The Intermediate PR Theorem

Today in class I talked about the Intermediate Value Theorem, so this is a good opportunity for me to talk about my favorite application thereof.

Imagine that you are racing an 800, and your coach tells you to run 80 seconds per lap. Well, you get all over-excited for the first lap and run a 70, and then you are tired and run a 90 for the second lap. Afterwards, your coach says, "that was the worst-paced 800 I've ever seen! Was there even a single lap in there that you did in 80 seconds?"

In fact, there is! Since you ran the first lap in 70 and the next lap in 90, there must have been some 400-meter stretch which you covered in exactly 80 seconds. Maybe it was from 200 to 600 meters, something like that.

To prove this with the Intermediate Value Theorem in a way that will generalize to my next example, let's do something slightly different. Instead of timing you for different 400-meter stretches and finding one where you ran it in 80 seconds, we'll look at different 80-second time periods and try to find one where you covered exactly 400 meters. We can do this using the following picture:


Now we'll make a function f(t) = the amount of distance covered between time=t and time=t+80. In mathematical terms, if your speed is some function v(t):



Then f(0) is the distance covered in the first 80 seconds. This is going to be more than 400 meters, since it took you only 70 seconds to cover the first lap:



Similarly, f(80) is the distance covered in the last 80 seconds (from 80 seconds to the end -- 80 + 80 = 160 seconds = 2:40), which is less than 400 meters, since it took 90 seconds to cover the last 400 meters:



And for some time in the middle, the distance covered is exactly 400 seconds. Maybe it's from 29 to 109 seconds, so f(29) = 400 meters:



Now here is a picture of f(t) itself. It starts out greater than 400 and ends up less than 400, so the Intermediate Value Theorem says that it must be exactly 400 at some point in the middle.



Okay, now for the fun part. I run a lot of races and I keep careful track of all of my personal records (PRs). Sometimes, I run a personal best for a shorter distance during a race of a longer distance -- say, a 3-mile PR while I'm running a 5k. In that case, the course is usually marked at the 3-mile point, and I can look at my watch and see what my time was. However, sometimes I want to just multiply my final time by a suitable factor.

Last weekend, I went through the first two miles of a 5k in 11:09. I converted this to a 3k time by multiplying by 1.864/2, and got 10:24 for the 3k distance. This is extremely close to my PR of 10:22. I'd like to be able to say, "I ran a 10:24 3k during my 5k!" But is this really true? The Intermediate Value Theorem says yes!

To see this, we set up a function as above, f(t) = the distance covered in the 10:24 starting at time=t. If I cover exactly 3k in the first 10:24 of the race, i.e. if f(0) = 3000m, then we're good! If not, then either it's more or less. Let's say f(0) > 3000m, so I was even faster than average at the beginning! No problem. But then there must have been a time in the last bit of the 2-mile period when I was slower than the average, so by inching t along, we can find a place where f(t) = 3000m exactly. Similarly, if I was slower at the beginning and f(0) < 3000, then there must have been a time at the end when I was faster than average, so we can inch t forward until f(t) = 3000m exactly.

So now you know: I lied to my class today when I told them my favorite application of the IVT was the "temperature at antipodal points" example. My real favorite example is this application to running times.

Downtown 5k

(Results) My time in this race was not as fast as I had hoped, but my place was higher than I would have dreamed! Before the race, I hoped I might be able to place in the top 25. However, I ended up placing 16th, 13th American. Last year, the 16th female ran 16:33! Who knows why many fewer elite women showed up this year, but I'll take it.

The morning was cold, so I wore warm-up pants and a jacket to walk down to the race area. I visualized my 18-minute race. My plan was to run with Joan Benoit Samuelson. I did my warm-up run in my long pants, and by the time I got back my leg muscles were thoroughly warmed up, as was the day itself. It turned out to be much warmer than I expected.

I lined up about 5 rows from the front. I was happy to meet my friend Jenn on the starting line. When I told her of recent PR and my plan to run with Joan, she said that I would probably be faster than Joan. She also told me that she herself would be lucky to run under 18 minutes. So I revised my plan and decided to try to run with Jenn. Also, I couldn't find Joan on the starting line, even though I knew what she was wearing. Only much later, when I watched the race video, did I realize that there were two parallel starting lines: Joan was on the other side of a barrier from me, so I couldn't see her.

The gun went off and I went out at a fast clip. I saw Jenn zip past me on the left and I caught up to her. A pudgy woman passed me on the downhill in the first 1/10 of a mile and called out happily to her friend. Whatever! Jenn was going quite fast and passing people left and right. I kept up with her for a while, but after about 1/4 mile I realized that she was going too fast for me, so I let her go. I also couldn't see Joan, so I had to just run on my own.


Here is a picture of me and Henry in the first mile. I should not have been ahead of Henry -- he finished in 17:28. Photo by Scott Mason.

I passed the mile in 5:28. I had expected the first mile to be fast, and it was! Just then, Joan came zipping by on my left. I caught up to her and ran behind her for a few paces, but I was unable to match her speed, and I watched her open a gap ahead of me as we ran past the Wild Colonial. In the middle of the third mile, Alan was on the side of the course and told me I was in 15th place. I was very surprised to be at such a high position! I wondered if a horde of women was going to pass me in the second half of the race, since I had gone out too fast.

I tried to keep up a good pace for the second mile, but it was very difficult. I felt that I had gone too fast, and my legs had no pep left. So I tried to keep even with the men around me, though some men passed me. I got to the 2-mile mark in 11:28, for a 6:00 second mile, and 19 seconds slower than my 2-mile time last week. Darn. In the third mile, one woman passed me, and I had no response. I just wanted to maintain my position as well as possible, and I was very fortunate that the hordes of women I feared would pass me never materialized.


Here I am at about 2.8 miles, before turning to go up the ramp.

I was very happy to see the approaches to the mall and know that the race was almost over. I hit the 3-mile clock in 17:26, so I thought that I would finish around 18:00, since last week I discovered that it is about 33 seconds to the finish from 3 miles. So I was unhappy to see the clock saying 18:04, 18:05 when I was not even close yet! I crossed the line in about 18:10. Kevin told me later that the race winner (Molly) had taken about 42 seconds from 3 miles to the end, so my 44 seconds was not too bad.

Given the hot conditions, the many turns on the course and the fact that I went out too fast, 18:10 was a fine time. I was not disappointed. After all, if I hadn't run last week, it would be an 11-second PR. Plus, to finish 16th at the national championships was beyond my wildest dreams! (It all depends on who shows up...)

In the past week I have joined New Balance Boston, so I went to chat with the team after the race. I met some of the other runners on the team, who were all very nice. After a while we headed over to the awards ceremony, and we were happy to learn that the team had placed first! Three women's teams showed up, and only one men's team. Like Steve said, "90% of life is showing up." So we went up and received medals for this feat.


Here is a picture of our three runners. You can see that I am wearing the medal; the others took theirs off. I was very happy to be able to contribute to NBB's third straight team title.

There was cash and special medals for the top 10 Americans, so I was only three places out of such an honor. Crazy! I stayed until the very very bitter end of the awards ceremony, when only about 10 people were left, and Brown won the College Challenge competition, as well (the only college to field a full team). So it was a day for lots of winning.

JCCRI 5k

(Results) I cannot say enough good things about this race. It was great! It was super! At various times today after the race, people asked me, "so, how was the race?" and I could not help but say, "it was awesome! It was so great!" because truly, it was.

I prepared for this race. I made a plan, and I visualized the race for an entire 18 minutes beforehand, imagining all the turns and the vistas and the times and the thoughts.

My goal was to break 18:00. Last fall, my goal was to break 19:00, and I did, with an 18:58 at the Downtown 5k in September. I improved it to 18:26 in October, and all winter I tried to break 18:00. I even gave up ice cream for about six weeks preceding the USATF-NE indoor track championships, where I finished in a disappointing 18:29. In March I ran 18:24 at St. Pats, chipping two seconds from my PR. In April I ran 18:21 at the Scott Carlson 5k, chipping off another three seconds. Such tiny improvements were frustrating. This summer I ran a lot, and at the end of the summer my comfortable training pace suddenly dropped from 7:45 to 7:15. This told me that I was probably ready for a big improvement in the fall racing season, so I set my goal for this race at sub-18.

Race plan: Run with Katie. Run on her shoulder, and don't let her get away. If she gets away, reel her back in. Stick like glue to Katie as long as possible.

Time plan: Aim for 5:40 for the first mile and under 11:30 for two miles. Try to get to three miles before 17:30, because 30 seconds for the last 0.11 is really quite fast. The first mile has a downhill and the second and third miles both have slight uphills.

The course is quite nice. Basically, you leave the JCC, go to Blackstone Boulevard and run a ways down it, turn around close to the end (at Lloyd Street), and come about a mile up it, and then come back all the way on Elmgrove.

I lined up on the starting line. Steph and Katie were both there, so I expected to get third. Someone sang the national anthem and then I thought we would start, but after the national anthem someone sang what sounded like a Hebrew song of some kind. Mayor Cicilline blew the air horn and we were off. My reaction time was a little faster than most other people's, because I was looking directly at the Mayor instead of listening for the horn. Steph took off in the lead pack with the guys and I stayed at a reasonable pace with Katie. I glanced at my watch at the 3-mile line and saw that it was 33 seconds from there to the start/finish line, so I would have to get there in about 17:26 or faster. We turned a corner at 1/4 mile and Dave S. was right with us. Before the 1/2 mile turn onto Blackstone, we passed him. I have never beaten Dave before, so I confess I was a little excited to be ahead of him in a race.

The first stretch of Blackstone was perfectly straight. Katie and I tucked in behind three guys. I had a sense that she wanted to get around them, but it was impossible because we were running in the bike lane and there were moving cars on the left and parked cars on the right, so we couldn't go anywhere. It was a good thing we didn't, too, because as it was we passed the mile in 5:32. Alan had said to go through the mile no faster than 5:40, but I was sticking to Katie no matter what happened.

Soon we got to the turnaround point. This broke up our little group and we were able to get past two of the guys. Just before we did, a spectator shouted something like, "that's right, get those guys!" and then a few seconds later when we passed them (two of them seemed to be younger, and friends) one of the guys groaned, "oh no..." because he was passed by a female. Actually, two females. (He probably didn't see Steph way up ahead.)

Now we had to run up the gradual hill of Blackstone Boulevard. Katie got a little ahead of me, and I surged for a few strides to catch up again. Then she got a little ahead of me again, and I surged to catch her again. This probably happened five times. I was happy to see that it wasn't too hard to catch back up each time; the hard part was just to stay even and not fall behind.

At this point, there was a guy with long hair (not tied back, just loose) running with us. Every time I surged to catch back up to Katie, he appeared to take it personally, as he put in a big effort to get back in front of me. I tried to ignore his antics, but it was very distracting. After the third time I was getting fed up. I surged to get away from him, but again he got right back in front of me. I was not interested in playing games; I had a race to run! "I'm not racing you," I said. "That's fine," he shouted back. The next time I surged ahead, I was able to leave him behind.

By this time Katie had gotten a bit ahead of me, and we were coming up to the two-mile mark. I passed two miles in 11:11, for a 5:39 second mile. Wow! I was only hoping for 11:30, and here I was 19 seconds under my goal. Pretty exciting! I kept keying off of Katie for the third mile, as we made a sharp turn onto Elmgrove back towards the finish line. I could easily see her ahead of me, and I tried to keep the distance between us constant. I focused on increasing my turnover and speeding up whenever I felt that I was faltering.

Finally I saw cones and flashing lights up ahead. Was it really so soon? I glanced at my watch at the line marked 400/800 (for the children's races) and it said 16:11. With a quarter of a mile to go, I was almost certain to break 18:00!

I kicked hard from there. At the 3-mile mark, where I had hoped to sneak in under 17:30, I was thrilled to see a 17:09. Wow, I was really going to do it! This was a great race! I kicked kicked kicked and watched the clock tick 17:38, 17:39, and then as I crossed the line, the announcer said my name and "17:40." Perfect!

What does 17:40 mean to me?
1. As I said above, it is super! awesome! great!
2. It starts with "17:" which is quite amazing and wonderful.
3. It is the highest performance index I have ever run:

4. I broke 18:00, so I can try to join New Balance Boston!

Also, we had a Brown Running Club team, which was the second team overall and the first all-women's team. The prize is that we get our team name engraved on a plaque in the JCC. Cool! I will have to go there and look at it after it is engraved. So, thanks to Carmen and Emily for being a great team.

Also, there was a raffle afterwards. I nearly always win a prize in post-race raffles. I had three race numbers -- my teammates had to leave early -- and yet, when one of the numbers was called, it was mine. I won a case of beer and a yoga mat. Not bad!

Oh, and ALSO, I was expecting that since I was third, I would receive $50 (the prizes went $150-$100-$50) but it turns out Steph is in another age category, so I got $100 instead of $50! That was very nice. So, a good day all around.

Running on Deer Isle: Part 1

This fall, a series of four essays is running in the Island Ad-Vantages, the local newspaper for Deer Isle and Stonington, ME. After each of my essays appears in the paper, I will also post it here. I have added a photo to jazz up this blog coverage.

I’m a runner, and I’ve been running on Deer Isle for 15 years. It’s a beautiful place to run, and I recently realized that there were a lot of places on the island where I had never been, so in the fall of 2009 I decided to run every road on Deer Isle. During the past year, each visit to Deer Isle has been an exploratory adventure: I pick out a road on the map, drive my car to the spot, and run down yet another collection of small dirt roads, not knowing what I will find at the end or along the way.

I first came to Deer Isle when I was two months old, and I first ran on the island the summer when I was ten, in preparation for joining the cross country team in fifth grade. For the next decade or so, my runs on the island were predictable: I would run out my front door and run out to Route 15 and back a few times, or maybe run to the end of the Sunset Crossroad and back.

Of course, I also ran the Fourth of July Fun Run in Stonington. Some of you may remember my epic duel with Wally Fifield back in 1996: I was 11; he was about 45; we shared the lead all along Cemetery Road and Route 15, until I made a wrong turn at the harbor. Wally got a little ahead of me, and though I sprinted to catch back up, he was able to just out-lean me at the finish line. I ran the Fun Run for four years in a row in the late ‘90s, and now I run the Stonington Six every year.

Once, just before I left the island for my senior year of college, I “ran around the island,” completing a 17-mile loop of Route 15, 15A, and Sand Beach Road. Three years later, I decided to run all the roads on the island. I printed a map of Deer Isle, and highlighted all the roads I had run. Most of the main roads were done, but clearly visible on the map were the hundreds of roads I knew nothing about, like little fingers stretching from the main roads towards the ocean.

That summer, anyone who ran with me joined me on a new road. On Fourth of July weekend, my brother and his wife joined me on a 9-mile run looping around Dunham Point Road and Pressey Village Road. On Labor Day, my boyfriend joined me on a 7-mile run of the Oceanville Road. These are beautiful places! My sister-in-law loved how the Dunham Point Road repeatedly brought us to beautiful views of the water, with the Sylvester’s Cove beach as well as some little private inlets. Why had none of us ever run out there before? We didn’t know what we were missing!

Last winter, I started running the little roads in earnest. I would look at the map of the island, find a main road with a lot of little roads off it, and then set off to run them, highlighting the roads on my map when I was done. The Sunday before Christmas, I ran home from church in a blizzard, and ran all of the little roads off of 15A on the way. On Christmas day, a mild, dry day without much snow on the ground, my boyfriend and I ran the eastern half of the Reach Road, 13 miles of little dirt roads stretching towards the Eggemoggin Reach. A few days later, we ran the Dow Road in the pouring rain, a place where I had never even driven.


Here I am after running home the Sunday after Christmas in a blizzard.

I doubt that anyone has ever run all of the roads on Deer Isle; it just seems too unusual. Some people may have been to all the roads on Deer Isle – especially Hubert Billings, the island’s code enforcement officer. But many people haven’t seen every road, so I would like to share with you my experiences running everywhere on the island, since you have likely never been to some of the places I’ve visited. Over the next few weeks, I will write several more columns about the strange and beautiful things I’ve seen, and I hope you enjoy them nearly as much as I have.