Application of Derivatives Set-6

Q1. Use linear approximation (first-order Taylor expansion) of $f(x)=\sqrt{x}$ about $x=49$ to estimate $\sqrt{50}$.

Q2. Water is draining from an inverted right circular cone of height $10\,$m and base radius $5\,$m. If the volume of water is decreasing at the rate $2\ \text{m}^3/\text{min}$, how fast is the water level falling when the depth of water is $6\,$m? (Take the outflow rate as $-2\ \text{m}^3/\text{min}$.)

Q3. Find the equations of the tangents to the curve $y=x^{3}-6x^{2}+9x$ which are parallel to the line $y=3x+1$.

Q4. Let $f$ be a real function for which derivatives up to order three may or may not exist at a point $x_{0}$. Consider the statements:

A: There exists a function $f$ such that $f'(x_{0})=f''(x_{0})=0$ and $x_{0}$ is a point of inflection.

R: For any function $f$, if $f'(x_{0})=f''(x_{0})=0$ then $f'''(x_{0})\neq 0$.

Q5. Define f(x)=\begin{cases}x^{2}\sin\!\left(\dfrac{1}{x}\right), & x\neq 0,\$$4pt]0, & x=0.\end{cases} Which one of the following is true about $f$ at $x=0$?

Q6. Find the points on the curve $y=x^3-3x^2+2$ at which the tangent is horizontal.

Q7. For the curve $y=e^x$, the tangent at a point $(a,e^a)$ meets the coordinate axes and forms a triangle. The maximum possible area of such a triangle is

Q8. An open-top rectangular box has a square base of side $x$ (m) and height $h$ (m). If its volume is $32\ \text{m}^3$, the value of $x$ that minimizes the total surface area (base + four sides) is

Q9. Assertion (A): For any real constants $a,b$, the equation $e^x = ax + b$ has at most two real roots.
Reason (R): If $g(x)=e^x-ax-b$ then $g''(x)=e^x>0$ for all $x$, so $g'$ is strictly increasing and hence can have at most one real zero; by Rolle's theorem this implies $g$ can have at most two real zeros.

Q10. Assertion (A): If $f$ is differentiable on $\mathbb{R}$ and $|f'(x)|\le M$ for all $x\in\mathbb{R}$ (for some constant $M$), then $f$ is uniformly continuous on $\mathbb{R}$.
Reason (R): By the Mean Value Theorem, for any $x,y\in\mathbb{R}$ there exists $c$ between $x$ and $y$ with $|f(x)-f(y)|=|f'(c)||x-y|\le M|x-y|$, so $f$ is Lipschitz and hence uniformly continuous.

Q11. Using the linear approximation of $f(x)=\sqrt{x}$ about $x=4$, estimate $\sqrt{4.1}$.

Q12. A rectangular sheet of metal of size $20\ \text{cm}\times 12\ \text{cm}$ is made into an open-top box by cutting congruent squares of side $x$ from the corners and folding up the sides. For $0<x<6$, the value of $x$ that maximizes the volume is

Q13. For the function $y=xe^{-x^{2}}$, the maximum value for $x>0$ is

Q14. Assertion (A): For $f(x)=\lvert x^{3}-3x\rvert$ the point $x=0$ is a stationary point of $f$.
Reason (R): A stationary point of a function (when defined in the usual sense) is a point where the derivative equals $0$.

Q15. Points on the parabola $y^{2}=4x$ are considered. How many points on this parabola are closest to the point $(3,0)$?