7.1.1If I speed up, will you slow down?

Using a closed loop of yarn, have two people in your team make a rectangle by holding two corners as shown in the picture at right. Start with your fingers together so that the length, $l$, is close to zero. Move your hands apart at a constant rate.

1. As the length increases at a constant rate, how does the width change?

2. If $\frac { d w } { d t }$ is the rate the width is changing and $\frac { d l } { d t }$ is the rate the length is changing, write an equation relating these two rates.

3. As the length increases, how does the perimeter, $P$, change? What is $\frac { d P } { d t }$?  

4. As the length increases, what happens to the area? Is the rate of change for the area a constant?
   
   Changes in the length and width also affect other measurements, such as the area, perimeter, and the length of the diagonals. These rates are dependent, since they depend on the rate of the changing dimensions. Because of this relationship, we call them related rates.

Translate each of the expressions below into a complete sentence. Determine if each rate is positive, negative, zero, or some combination. Then determine is the rate is constant or not. For example, $\frac { d a } { d t }$ represents the rate at which your age changes over time (you can also say “compared to time,” or “with respect to time” or “as time changes”). 

In each expression below:

The amount of gas a car uses each hour depends on how fast the car is traveling. If $x$ represents the distance a car travels and $g$ represents the number of gallons of gas the car has consumed, analyze these related rates.

1. What do $\frac { d x } { d t }$ and $\frac { d g } { d t }$ represent?

2. Assume a car uses $12$ gallons of gas to travel $360$ miles. What is $\frac { d g } { d t }$ when the car is traveling $60$ mph? $30$ mph? $10$ mph

3. Notice that $\frac { d x } { d t }$ varies directly with$\frac { d g } { d t }$. Write an equation relating $\frac { d x } { d t }$ and $\frac { d g } { d t }$ for the car described in part (b) above.

Consider a coffee cup shaped like the one shown at right. Suppose coffee is poured into the cup at a steady rate

1. Sketch a graph that shows $h\left(t\right)$, the height of coffee in the cup after $t$ seconds. Is $\frac { d h } { d t }$ positive, negative, or neither? Does $\frac { d h } { d t }$ change at a constant rate? Explain how your answer relates to the shape of the cup. 

2. Does volume of coffee in the cup vary directly with time? Sketch a graph that shows $V\left(t\right)$, the volume of coffee after $t$ seconds. Is $\frac { d V } { d t }$ positive, negative, or neither? Does $\frac { d V } { d t }$ change at a constant rate? Explain. 

3. For the coffee cup, how is $\frac { d h } { d t }$ related to $\frac { d V } { d t }$? Discuss this with your team and write a complete description.   
   
   These last few problems all focused on rates of change. Often, when something changes, several other measures change accordingly, causing their rates to be related. We call these related rates.

RELATED RATES

When two rates are related, we can describe their relationship with a related rate statement. It is important to remember that related rates are not always proportional. Consider the situations below.

The sketch at right shows a man pulling at a rope that is tethered to the bow of a boat. He pulls at a slow and steady rate and as he pulls, the boat moves toward the dock.

1. Make a prediction. Will the boat approach the dock at a constant rate? If you have never been on a boat, talk to someone who has before stating your prediction.

2. What do $x$ and $d$ represent? What do $\frac { d x } { d t }$ and $\frac { d d } { d t }$ represent? Explain.

3. Are $\frac { d x } { d t }$ and $\frac { d d } { d t }$ each positive or negative? Explain.

4. Do you think that $\frac { d x } { d t }$ and $\frac { d d } { d t }$ each are increasing, decreasing, or constant? Explain.

5. Are the related rates in this situation directly proportional? Explain.

Differentiate each equation with respect to $x$. Leave your answers in terms of only $y$ and $x$. Homework Help ✎

5. What is $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for the function in part (a)? Write your answer in terms of $x$ only.

6. Evaluate $\frac { d ^ { 2 } y } { d x ^ { 2 } } | x = 3$ for the function in part (a).

7. Write the equation of the tangent line at $x = 3$ for function in part (a).

8. If the tangent line is used to approximate the function at $x = 3.01$, will it give an underestimate or an overestimate? Use part (f) to determine your answer.

Integrate. Homework Help ✎

For each function below, calculate the average value over the given interval and state the value of $t$ such that $g\left(t\right)$ equals the average value. Homework Help ✎

Sketch the region bounded by $y = x^4$ and $y = 1$. Calculate the exact area of the region. Homework Help ✎

Find two positive numbers whose sum is $8$, such that the sum of the square of one number and the cube of the other number is a minimum. Use calculus to justify your solution. Homework Help ✎