Concept explainers
Cobb-Douglas Production FunctionConsider the Cobb-Douglas production function
(a) Show that
(b) Show that
Trending nowThis is a popular solution!
Chapter 13 Solutions
Calculus
- Hollings Functional Response Curve The total number P of prey taken by a predator depends on the availability of prey. C.S. Holling proposed a function of the form P=cn(1+dn) to model the number of prey taken in certain situations. Here n is the density of prey available, and c and d are constants that depend on the organisms involved as well as on other environmental features. Holling took data gathered earlier by T. Burnett on the number of sawfly cocoons found by a small wasp parasite at given host density. In one such experiment conducted, Holling found the relationship p=21.96n1+2.41n, Where P is the number of cocoons parasitized and n is the density of cocoons available measured as number per square inch. a Draw a graph of p versus n. Include values of n up to 2 cocoons per square inch. b What density of cocoons will ensure that the wasp will find and parasitize 6 of them? c There is a limit to the number of cocoons that the wasp is able to parasitize no matter how readily available the prey may be. What is this upper limit?arrow_forwardCalculus Define T:P2R by T(p)=01p(x)dx What is the kernel of T?arrow_forwarddiiferential of a function: f(x,y)=ye^x at (0,-2)arrow_forward
- Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at that point. f(x,y)=y+sin(x/y),(0,3)arrow_forwardA function f(x, y) = x3 − 3xy2 + y3 is homogeneous of degree n when f (tx, ty) = tnf (x, y). (a) show that the function is homogeneous and determine n, and (b) show that xfx(x, y) + yfy(x, y) = nf (x, y).arrow_forwardDetermine wether the function f(x, y) = tan(x + y) is homogeneous, and if it is, determine its degree. A function f(x, y) is homogeneous of degree n if f(tx, ty) = tnf(x, y)arrow_forward
- Find div F if F(x, y, z) = (x2 + y) i + z2 j + (ey - z) k. A . 2xy i + k B. 2x - 1 C. 2xy - 1 D. 2x i - karrow_forwardJoint PDF Hard Problem: If f_X,Y (x,y) = 6e^-(2x + 3y) for x, y >=0 then what is P[X+Y <=1]?arrow_forwardDetermine whether the statement is true or false. Give a proof or counter example. Suppose f(x) is differentiable and has a stationary point at c. If c is a local minimum point then f''(c) > 0arrow_forward
- World oil production as shown show data for the rate of world crude oil production (in billions of barrels>yr) over a 16-year period. If the rate of oil production is given by the (assumed to be integrable) function R, then the total amount of oil produced in billions of barrels over the time period a ≤ t ≤ b is Q = ∫ba R(t) dt. Use the Midpoint and Trapezoid Rules to approximate the total oil produced between 1995 and 2011.arrow_forwardLevel curves: Sketch the level curves f (x, y) = c of the followingfunctions for c = −3, −2, −1, 0, 1, 2, and 3:1. f (x, y) = −2y/x2. f (x, y) = y2 − x23. f (x, y) = x −sin−1 y 4. f (x, y) = x sec yarrow_forwardIntegral of the constant function f(x) = k is: C 0 kx+C k+Carrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning