Calculus Answers

Johnny is designing a rectangular poster to contain 24in^2 of printing with a 3-in margin at the top and bottom and a 2-in margin at each side. what overall dimensions will minimize the amount of paper used?

HardyWeinberg equilibrium problem if the three genotypes AA, Aa, and aa have respective frequencies pAA =θ^2, pAa = 2θ(1 − θ), and paa = (1 − θ)^2, n! / (divide) n1! n2! n3! PAA^n1 PAa^n2 Paa^n3 where n = n1 + n2 + n3. This probablity depends on θ. There is a method called the maximum likelihood method, that can be used to estimate θ. The principle is simple: We find the value of θ that maximizes the probability of the observed data. Since the coefficient n! / (divide) n1! n2! n3! does not depend on θ, we need only maximize L(θ) = PAA^n1 PAa^n2 Paa^n3 (a) Show that if L(θ) is maximal for θ = ˆθ then ln(L(θ)) is maximal for θ = ˆθ. (Note that L is strictly positive and twice differentiable.) (b) Use the result in (a) to find the value ˆθ that maximizes L(θ)

Which of the following statements are true? Give reasons for your answers, in the form of a short proof or a counterexample. i) 2 2 2 dx dy dx d y       = ii) The inverse function of 3x y = e is ln x 3 1 y = . iii) If f is increasing and 0 )x(f > on an interval I, then )x(f 1 )x(g = is decreasing on I. iv) An equation of the tangent line to the parabola 2 y = x at )4,2 (− is y − 4 = 2 x(x + )2 . v) If f is one-one onto and differentiable on R , then f )6( 1 f( )6()' 1 ′ = − .