Embibe Experts Solutions for Chapter: Applications of Differential Calculus, Exercise 1: Tamilnadu Board-2018

If $w=x+2y+z^2$ and $x=\cos t; y=\sin t; z=t$ find $\frac{dw}{dt}$ by using chain rule. Also find $\frac{dw}{dt}$ by substitution of $x, y$ and $z$ in $w$ and hence verify the result.

If $f\left(1\right)=10$ and $f^{\text{'}}\left(x\right)\ge 2$ for $1\le x\le 4$ how small can $f\left(4\right)$ possibly be?

Let $P$ be a point on the curve $y=x^3$ and suppose that the tangent line at $P$ intersects the curve again at $Q$. Prove that the slope at $Q$ is four times the slope at $P$.

Trace the curve $y=x^3$.

Find the length of the curve ${\left(\frac{x}{a}\right)}^{\frac{2}{3}}+{\left(\frac{y}{b}\right)}^{\frac{2}{3}}=1$.

Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius $\text{'}a\text{'}$ is $\frac{8}{27}$ (volume of the sphere).

Which one of the following statements is true about the curve $y=x^{\frac{1}{3}}$?

Show that the function $f\left(x\right)={\tan }^{-1}\left(\sin x+\cos x\right), x>0$ is strictly increasing in the interval $\left(0, \frac{\mathrm{\pi }}{4}\right)$.