Sample problem sets from Calculus for the Life Sciences 1. These problems were designed to be solved by groups of students in 50-minute, twice-weekly 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 single-celled 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. Write an equation describing reproduction in a bacterium that can reproduce itself by binary fission.
  2. Using this equation, calculate how many bacteria there would be after 1 hours and after 8 hours, if the bacteria could reproduce itself by binary fission every 40 minutes.
  3. What does this equation assume about the bacteria?
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
  • N(t) = number of individuals in population at time t
  • N0 = initial population (number of individuals in population at time 0)
  • r = intrinsic rate of population growth
  • t = time
    1. Write an equation using the same variables to describe linear population growth.
    2. Give an example of a situation where you would expect exponential population growth. Why would this population not experience linear population growth?
    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.
    1. Graph the estimated pond snail population over 5 years for the intrinsic population growth rates of r = 0.66 and r = 1.5. Remember to label your axes and include units. r = 0.66 r = 1.5
    2. How does the different r value impact the projected snail population? How might adding a predator to the pond and limiting the amount of algae present change the equation used to project the future snail population?
    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)=4t-0.02t^2
  • where s(t) is measured in L and t is time in minutes.
  • (1pt) What is the rate of sap uptake by an aphid at 5 minutes of feeding?
    • 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
  • P(N)=1-0.1N
  • where P(N) is the probability of survival for each chick and N is the clutch size.
  • (1pt) In words, describe what happens to the likelihood of survival for each chick as the number of eggs laid increases?
    • 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 4-7. 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:
  • C(i)=5i^2
  • At the same time, krill density depends on ocean productivity, and this relationship is described by the following equation:
  • K(i)=(C(i)) ^ 0.8
  • Krill density, in this case, is measured as kilograms of krill per season.
    1. Write the function for the instantaneous rate of increase of krill density using the techniques to finding the derivative discussed in section 4.1. (1pt)
    2. Graph the equation for the relationship between sea ice extent and krill density, then graph the first derivative of the function. (Graph in the domain i [0,35] using 5 unit intervals. Make sure to label axes and units.) (2pts)
    3. Does K(i) have a minimum or maximum? If so, which, and for what value of i?
    4. How do the features of K(i) relate to the values of K(i)?
    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.
  • Krill is the basic diet of Adlie penguins (Pygoscelis adeliae). A colony of about 25000 individualscatch approximately 28.5 metric tons of krill per day during the breeding season, which lasts around 90 days. (1 metric ton = 1000 kg). Write a function relating krill density and penguin feeding to find the maximum size of a colony. (Hint: start by determining how many kg of krill are required for each penguin per day.) Make sure to define your variables and keep track of units. (1pts)
    • 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.
    1. How much will the algae population at the South Pole have grown in 100 years? In 500 years? In 1000 years? (Put your answer in terms of N(t) = a constant * initial algae numbers) (1pt)
    2. Find the equation describing the rate of change in algae population growth over time (assume the mean rate of warming at the South Pole remains the same). (1pt)
    3. What happens to the population growth rate over time? (1pt)
    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.
  • Find the rate of change of the fish population when there are 10 algae in a cm3 of water and 100 algae in a cm3 of water. Provide units. (3pts)
    • 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:
  • P(t)=(3t^2)/ ((230+ (3t^2))
    1. Determine the function that describes the rate of change in the feeding behavior of the osprey parents. (2pts)
    2. What can we learn about the biology of osprey through the analysis of this function? (1pt)
    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.
  • How could algae blooms harm an ecosystems trophic interactions and life cycles based on what youve learned above? (1pt)
    • 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
  • L(t)=12 - 4.5 cos(t)
  • where t represents time measured in units, 2 corresponds to one year, and the shortest day is December 21st (the winter solstice). A plant puts out leaves in the spring in response to the change in day length.
    1. Find the derivative of L(t). (1pt)
    2. Graph L(t) and L(t) on the same graph for the domain [0,2]. Label axes and units. (2pts)
    3. Find the domain of t values where the numbers of hours of daylight are increasing, and also where the numbers of hours of daylight are decreasing. What properties of L(t) tell us that the original function L(t) is either increasing or decreasing? (1pt)
    4. What are the longest and shortest days? How many hours of daylight do they have? (1pt)
    5. At what times of year would it be easiest for the plant to detect changes in day length and why? (1pt) (Hint: where the day length changes most rapidly)
    7 2 The length of the monkey face prickleback, a West Coast game fish can be approximated by L(t)=71.5(1-e^(-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.
    1. What is the approximate length of a 6-year-old monkey face? (1pt)
    2. How fast is a 6-year-old monkey face growing in length? Make sure to provide units. (1 pt)
    3. Find the approximate weight of a 6-year-old monkey face. (1 pt)
    4. How fast is the weight of a 6-year-old monkey face changing? Provide units carefully! (1pt)
    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 3-7 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 3-7 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.
  • Researchers have found that the survival of these finches depends on their ability to eat the larger cactus seeds or smaller thistle seeds, and finches with different beak lengths will eat seeds of corresponding size. In finches, fitness (survival and reproduction) is determined by beak length. They measured a relative fitness term (F) that is dependent on the difference (D, in mm) in the beak length from the average (1.5 cm). Their data are described by the equation:
  • F(D) = (-0.1) D4 + (2/30) D3 + (1.5) D2 *Note: fitness is relative to average beak length
  • D is the difference in beak length from the average of 1.5 cm. Finches were only found to vary in beak length between 1.0 and 2.0 cm. Using this information, determine the domain of F. Pay attention to units!! (Hint: D is the independent variable)
    • 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.
    1. Sketch the tank and label its dimensions. Recall that V=lwh and SA=2(lw+wh+lh).
    2. We want to minimize costs to build the tank. The material used to build the top and bottom of the tank costs $10/ft2 and the material used to build the sides costs $6/ft2. Find the cost equation and simplify.
    3. What is the constraint (what is your limiting factor) and what is its equation?
    4. What is the minimum cost to build the tank? (Hint: solve the constraint for one variable, plug in the solution to the cost equation, find the first derivative and then the minimum.)
    5. Verify that this is the minimum cost to build the tank using the second derivative test to find concavity.
    6. What are the dimensions (length, width, height) of the tank?
    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.
    1. The average daily metabolic rate for infant nutria can be expressed as a function of weight by: m/86=w^0.5 where w is the weight of the nutria (in kg) and m is the metabolic rate (in kcal/day). Suppose that the weight of the nutria is changing with respect to time at a rate dw/dt. Find dm/dt.
    2. Determine dm/dt for a 0.5kg infant nutria gaining weight at a rate of 0.1 kg/day.
    3. What are the units for dm/dt?
    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. Right an equation showing the surface area (S) of the pond.
    2. Take the implicit derivative and solve for dS/dt.
    3. If the pond border moves outward, making the radius of the pond grow at the rate of 5 inches per week when the radius is 20 feet, approximately how fast is the surface area of the pond growing at that time? Provide units.
    1 1 Exponeential functions bacterial cell division Many bacteria reproduce by binary fission, in which a single-celled bacterium divides into two bacteria, which each have identical genetic information.
    1. Write an equation describing reproduction in a bacterium that can reproduce itself by binary fission.
    2. Using this equation, calculate how many bacteria there would be after 3 hours and after 10 hours, if the bacteria could reproduce itself by binary fission every 30 minutes.
    3. What does this equation assume about the bacteria?
    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
  • N(t) = number of individuals in population at time t
  • N0 = initial population (number of individuals in population at time 0)
  • r = intrinsic rate of population growth
  • t = time
    1. Write an equation using the same variables to describe linear population growth.
    2. For what values of the intrinsic growth rate r will the exponential growth equation overtake the linear growth equation over time?
    3. Give an example of a situation where you would expect exponential population growth. Why would this population not experience linear population growth
    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:
  • mbLg = 3.75 + 0.90 log(D) + log(A/T) for 0.5 D 4.0
  • mbLg = 3.30 + 1.66 log(D) + log(A/T) for 4.0 D 30
    • where mbLG is magnitude of the surface wave, D is the distance in geocentric degrees, A is the amplitude, and Tis the period.
    1. Using the formulas above, fill in the missing magnitudes in the table
    2. Using trigonometry, fill in the missing distances from the table
    3. Earthquakes do not necessarily travel in an exact circle. Does it appear that the earthquake was stronger in any particular direction? Explain your reasoning. From the data in the table, is there any other geological aspect that may have decreased the magnitude of the waves in a given area?
    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:
  • N(t)=(850t^2)/(420+2t^2)-20
    1. Using this formula, how many stink bugs would you expect to find in an orchard with 50 trees? With 100 trees?
    2. What is the average increase in stink bugs between these two orchards?
    3. Is there a limit to how many stink bugs are found in a given orchard? If so, what is the limit?
    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).
    1. Each pulse of sound has a carrier frequency proportional to male body size, and can be modeled by the equation f0 (in Hertz)=625/(2pi*mass in grams). Males range in weight from 19 to 25 milligrams. Determine the carrier frequency (in Hertz, abbreviated Hz, where 1 Hz = 1s1) for a male of mass 19 mg and 25 mg.
    2. If females prefer songs with lower (smaller) carrier frequencies, which size males will they find most attractive? Determine the period for the songs of the most attractive males and write an equation to describe the sine wave for these males (amplitude = 20, no phase shift or vertical shift). Why do you think a female might have a preference for these males?
    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)=4t-0.02t^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)=1-0.1N, where P(N) is the probability of survival for each chick and N is the clutch size.
    1. In words, what happens to the likelihood of survival for each chick as the number of eggs laid increase?
    2. Determine the average rate of change of chick survival if the typical number of eggs in a clutch ranges from 3-7 Explain your answer.
    3. The total number of offspring that are likely to survive, S(N), can be modeled as the product of the clutch size and the probability of survival. What is the instantaneous rate of change at N=4, N=5, and N=6?
    4. Based on your answer from the previous question and what you know about instantaneous rates of change, predict the optimal clutch size for a female bird? Explain how you came to this conclusion.
    5. How many chicks can you expect to survive in a clutch of the optimal 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.
    1. Find the first derivative of this function and explain its meaning. Provide units in your answer.
    2. Find the derivative of your answer in part a and explain its meaning. Provide units in your answer.
    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+6t-4 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.

    1. Find the rate of change of the spread of teh virus. Provide units for your answer.
    2. When does the number of infected people reach a maximum (i.e whendoes the rate of chage =0)?
    3. What is the maximum number of people infected?
    4. Find the average rate of change of V(t) on the interval [1,3] and explian your answer.
    5. Find the instatnaeous rate ofchange of V(t) at t=4 and explain your answer. What do you think could be going on with the spread of the virus at this time?
    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 ice-edge 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 Chlorophyll-a concentration, since Chlorophyll-a is a good proxy for the amount of algae present in the water. Chlorophyll-a concentration, (mL/Kg^2), depends on the extent of sea-ice, kg, during the winter, and we can use the following equation to describe this relationship:
  • C(i) = 5i^2 + 10.
  • At the same time, krill density, K, depends on ocean productivity, C(i), and this relationship is described by the following equation: K(C(i))=(C(i))^0.8. Krill density in this case is measured as Kilograms of Krill per season.
    1. Write the function for the instantaneous rate of increase in krill density.
    2. Graph the equation for the relationship between sea-ice extent and krill density, and its first derivative (graph them in the domain [0, 35], using 5 units intervals).
    3. Does this function (not the derivative) have a minimum or maximum? If so, which and for what values of i?