## Part I ### Problem 1 (This question has already been solved in the lecture) A circular balloon has radius \ $r$ cm and volume $$V=\frac{4}{3}\pi r^3$$ Its radius is increasing at \ $0.50$ cm/s. Find \ $\frac{dV}{dt}$ when \ $r=6$ cm. ### Problem 2 (This question has already been solved in the lecture) A 10 m ladder rests against a vertical wall. The foot of the ladder is pulled away from the wall at 1.2 m/s. How fast is the top of the ladder moving when the foot is 6 m from the wall? ### Problem 3 The sides \ $x$ and \ $y$ of a rectangle change with time. At one instant, $$x=8\text{ cm},\quad y=5\text{ cm}$$ $$\frac{dx}{dt}=0.7\text{ cm/s},\quad \frac{dy}{dt}=-0.2\text{ cm/s}$$ Find the rate of change of the area and state whether the area is increasing or decreasing. ### Problem 4 Two cars are approaching the same roundabout on perpendicular roads. Let \ $x$ and \ $y$ be their distances from the roundabout. At one instant, Car A is 0.6 km from the roundabout and moving toward it at 40 km/h, while Car B is 0.8 km from the roundabout and moving toward it at 30 km/h. How fast is the straight-line distance between the cars changing at that instant? ### Problem 5 For a differentiable function \ $f$, the linearization at \ $x=a$ is $$L(x)=f(a)+f'(a)(x-a)$$ (a) Find the linearization of \ $f(x)=\sqrt{x+5}$ at \ $a=4$. Then use it to approximate \ $\sqrt{9.2}$. (b) Find the linearization of \ $f(x)=\cos x$ at \ $a=0$. Use it to estimate \ $\cos(0.06)$. (c) Find the linearization of \ $f(x)=e^x$ at \ $a=0$. Use it to estimate \ $e^{0.03}$. ### Problem 6 (a) For \ $y=x^3-2x+1$ at \ $x=2$, calculate \ $\Delta y$ and \ $dy$ when \ $dx=0.05$. Compare the two values. (b) For \ $y=x^3+2x^2-5x+4$ at \ $x=1$, calculate \ $\Delta y$ and \ $dy$ when \ $dx=0.04$. Compare the two values.

Answer: $$\frac{dV}{dt} = 72\pi\text{ cm}^3\text{/s} \approx 226.19\text{ cm}^3\text{/s}$$

Step-by-step solution

Solve your own maths question free →