Application of Derivatives Set-4

Q1. For the curve $y = x^3 - 6x^2 + 9x + 2$, the tangents parallel to the line $y = 3x + 5$ occur at the $x$-coordinates:

Q2. A ladder of length $13\ \text{m}$ rests against a vertical wall. The foot of the ladder is pulled away from the wall at $0.6\ \text{m/s}$. At the instant when the foot is $5\ \text{m}$ from the wall, the top of the ladder is sliding down the wall at the rate:

Q3. From a $30\ \text{cm} \times 20\ \text{cm}$ rectangular sheet, squares of side $x\ \text{cm}$ are cut from each corner and the flaps are folded to form an open-top box. The value of $x$ that maximizes the volume of the box is:

Q4. Assertion (A): The function $f$ defined by $f(x)=x^2\sin\frac{1}{x}$ for $x\ne0$ and $f(0)=0$ is differentiable for all real $x$ and $f'(0)=0$.
Reason (R): For $x\ne0$, $f'(x)=2x\sin\frac{1}{x}-\cos\frac{1}{x}$ and $\lim_{x\to0}f'(x)$ does not exist, hence $f'$ is not continuous at $0$.

Q5. A right circular cone has height $12\ \text{cm}$ and base radius $6\ \text{cm}$. A right circular cylinder is inscribed in the cone with its axis coincident with the cone's axis. The values of radius $r$ and height $h$ of the cylinder that maximize its volume are:

Q6. Using linear approximation (differential), estimate $\sqrt{4.1}$ by linearizing $f(x)=\sqrt{x}$ at $x=4$.

Q7. A closed cylindrical can (with top and bottom) is to have volume $2000\pi\ \text{cm}^3$. Find the radius $r$ (in cm) that minimizes the total surface area.

Q8. Let $f(x)=x\sqrt{1-x}$ for $x\in[0,1]$. The maximum value of $f(x)$ on $[0,1]$ is

Q9. Assertion (A): If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with $f'(x)\neq 0$ for all $x\in(a,b)$, then $f$ is either strictly increasing or strictly decreasing on $[a,b]$.
Reason (R): Since $f'(x)\neq 0$ on $(a,b)$, $f'$ is continuous on $(a,b)$ and therefore cannot change sign on $(a,b)$.

Q10. For the curve $y^2=x^3-x$, the number of points at which the tangent is horizontal is

Q11. Use the linear approximation of $f(x)=\sqrt{x}$ at $x=4$ to estimate $\sqrt{4.1}$.

Q12. An open-top box with a square base has volume $32\ \text{cm}^3$. Find the side of the base and the height that minimize the total surface area (give dimensions as base side $\times$ height).

Q13. Water is poured into an inverted right circular cone of height $12\ \text{cm}$ and base radius $6\ \text{cm}$ at the rate $100\ \text{cm}^3/\text{min}$. How fast is the water level rising when the depth of water is $8\ \text{cm}$?

Q14. Evaluate the limit using appropriate applications of derivatives (L'Hospital/Taylor reasoning): $\displaystyle \lim_{x\to 0}\frac{\sin x - x + \dfrac{x^3}{6}}{x^5}$.

Q15. Assertion (A) and Reason (R):
A: $x=1$ is not a point of inflection of the function $f(x)=x^4-4x^3+6x^2$.
R: $f''(1)=0$.