Sample problem sets from Calculus for the Life Sciences 1. These problems were designed to be solved by groups of students in 50minute, twiceweekly recitation sessions. Problem sets are meant to demonstrate the utility of mathematical approaches for solving biological problems and are taken from a wide variety of biological contexts.
Project Number  Problem Number  Math Topic  Biology Topic  Problem text 
1  1  Exponential Functions.Logarithmic Functions Applications: Growth and Decay.  Bacterial, binary fission  Many bacteria reproduce by binary fission, in which a singlecelled bacterium divides into two bacteria, which each have identical genetic information. These two bacteria then both divide and you have a total of 4 bacteria, and so on.

1  2  Exponential Functions.Logarithmic Functions.Applications: Growth and Decay.  Bacterial, binary fission  Consider the exponential equation of population growth: N(t)=N_0 e^rt where

1  3  Exponential Functions.Logarithmic Functions.Applications: Growth and Decay.  Snails, algae (predatory/prey), population dynamics  A male and a female snail are introduced to a pond with no predators and an unlimited food supply in the form of algae.

3  1  Limits.Continuity.Rates of Change.  Plant/insect interactions 

3  2  Limits.Continuity.Rates of Change.  Natural selection  One of the fundamental concepts of evolutionary biology is that of natural selection where organisms with genes that are best suited for their environment have a better chance of survival and are more likely to reproduce. The more offspring an organism has, the greater the frequency of their genes. However, in organisms where parental care is costly (e.g. humans) there is a tradeoff between having more offspring and being able to provide sufficient resources to ensure the survival of each offspring. In birds, the size of the clutch (# of eggs in the nest) is a good example of this tradeoff, and can be modeled by the equation

3  3  Limits.Continuity.Rates of Change.  Natural selection  (2pts) Determine the average rate of change of chick survival if the typical number of eggs in a clutch ranges from 47. Explain your answer. 
3  4  Limits.Continuity.Rates of Change.  Natural selection  (3pts) The total number of offspring that are likely to survive S(N) can be modeled as the product of the clutch size and the probabiltity of survival. What is the instantaneous rate of change at N=4, N=5, and N=6? 
3  5  Limits.Continuity.Rates of Change.  Natural selection  (2pts) Based on your answer from the previous question and what you know about instantaneous rates of change, predict the optimal clutch size female bird? Explain how you came to this conclusion. 
3  6  Limits.Continuity.Rates of Change.  Natural selection  (1pt) How many chicks can you expect to survive in a clutch of the optimal size? 
5  1  Techniques for Finding Derivatives.  Marine food webs  Antarctic krill (Euphausia superba), a widespread species with circumpolar distribution, is central to the Antarctic marine food web, as most organisms are either direct predators of krill or are just one tropic level removed. Krill are tightly coupled with the marginal iceedge zone to forage on sea ice algae in summer and winter, and juvenile krill rely on under ice habitat for overwintering and as a refuge from predators.Ocean productivity is usually measured as Chlorophylla concentration since Chlorophylla is a good proxy for the amount of algae present in the water. Chlorophylla concentration (mL/kg2) depends on the amount of sea ice (kg) during the winter, and we can use the following equation to describe this relationship:

5  2  Techniques for Finding Derivatives.  Marine food webs  Areas of the highest krill concentration are often close to the landbased breeding colonies of penguins. These colonies depend on nearby krill populations to feed and rear their offspring during the Antarctic summer.

6  1  The Chain Rule.Derivatives of Exponential Functions.Derivatives of Logarithmic Functions.  climate change, population dynamics  Imagine that the warming at the South Pole causes an exponential increase in sea ice algae due to the increased temperature and resulting longer growing season according to the equation N(t)=N_0*e^(0.06rt) where r is 0.037, the mean rate of warming (at the South Pole) in degrees/year and N0 is initial amount of sea ice algae.

6  2  The Chain Rule.Derivatives of Exponential Functions.Derivatives of Logarithmic Functions.  climate change, population dynamics, algae  Algal blooms occur in other parts of the world and spur many ecosystems growth. Fish feed on the algae and have a resulting population of
F(a)=ln(a+5)*(a+200) where a is the amount of algae present in a cm3 of water.

6  3  The Chain Rule.Derivatives of Exponential Functions.Derivatives of Logarithmic Functions.  climate change, population dynamics, algae  This abundance of algae and the resulting growth of fish populations coincide with many species breading seasons especially the osprey, whose nests you often see perched on tall docking posts by the water. As the breeding season progresses, osprey parents have to devote a bigger proportion of their catch intake to feed their chicks, until they fledge and are able to feed themselves. The following equation describes this phenomenon, where P is the proportion of prey given to chicks and t is time in weeks over the course of the breeding season:

6  4  The Chain Rule.Derivatives of Exponential Functions.Derivatives of Logarithmic Functions.  climate change, population dynamics, algae  Harmful algae blooms have been occurring off the Gulf of Mexico in the Mississippi River Basin. Large amount of nutrients are available for algae to feed on, but then they grow to such numbers that they can block out the sun and they die off themselves, adding to more nutrient loading into the system.

7  1  Derivatives of Trigonometric Functions  photosynthesis  Plants are photoautotrophic which means they produce their own food using light. Photosynthesis is the mechanism by which plants produce food using water, a carbon source, and sunlight. In London, the number of hours of daylight follows roughly

7  2  The length of the monkey face prickleback, a West Coast game fish can be approximated by L(t)=71.5(1e^(0.1t))
and weight is approximated by W(L)=0.01289L^2.9 where L is the length in cm, t is the age in years, and W is the weight in grams.


9  1  Absolute Extrema.  Natural selection  On an island in the Galapagos, a group of finches with beaks about 1.5 cm long eats primarily seeds that are 37 mm long. However, the plant that makes seeds of this size will only reproduce in wet years (> 30 cm rain). After a series of unusually dry years, there are no seeds 37 mm long anymore. However, plants that produce smaller seeds (thistles, seeds <2 mm) and longer (cactus, seeds >10 mm) seeds are still able to reproduce during the dry years.

9  2  Absolute Extrema.  Natural selection  For the domain of F you determined, find all critical points of the function F(D). 
9  3  Absolute Extrema.  Natural selection  What are the increasing and decreasing intervals of F(D)? 
9  4  Absolute Extrema.  Natural selection  For each of the relative extrema found above, use the first or second derivative test to determine if it is a relative minimum or relative maximum. 
9  5  Absolute Extrema.  Natural selection  What is the absolute minimum of F(D) within this domain? How did you know? 
9  6  Absolute Extrema.  Natural selection  What is the absolute maximum of F(D) within this domain? How did you know? 
9  7  Absolute Extrema.  Natural selection  F(D) represents fitness. Natural selection for a given trait, such as beak length, tends to maximize fitness for that trait. Who are the most fit among the finches, and what are their beak sizes? 
9  8  Absolute Extrema.  Natural selection  Given this information, what will happen to the finch population and what seeds will be eaten most in the coming years if the environment stays dry? 
9  9  Absolute Extrema.  Natural selection  Using your answers above for the increasing/ decreasing intervals, extrema, and concavity, Graph F(D) on the domain found in question 1. Label axes & indicate scale. 
10  1  Applications of extrema  captive breeding  When transporting animals from a controlled environment (e.g. zoo, aquarium), to be released into the wild, perhaps after rehabilitation or as a product of a captive breeding program, a minimum cage or tank size must be established so that the animal has ample space. Consider a shark that requires a volume of 50ft3 during transport.
The shark will be transported in a rectangular tank whose base length is 3 times the base width.

11  1  Implicit Differentiation.Related Rates.  Invasive species  Nutria (Myocastor coypus), or river rats, are a large, herbivorous, semiaquatic rodent that were initially introduced from the original temperate climate of South America to the Gulf of Mexico coast for their fur value. Nutria are now considered an invasive species in the States because of their destructive feeding on shoreline plants and burrowing habits that encourage erosion and the destruction of wetlands. A large effort has sprung up in Louisiana to protect the precious wetlands that provide many ecosystem services to the community. Part of the process of creating a management plan has been to understand the growth and metabolic rates of baby nutria and how nutria impact their surrounding environment.

11  2  Implicit Differentiation.Related Rates.  Invasive species  Suppose that we are studying nutria living in a circular pond. The nutrias dining on the surrounding shoreline water plants has caused erosion to occur on each side of the pond, so that as the sediment is falling to the depths of the pond, the displaced water is expanding the ponds surface area.

1  1  Exponeential functions  bacterial cell division  Many bacteria reproduce by binary fission, in which a singlecelled bacterium divides into two bacteria, which each have identical genetic information.

1  2  Exponential Functions.Applications: Growth and Decay.  Exponential and linear population growth  Consider the exponential equation of population growth: N(t)=N_0 e^(rt) where

1  3  Exponential Functions.Logarithmic Functions.Applications: Growth and Decay.  Predatory/prey population dynamics  A male and a female rabbit are introduced to an island. Graph the estimated island rabbit population over 5 years for each of the following intrinsic population growth rates: r = 0.5 and r = 2. Remember to include your axes and labels. What does the different r value imply for the rabbit population? 
2  1  Logarithmic functions; trigonometry  Earthquakes  On the afternoon of August 23, the east coast of the United States experienced an earthquake centered near Mineral, Virginia. Unlike earthquakes in California, this quake was felt from hundreds of miles away. To determine the magnitude of an earthquake, the USGS averages the magnitude calculated from seismographs surrounding the earthquake. Readings from some of these seismographs are recorded in the table below. Earthquakes send out seismic surface waves measured by a seismogram. The seismogram translates the waves into a chart, graphing the movement of the earth at that location over time. Researchers can then measure the amplitude and period of the waves, and use those values to calculate the magnitude of the earthquake using the following functions:

2  2  Limits  Population dynamics  Dr. Shrewsbury is studying the spread of brown marmorated stink bugs through the orchards of Maryland. So far, he has found that the number of stink bugs (in thousands) is a function of the number of trees in the orchard:

2  3  Trigonometric Functions  Circadian rhythms and characteristics  Ultradian rhythms have periods of less than one day, and can be used to describe insect sounds. Male crickets produce a song with a pulsed, sinusoidal sound wave by stridulating (rubbing their wings together).

3  1  Limits.Continuity.Rates of Change.  Plant/insect interactions  Aphids feed on the sap of a common ornamental garden plant called Lupine. The aphids puncture phloem vessels in the lupine and feed on its sap. If the total amount of sap eaten by an aphid is given by: s(t)=4t0.02t^2 where s(t) is measured in L and t is time in minutes. What is the rate of sap uptake by an aphid at 10 minutes of feeding? 
3  2  One of the fundamental concepts of evolutionary biology is that of natural selection, where organisms with genes that are best suited for their environment have a better chance of survival and are more likely to reproduce. The more offspring an organism has, the greater the frequency of their genes. However, in organisms where parental care is costly (e.g., humans!), there is a tradeoff between having more offspring and being able to provide sufficient resources to ensure the survival of each offspring. In birds, the size of the clutch (# of eggs in the nest) is a good example of this tradeoff, and can be modeled by the equation P(N)=10.1N, where P(N) is the probability of survival for each chick and N is the clutch size.


4  1  rates of change  velocity of rain drops  A raindrop falls froma leaf according to the function S(t)= 16t^2+V(0)t+S(0) where t=time in seconds, V(0)=initial velocity in ft/s and S(0)=initial distance in meters.

4  2  instantaneous and average rate of change  spread of a virus 
The spread of a virus can be modeled by V(t)=t^2+6t4 where V(t) is the number of people in hundreds with the virus and t is the number of weeks since the first case was observed.

5  1  Rates of Change. Definition of the Derivative. Graphical Differentiation.  Antarctic krill (Euphausia superba), a widespread species with circumpolar distribution, is central to the Antarctic marine food web, as most organisms are either direct predators of krill or are just one trophic level removed. Krill live at the marginal iceedge zone, where they forage on sea ice algae, and juvenile krill rely on under ice habitat for overwintering and as a refuge from predators. Southern ocean productivity is usually measured as Chlorophylla concentration, since Chlorophylla is a good proxy for the amount of algae present in the water. Chlorophylla concentration, (mL/Kg^2), depends on the extent of seaice, kg, during the winter, and we can use the following equation to describe this relationship:
