A bunch of practice questions January 31, 2008

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1 A bunch of practice questions January 3, 28. Which of the following functions corresponds to the one plotted to the right? 7 6 (a) f(x) = x 3 4x 2 +, (b) f(x) = 2x 3 3x 2 +, (c) f(x) = x 3 4x 2 + 2x +, (d) f(x) = x 2 + 6x +. f(x) x 2. Match the plotted functions to the functions given by A(x) = x 2, B(x) = x 6 and C(x) = x /3? Given expression Number on plot A(x) = x 2.8 ().6 B(x) = x (2) (3) C(x) = x /3 3. Let f(x) = x 2 3x. (a) Calculate the slope of the secant line to the graph of f(x) between the points x = and x = 2. (b) Calculate the slope of the secant line to the graph of f(x) between the points x = and x = +h. (c) Using the definition of the derivative, calculate the derivative of the function f(x) at x =. 4. Let g(s) = s 3 4s. (a) Find the zeros of g(s), that is, the values of s at which g(s) =. (b) Find the locations of all local maxima and minima of g(s). For each one, be sure to state how you know that it is a maximum or a minimum. (c) Find all inflections points of g(s). On which intervals is g(s) concave up? On which intervals is g(s) concave down? (d) Using your answers to (a), (b) and (c), sketch the graph of g(s). Be sure to label all zeros, maxima, minima and inflections points.. Show that the sum of the x and y intercepts of any tangent line to the curve x + y = c is equal to c. Sketch the curve. Note: c is a constant. x

2 6. Hill functions A single protein can undergo three distinct enzymatic reaction depending on the other reactants present. Each reaction proceeds at a rate given by a Hill function of the form v = K max c n k n m + c n where c is the concentration of the protein and v is the reaction rate. The three Hill functions, one for each reaction, are plotted below. Match each Hill function (A, B, and C) with the correct set of parameters from the list below. 4 (a) K max = 4, k m =, n =. (b) K max = 2, k m =, n =. (c) K max = 2, k m =, n = 3. (d) K max = 4, k m =, n = 3. (e) K max = 4, k m = 2, n =. Rate of reaction (mol/sec) A B C (f) K max = 2, k m = 2, n =. 2 Concentration of protein (µ M) 7. Find the tangent line to the graph of the function h(x) = 3x 2 x + 2 at the point x =. Use this to estimate the value h(.). Is your approximation an underestimate or an overestimate? 8. Find all maxima and minima for the function f(x) = ln(x 2 +) 2 x2. Make a table to summarize your answer listing the x and f(x) values for each minimum and maximum as well as f (x) and whether the point is minimum or maximum. Use this information to sketch the graph of the function. 9. Suppose a population of rabbits reproduces at a rate proportional to the population size with constant of proportionality b = ln(3) individuals per capita per month. However, the population also has a death rate proportional to the population size with constant of proportionality m = ln(2) individuals per capita per month. Suppose the population starts with 6 members. After 4 months, how many rabbits will there be?. A colony of bacteria consists of two strains, one susceptible (S) to a particular antibiotic and the other resistant (R) to the same antibiotic. Upon treatment with the antibiotic, the population size of the susceptible strain as a function of time is S(t) = 2 ( t ) while the population size of the resistant strain is R(t) = 2 t. Here, t is measured in months with t = corresponding to the beginning of treatment. (a) How many of each strain of bacteria are there at the beginning of the treatment? (b) At what time will the resistant strain consist of 8% of the total colony (of the total colony at that time, not of the initial colony)? (c) How big are the two strains at that time?. A microtubule (MT) is a linear polymer that grows by addition of tubulin subunits at its tips. Tubulin subunits are placed in a rectangular microfabricated chamber (a micron-scale hole etched out of glass) with dimensions 2µm 8µm 8µm at an initial concentration of µm. No microtubules are initially present. At a particular moment in time, six MTs spontaneously form, each with a negligible (essentially 2

3 zero) length. From that time on, tubulin continually adds to and falls off the two tips of each polymer the rate of addition at each tip is proportional to the tubulin concentration with rate constant k on = subunits/(second µm) and the rate of removal is a constant rate k off = 2 subunits/second. Some assumptions that will be useful: () Addition of a single tubulin subunit increases the length of the polymer by approximately δ =. nm. (2) Tubulin subunits are conserved meaning that their total number never changes; they simply change from soluble form (measured in concentration) to polymer form (measured as length) and back. (3) Avogradro s number= (4) Although addition of subunits increases the length in discrete increments, it is reasonable to treat length as a continuous variable. (a) Write down a differential equation for the total length of polymer as a function of time, l(t). What is the appropriate initial condition? This question requires you to think carefully about conversion of units! (b) What is the total polymer length after a long period of time? Do the polymer fit in the chamber without having to bend? (c) What must I mean in the previous question when I say a long period of time? That is, a long time compared to what? (d) Simplify and solve the equation. 2. Skibbens et al. (993) measured the movement of a chromosome during mitosis before capture by the spindle was complete (plotted below). Their data show a dramatic oscillation in the position of the chromosome. (a) Use the data on the first graph below to sketch the velocity of the chromosome as a function of time on the second (blank) graph. Be sure to fill in a few values on the vertical axis and indicate the units in the parentheses. (b) On the first graph, circle one location where the acceleration is positively large and one location where the acceleration is negatively large. Ve locity ( ) Time (sec) 3. When fireworks explode, the vertical position of each flaming piece emerging from the explosion undergoes vertical motion described by y(t) = y + v t 2 gt2 where y is the height at which the firework explodes, v is the initial velocity of the flaming piece in question and g m/s 2 is the gravitational constant. (a) Find an expression for the time at which the flaming piece reaches its maximum height in terms of the initial velocity v. Be sure to consider all possibilities, that is v and v <. (b) Give an expression for the maximum height achieved in terms of v (again, consider all possibilities). 3

4 4. Suppose that a population of fish, measured as N in thousands of individuals, has a natural growth rate that depends on its density as described by the function G(N) = N(N )(3 N). If the population size is N = thousand and a few fish are removed, do you predict that the population size recover to? If not, what does it do? What if the population size is N = 3 thousand - will it recover from a similar removal of a few fish? When the population size is at we refer to it as steady but unstable. The population size at 3 is referred to as steady and stable. This should make sense based on your answer to the previous question. If you remove fish at a rate EN so that the total growth rate is now determined by the natural growth rate and the fishing rate, find the new stable steady population. Note that this population size is a function of E; call it N ss (E). Larger E means more fishing and hence a smaller stable steady population - check that your expression for N ss (E) decreases with larger E as one might intuit. What is the upper limit on E for which there is a stable steady population? What happens above this value of E? What is the yield (fish taken per unit time) when fishing occurs with rate constant E? Find the fishing rate constant, E max at which the yield is largest. What is the maximum sustainable yield (i.e. maximum yield that does not cause the population to crash)? Check that your answer for E max is within the allowable range. Try to provide a graphical interpretation of your answers.. Suppose you have just taken a spherical allergy pill with radius r. The volume of the pill gradually decreases as your stomach digests it but it remains spherical throughout. Suppose the rate of change of volume (V ) of the remaining portion of the pill is proportional to its surface area (S) with rate constant k. Derive an equation for the radius of the pill as a function of time. Solve the equation and calculate the length of time required to completely digest the pill or if more appropriate the time constant associated with the pill s dissolving. 6. Comparing functions It has been observed that wildebeest often migrate in extremely large herds (numbering in the thousands). Assume that the herds tend to be circular in form and the number of individuals per unit area is the same in all herds. If hyenas, the primary predator of wildebeest, can only reach members of the herd at its outer edge, explain why it is safer for the entire herd (on average) to be larger rather than smaller. Do this by writing down and comparing expressions for the total number of individuals in a herd and the number of exposed members at the outer edge. Explain your observation in words and include a sketch to illustrate your point. A herd of wildebeest grazing. 7. As an individual bacterium grows, its surface area increases and so it must produce new cell-wall material. In an attempt to measure this rate of cell-wall production using a microscope, we can look at two cross sections of the bacterium and measure the rates of cross-sectional area increase in the plane of the cross section (see grey areas in the figure below). The bacterium is shaped like a rod, composed of a cylindrical body capped by a half-sphere on each end. The measured rate of increase for cross-section is µm2 /min and for cross section 2 is µm2 /min. Calculate the rate of cell-wall (i.e. surface area) production. There is a way to solve this without knowing the radius and length - try eliminating those variables as early as possible. 4

5 Cross section Cross section 2 R L