Class 12

Math

Calculus

Application of Derivatives

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height $h$is $31 h$

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Using differentials, find the approximate value of each of the following up to 3 places of decimal.(i) $25.3 $ (ii) $49.5 $ (iii) $0.6 $ (iv) $(0.009)_{31}$ (v) $(0.999)_{101}$(vi) $(15)_{41}$

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:(i) $f(x)=x_{2}$ (ii) $g(x)=x_{3}−3x$ (iii) $h(x)=sinx+cosx,0<x<π2$ (iv) $f(x)=sinx−cosx$,$0<x<2π$(v)f(x) = $x_{3}−6x_{2}+9x+15$(vi) g(x) =$2x +x2 $ , $x>0$(vii) g(x) = $x_{2}+21 $(viii) $f(x)=x1−x $ , $x>0$

Prove that the function given by $f(x)=cosx$is(a) strictly decreasing in $(0,π)$(b) strictly increasing in $(π,2π)$, and(c) neither increasing nor decreasing in $(0,2π)$

Show that the tangents to the curve $y=7x_{3}+11$ at the points where $x=2$ and $x=−2$ are parallel.

Find the slope of the tangent to the curve $y=x−2x−1 ,x=2$ at $x=10$.

Find the maximum value of $2x_{3}−24x+107$ in the interval [1, 3]. Find the maximum value of the same function in $[3,1]˙$

Find absolute maximum and minimum values of a function f given by $f(x)=12x_{34}−6x_{31},x∈[−1,1]$.

Find the intervals in which the function f given by $f(x)=x_{3}+x_{3}1 ,x=0$is (i) increasing (ii) decreasing.