Final Exam, Calculus with Life Sciences Applications Spring 2005, Monday May 9, 2005 Please show all your work, not a calculator answer. And relax, relax, relax, ... 1. Twelve persons are to be equally divided among four laboratories. How many ways can this be done? 2. Suppose 40% of the seeds of a plant germinate. A researcher plants 7 pots with 8 seeds in each. What is the probability that at least one pot will have no seeds germinate? 3. Suppose the rainfall in a region is normally distributed with mean 100 cm. and standard deviation 15 cm. What is the probability that in a given year the rainfall does not exceed 70 cm.? You may use the following table of areas under the standard normal curve to the left of the given z-value: z Area 0 0.50 0.5 0.69 1 0.84 1.5 0.93 2 0.98 2.5 0.99 4. Let X be a random variable with probability mass function described by the following table: x P(X=x) -1 0.1 0 0.3 2 0.2 3 0.1 4 0.2 6 0.1 Find E(X) and Var(X). 5. Suppose we cross dominant phenotype red flowering pea plants with recessive phenotype white flowering pea plants. Suppose that the offspring are 20% white flowering. What percentage of the original red flowering plants had a pure RR genotype? 6. Let X be a continuous random variable with density function f(t) = 0.5 exp(-0.5 t) for t >0 and f(t) = 0 otherwise. (a.) Verify that this indeed has the properties of a probability density function. (b.) Find E(X). 7. To solve a system of linear equations efficiently on a computer, we set up an augmented matrix and use row reduction. Suppose we have the system of equations 3 x2 yz9, 2 x4 y3 z8, 5 y2 z1 (a.) Set up the augmented matrix used to solve this system of linear equations. (b.) Carry out the first two steps of the row reduction for this matrix. 8. Here is a slope field: (a.) Does the origin represent a stable or unstable equilibrium? (b.) It corresponds to one of the following four sets of differential equations. Which one, and why? (i.) d x1 / dt = +0.5 x1 + 0.8 x2 d x2 / dt = +0.5 x1 – 0.8 x2 (ii.) d x1 / dt = +0.5 x1 – 0.8 x2 d x2 / dt = +0.5 x1 + 0.8 x2 (iii.) d x1 / dt = -0.5 x1 – 0.8 x2 d x2 / dt = -0.5 x1 + 0.8 x2 (iv.) d x1 / dt = -0.5 x1 + 0.8 x2 d x2 / dt = +0.5 x1 + 0.8 x2 9. In the Gompertz growth model, the per capita growth rate depends on the population density. Gompertz argued that the population N(t) satisfies a differential equation dN / dt = k N ( ln(K) – ln(N) ) with initial condition N(0) = N0 where k and K are positive constants. He then showed that the solution to this equation is given by N(t) = K exp( - ln( K/N0 ) exp( -k t) ) Verify that this N(t) is a solution to Gompertz’ differential equation and initial condition. 4 10. Find z 2 z5 dz 3 5 1 dx 11. Find 2 x 2 x 3 12. Use the midpoint rule with four boxes to estimate this integral: 0.3 2 x 2 e dx -0.1