Tag Archives: calculus

The melting season for Arctic Sea Ice has started with a quick drop in ice. The total ice is at a record low for this time of year (orange line in chart). But, how this plays out throughout the melting seasons is hard to predict based solely on past seasons. For instance, 2012 is the year of the record low (dashed line), but numerous seasons have been lower than 2012 at this time of year (2016 – yellow, 2015 – green, 2007 – blue shown here).  Arctic Sea Ice extent is updated daily on the Charctic Interactive Sea Ice Graph by NSIDC. This graph allows the user to select years, download the image, and choose between Arctic and Antarctic ice extent. NSIDC posts the data and there is a project on both the Calculus and Statistics page using this data, as well as an interactive graph.

The Guardian article, Plummeting insect numbers ‘threaten collapse of nature’ by Damian Carrington (2/10/19) reports on the recent Biological Conservation paper Worldwide decline of the entomofauna: A review of its drivers by Francisco Sánchez-Bayo and Kris A.G.Wyckhuys (1/20/19).

More than 40% of insect species are declining and a third are endangered, the analysis found. The rate of extinction is eight times faster than that of mammals, birds and reptiles. The total mass of insects is falling by a precipitous 2.5% a year, according to the best data available, suggesting they could vanish within a century.

(Note percentage rate of change in the quote.) Why?

The analysis, published in the journal Biological Conservation, says intensive agriculture is the main driver of the declines, particularly the heavy use of pesticides. Urbanization and climate change are also significant factors.

So what?

One of the biggest impacts of insect loss is on the many birds, reptiles, amphibians and fish that eat insects. “If this food source is taken away, all these animals starve to death,” he said. Such cascading effects have already been seen in Puerto Rico, where a recent study revealed a 98% fall in ground insects over 35 years.

and

“If insect species losses cannot be halted, this will have catastrophic consequences for both the planet’s ecosystems and for the survival of mankind,” said Francisco Sánchez-Bayo

The Guardian article is a good QL resource. The paper has nice graphs and data but is behind a paywall.

The EIA Today in Energy report, EIA forecasts renewables will be fastest growing source of electricity generation (1/18/19), provides projections for electricity generation.

EIA expects non-hydroelectric renewable energy resources such as solar and wind will be the fastest growing source of U.S. electricity generation for at least the next two years. EIA’s January 2019 Short-Term Energy Outlook (STEO) forecasts that electricity generation from utility-scale solar generating units will grow by 10% in 2019 and by 17% in 2020. According to the January STEO, wind generation will grow by 12% and 14% during the next two years. EIA forecasts total U.S. electricity generation across all fuels will fall by 2% this year and then show very little growth in 2020.

The good news is more renewables, but “fastest growing” can be misleading. According to the chart (copied from the article) nonhydro renewables are projected to go from 10% in 2018 to 13% in 2020, and so their share of electricity generation is still small.  This is good good discussion for a calculus class or any QL based course.  The article includes two other charts and one is a complex bar chart that could be the focus of a class period.

The Royal Society research article, The broiler chicken as a signal of human reconfigured biosphere by Carys E. Bennett et. el. (12/12/18) , provides the evidence of how human intervention has changed a species.

Breeding by natural selection has been modified by human-directed selection. While the size of the domesticated chicken in historical times was little different to the red jungle fowl (figure 3), domestic chicken bone morphology shows that selective breeding practices took place as early as the sixteenth century [53,54]. Chickens from the late twentieth century are markedly different in terms of size (figures 3 and 4), growth rate (figure 5) and body shape. The change in body mass and body shape has been visually documented by photographs of broiler breeds throughout ontogeny from 1957, 1978 and 2005 [14]. Broilers from a 1957 breed are between one-fourth and one-fifth of the body weight of broilers from a twenty-first century breed [13,14]. The modern broiler is a distinctive new morphotype with a relatively wide body shape, a low centre of gravity [13] and multiple osteo-pathologies. If left to live to maturity, broilers are unlikely to survive. In one study, increasing their slaughter age from five weeks to nine weeks resulted in a sevenfold increase in mortality rate [55]: the rapid growth of leg and breast muscle tissue leads to a relative decrease in the size of other organs such as the heart and lungs, which restricts their function and thus longevity [56]. Changes in the centre of gravity of the body, reduced pelvic limb muscle mass and increased pectoral muscle mass cause poor locomotion and frequent lameness [15]. Unlike most other neobiota, this new broiler morphotype is shaped by, and unable to live without, intensive human intervention.

The article includes a number of interesting charts including the one copied here and in reference to the figure they refer to a derivative:

Chicken-meat consumption is growing faster than any other meat type and is soon to outpace pork

Data used in the paper is available under Supplemental Material (left side bar).

The recent BBC article Climate change: Oceans ‘soaking up more heat than estimated’  by Matt McGrath (11/1/18) reports the result from a recent study showing that the oceans have warmed more than previously thought.  How did they do it? In short, from the BBC article:

The key element is the fact that as waters get warmer they release more carbon dioxide and oxygen into the air.

“When the ocean warms, the amount of these gases that the ocean is able to hold goes down,” said Dr Resplandy.

“So what we measured was the amount lost by the oceans, and then we can calculate how much warming we need to explain that change in gases.”

The image here is copied from the original article in Nature, Quantification of ocean heat uptake from changes in atmospheric O₂ and CO₂ composition by Resplandy et. el (10/31/18) . The abstract to the paper provides a nice summary:

The ocean is the main source of thermal inertia in the climate system¹. During recent decades, ocean heat uptake has been quantified by using hydrographic temperature measurements and data from the Argo float program, which expanded its coverage after 2007^2,3. However, these estimates all use the same imperfect ocean dataset and share additional uncertainties resulting from sparse coverage, especially before 2007^4,5. Here we provide an independent estimate by using measurements of atmospheric oxygen (O₂) and carbon dioxide (CO₂)—levels of which increase as the ocean warms and releases gases—as a whole-ocean thermometer. We show that the ocean gained 1.33 ± 0.20  × 10²² joules of heat per year between 1991 and 2016, equivalent to a planetary energy imbalance of 0.83 ± 0.11 watts per square metre of Earth’s surface. We also find that the ocean-warming effect that led to the outgassing of O₂ and CO₂ can be isolated from the direct effects of anthropogenic emissions and CO₂ sinks. Our result—which relies on high-precision O₂ measurements dating back to 1991⁶—suggests that ocean warming is at the high end of previous estimates, with implications for policy-relevant measurements of the Earth response to climate change, such as climate sensitivity to greenhouse gases⁷ and the thermal component of sea-level rise⁸.

The paper has other interesting graphs that could be used in a QL based class. For a calculus class, the graph here is an example of the use of the Δx notation in the “real world”.

The Yale Environment 360 article Redrawing the Map: How the World’s Climate Zones Are Shifting  by Nicola Jones (10/23/18)  provides animated maps, such as the one below, and quantitative statements about changing ecology including rates (great for a calculus class):

Lauren Parker and John Abatzoglou of the University of Idaho tracked what would happen to hardiness zones from 2041 to 2070 under future global warming scenarios, and found the lines will continue to march northward at a “climate velocity” of 13.3 miles per decade.

One study in northern Canada found that the permafrost around James Bay had retreated 80 miles north over 50 years. Studies of ground temperatures in boreholes have also revealed frightening rates of change, says Schafer. “What we’re seeing is 20 meters down, it’s increasing as high as 1-2 degrees C per decade,” he says. “In the permafrost world that’s a really rapid change. Extremely rapid.”

North America is seeing the opposite phenomenon: Its arable land is romping northward, expanding the wheat belt into higher and higher latitudes. Scientists project it could go from about 55 degrees north today to as much as 65 degrees North — the latitude of Fairbanks, Alaska — by 2050. That’s about 160 miles per decade.

The article includes potential ramifications of these changes along with other quantitative information.

Graphic: Hardiness zones in the U.S., which track average low temperatures in winter, have all shifted northward by half a zone warmer since 1990. SOURCE: UNITED STATES DEPARTMENT OF AGRICULTURE. GRAPHIC BY KATIE PEEK.

NASA’s Vital Signs of the Planet feature,  Keeping score on Earth’s rising seas by Pat Brennan (9/1918) summarizes a recent paper that  “ ‘closes’ the sea-level budget to within 0.3 millimeters of sea-level rise per year since 1993.”

A just-published paper assembles virtually all the puzzle pieces – melting ice, warming and expanding waters, sinking coastlines and a stew of other factors – to arrive at a picture of remarkable precision. Since 1993, global sea level has been rising by an average 3.1 millimeters per year, with the rise accelerating by 0.1 millimeter per year, according to the study published Aug. 28 in the journal, “Earth System Science Data.”

“Global mean sea level is not rising linearly, as has been thought before,” said lead author Anny Cazenave of France’s Laboratory for Studies in Geophysics and Oceanography (LEGOS). “We now know it is clearly accelerating.”

The above paragraphs can be used as calculus in the news and sea level data is available from NASA’s Sea Level page.

A recent NYT article, Antarctica Is Melting Three Times as Fast as a Decade Ago by Kendra Pierre-Louis (6/13/2018), states clearly that Antarctica is melting, well, three times faster than a decade ago, which is a rate of change statement. Rapid melting should cause some concern since:

Between 60 and 90 percent of the world’s fresh water is frozen in the ice sheets of Antarctica, a continent roughly the size of the United States and Mexico combined. If all that ice melted, it would be enough to raise the world’s sea levels by roughly 200 feet.

Any calculus student can roughly check the melting statement.  Antarctica ice data is available at NASA’s Vital Signs of the Planet Ice Sheets page. There you can download change in Antarctica ice sheet data since 2002. (Note: The NYT article has a graph going back to 1992, but ends in 2017 as does the NASA data.) A quick scatter plot and a regression line shows that the change is not linear and the data set is concave down. (The graph here is the NASA data and produces in R – the Calculus Projects page now has some R scripts for those interested.)  Now, a quadratic fit to the data followed by a derivative yields that in 2007 the Antarctica was losing 95 gigatonnes of ice per year and in 2017 it was 195.6 gigatonnes per year. Even with this quick simple method melting has more than doubled from 2007 to 2017. The NYT article states:

While that won’t happen overnight, Antarctica is indeed melting, and a study published Wednesday in the journal Nature shows that the melting is speeding up.

This is an excellent sentence to analyze from a calculus perspective. Given that the current trend in the data is not linear and at least about quadratic, then melting is going to increase each year.  On the other hand, maybe they are trying to suggest that melting is increasing more than expected under past trends, for example the fit to the data is more cubic than quadratic. In other words, is the derivative of ice loss linear or something else? If everyone knew calculus the changes in the rate of ice loss could be stated precisely.