Instantaneous Rate of Change as Slope of Tangent

Q.

If $V=\frac{4}{3}\pi r^3$, at what rate in cubic units is V increasing when r = 10 and $\frac{dr}{dt}=0.01$ ?

1. $\pi$

2. $4\pi$

3. $40\pi$

4. $4\frac{\pi }{3}$

Q.

A ladder20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

1. $\frac{1}{3}$

2. $\frac{2}{3}$

3. $\frac{4}{3}$

4. $\frac{5}{3}$

Q. If the curves $y=\frac{\ln x}{x}$ and $y=\lambda x^2$ (where $\lambda$ constant) touch each other, then $3e\lambda$ is

1. $e$
2. $e^2$
3. 1
4. 9