Class 12

Math

Calculus

Application of Derivatives

Find the equation of the tangent to the curve $y=(x−2(x−3)x−7 $ at the point where it cuts the x-axis.

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If the tangent at any point $(4m_{2},8m_{2})$ of $x_{3}−y_{2}=0$ is a normal to the curve $x_{3}−y_{2}=0$ , then find the value of $m˙$

Find the equation of the normal to the curve $y=∣∣ x_{2}−∣∣ x∣∣atx=−2.$

Let $P(x)$ be a polynomial with real coefficients, Let $a,b∈R,a<b,$ be two consecutive roots of $P(x)$ . Show that there exists $c$ such that $a≤c≤bandP_{prime}(c)+100P(c)=0.$

Find the locus of point on the curve $y_{2}=4a(x+as∈ax )$ where tangents are parallel to the axis of $x˙$

Draw the graph of $f(x)=x_{2}−xx_{2}−5x+6 $

Discuss the global maxima and global minima of $f(x)=tan_{−1}(g)_{e}x$ in$[3 1 ,3 ]$

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle is one-third that of the cone and the greatest volume of cylinder is $274 πh_{3}tan_{2}α˙$

A horse runs along a circle with a speed of $20km/h$ . A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. Find the speed with which the shadow of the horse moves along the fence at the moment when it covers 1/8 of the circle in km/h.