- Topics At a Glance
- Derivative as a Limit of Slopes
- Slope of a Line Between Two Points
- Slope of a Line Between Two Points on a Function
- Slope at One Point?
- Estimating Derivatives Given the Formula
- Estimating Derivatives from Tables
- Finding Derivatives Using Formulas
**Derivatives as an Instantaneous Rate of Change****Average Rate of Change**- Instantaneous Rate of Change
- Units, Words, and Notation
- Tangent Lines
- How Tangent Lines Look
- When Tangent Lines Don't Exist
- Tangent Lines and Derivatives
- Tangent Line Approximation
- Finding Tangent Lines
- Using Tangent Lines to Approximate Function Values
- Differentiability and Continuity
- The Derivative Function
- Graphs of
*f*(*x*) and*f*' (*x*) - Theorems
- Rolle's Theorem
- The Mean Value Theorem
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
- Appendix: Speed v. Velocity

A **rate** is a value that expresses how one quantity changes with respect to another quantity. For example, a rate in "miles per hour" expresses the increase in distance with respect to the number of hours we've been driving.

If we drive at a constant rate, the distance we travel is equal to the rate at which we travel multiplied by time:

Dividing both sides by time, we have

If you drive at 50 mph for two hours, the distance we'll travel is 50 mph × 2 hrs = 100 miles.

If we drive at a constant speed for 3 hours and travel 180 miles, we must have been driving

.

In real life, though, we don't drive at a constant rate. When we start our trip through Shmoopville, we first climb into the car, traveling at a whopping 0 miles per hour. We speed up gradually (hopefully), maybe need to slow down and speed up again for traffic lights, and finally slow down back to a speed of 0 when reaching our destination, The Candy Stand. We can still divide the distance we travel by the time it takes for the trip, but now we'll find our **average rate**:

To calculate the average rate of change of a dependent variable *y* with respect to the independent variable *x* on a particular interval, we need to know

- the size of the interval for the independent variable, and

- the change in the dependent variable from the beginning to the end of the interval.

Depending on the problem, we may also need to know

- the units of the independent and dependent variables.

Then we can find

The average rate of change of *y* with respect to *x* is the slope of the secant line between the starting and ending points of the interval:

Relating this to the more math-y approach, think of the dependent variable as a function *f* of the independent variable *x*. Let *h* be the size of the interval for *x*:

and let *a* be one endpoint of the interval, so the endpoints are *a* and *a* + *h*, with corresponding *y*-values *f*(*a*) and *f*(*a* +* h*):

Then the slope of the secant line is

This is the definition of the slope of the secant line from (*a*,* f*(*a*)) to (*a* + *h*, *f*(*a* + *h*)).

Example 1

According to Google Maps, it's 241 miles from Ann Arbor to Chicago. If Kevin made this trip in 4 hours, what was his average rate of travel during the trip? |

Example 2

Liana was 10 miles from home at 2 pm and 70 miles from home at 3:15 pm. What was her average rate of travel during the trip? |

Example 3

A famous author signed 200 books in two and a half hours. Find the average rate of change of the number of books signed with respect to the number of hours elapsed. |

Example 4

Thomas went mountain climbing and took some trail mix. After climbing 200 feet he had eaten three handfuls of trail mix. On average, how many handfuls of trail mix did he consume per foot? |

Example 5

Veronica has a tendency to forget about her cookies and leave them in the oven until the smoke alarm goes off. If she bakes her cookies at 300°F, it takes three-quarters of an hour for the smoke alarm to go off. If she bakes her cookies at 450°F (for those friends who like them...crispy), it only takes 10 minutes for the smoke alarm to go off. What is the average rate of change of time until the smoke alarm goes off with respect to oven temperature (in degrees Fahrenheit)? |

Exercise 1

Peggy drove from New York to Boston (a distance of 219 miles) in 4 hours. What was her average rate of travel?

Exercise 2

Lou was 20 miles from home at 1 pm and 120 miles from home at 3 pm. What was Lou's average rate of travel during that time?

Exercise 3

The graph shows John's starting and ending distances from home on a recent trip. Calculate his average rate of travel during the trip.

Exercise 4

The graph shows Suhaila's position during a trip. Calculate Suhaila's average rate of travel between 2 pm and 2:30 pm.

Exercise 5

Roger likes to eat jellybeans. One day he ate 56 jellybeans in 7 minutes. On average, how many jellybeans did he eat per minute?

Exercise 6

- When there are two mome raths in the forest, there are three borogoves. When there are twenty mome raths in the forest, there are 15 borogoves. What is the average rate of change of borogoves with respect to mome raths?

Exercise 7

An army is using catapults against their enemy. They go through the catapults one at a time, tossing them aside when they break. After the first catapult breaks they've thrown 20 stones. After the 19th catapultbreaks, they've thrown 220 stones. On average, how many stones does each catapult throw?

Exercise 8

Alice went to Wonderland and visited a succession of tea parties given by the Mad Hatter. The number of dormice at the tea parties changed depending on the number of teapots laid out. A party with 3 teapots would have only one dormouse, but a party with 14 teapots would have 24 dormice. What was the average rate of change of dormice with respect to teapots?