Class 12

Math

Calculus

Application of Derivatives

For the curve $y=4x_{3}−2x_{5},$find all the points at which the tangent passes through the origin.

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Let $f(x)=−sin_{3}x+3sin_{2}x+5$ on $[0,2π ]$ . Find the local maximum and local minimum of $f(x)˙$

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 $x=12mandthη=π/4.$

There is a point (p,q) on the graph of $f(x)=x_{2}$ and a point $(r,s)$ on the graph of $g(x)=x−8 ,wherep>0andr>0.$ If the line through $(p,q)and(r,s)$ is also tangent to both the curves at these points, respectively, then the value of $P+r$ is_________.

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 $φ_{prime}(x+a)=φ_{prime}(x)∈(0,2a)$

The two equal sides of an isosceles triangle with fixed base $b$ are decreasing at the rate of $3cm/s˙$ How fast is the area decreasing when the two equal sides are equal to the base?

Find the equation of the tangent to the $curvey={x_{2}xsin1 ,x=00,x=0aheorig∈$

Find the point on the curve $3x_{2}−4y_{2}=72$ which is nearest to the line $3x+2y+1=0.$

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).