Math 130-03, Lab 10

Exercise 1

Here are a few more derivative problems. Please find the derivatives, showing each step in your computation. For part (d), use the fact that $ (2+\sin(x))=e^{\ln(2+\sin(x))}$.

$\displaystyle \textbf{(a)\ \ }$ $\displaystyle \displaystyle \frac{d}{dx}\,2^x\log_{10}(x)$ $\displaystyle \textbf{(b)\ \ }$ $\displaystyle \displaystyle \frac{d}{dt}\,5^{\sec(\sqrt{t\,})}$    
$\displaystyle \noalign{\medskip } \textbf{(c)\ \ }$ $\displaystyle \displaystyle \frac{d}{dx}\,x2^x+x^23^{x^2}+x^34^{x^3}$ $\displaystyle \textbf{(d)\ \ }$ $\displaystyle \displaystyle \frac{d}{dx}\,\big(2+\sin(x)\big)^x$    

Exercise 2

This exercise tries to connect related rates with geometry. If a quantity changes with time, then the derivative with respect to time is the rate at which the quantity changes. When two or more quantities are related, then the rates at which they change are related. Often, this relationship reflects the geometry of the situation in an interesting way. For example, if the radius of a circle is changing with time, then so is its area. The radius and area are related by the equation $ A=\pi r^2$. If you take the derivative of both sides of this equation with respect to time, you get: $ \frac{dA}{dt}=2\pi r \frac{dr}{dt}$. Now, the interesting thing is that $ 2\pi r$ is the circumference of the circle. This is no accident. Think about it this way: If you change the radius by a small amount, $ \Delta t$, then a small strip is added to the area around the edge of the circle. The width of this strip is $ \Delta t$ and its length is the circumference of the circle, $ 2\pi r$. So, the rate at which the area changes is $ 2\pi r$ times the rate at which the radius changes.

(a) The volume of a cube of side $ s$ is given by $ V=s^3$. Suppose that $ V$ and $ s$ are changing with time. Differentiate both sides of the equation with respect to $ t$. Explain the answer in terms of geometry. Hint: What is added on to the cube when you increase $ s$ by a small amount?

(b) The volume of a sphere of radius $ r$ is given by $ V=\frac{4}{3}\pi r^3$. As we saw in class, differentiating this equation gives $ \frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$. What do you think the $ 4\pi r^2$ represents? Why?

(c) The area of a rectangle with width $ w$ and length $ l$ is given by $ A=lw$. Differentiate both sides of this equation with respect to $ t$, and explain the result geometrically. This exercise should make the product rule more believable. Why?

(d) The volume of a right circular cylinder of radius $ r$ and height $ h$ is given by $ V=\pi r^2h$. (This formula represents the area of the base times the height.) Differentiate both sides of this equation with respect to $ t$, and explain the result geometrically.

Exercise 3

Here is another of those standard related rates problems. Write out a careful solution to this problem, illustrating your solution with a diagram and explaining your work in full sentences and paragraphs. The answer to this problem is about 16.846 miles/hour (I think).

At 8:00 AM, a ship sails due west from a certain point in the ocean. It travels at a constant rate of 12 miles/hour. At 10:00 AM, another ship sails due south from the same point and travels at a constant 15 miles per hour. At what rate is the distance between the ships changing at 11:00 AM?

David Eck, April 2001