣∣ x_{2}−3∣∣ $ at their points of intersection when when x> 0, is