Scientific studies suggest that some animals regulate their intake of different types of food available in the environment to achieve a balance between the proportion, and ultimately the total amount, of macronutrients, consumed. Macro-nutrients are categorized as protein, carbohydrate or fat/lipid. A seminal study on the macro-nutrient intake of migratory locust nymphs (Locusta migratoria) suggested that the locust nymphs studied sought and ate combinations of food that balanced the intake of protein to carbohydrate in a ratio of 45:55 [1]. Assume that a locust nymph finds itself in an environment where only two sources of food are available, identified as food X and food Y . Food X is 20% protein and 80% carbohydrate, whereas food Y is 70% protein and 30% carbohydrate. Assuming that the locust eats exactly 100 mg of food per day, determine how many milligrams of food X and food Y the locust needs to eat per day to reach the desired intake balance between protein and carbohydrate.

Trigonometric Functions.

Why do they have asymptotes and why do they have intervals?

For the following function, find the local maximum and minimum values; and saddle point(s).

f(x,y)=(8x−8y)/(ex2⋅ey2)

Find the absolute maximum and the absolute minimum of the function f(x,y) = 6 - x² - y² over the region R = {(x,y) | -2 <= x <= 2, -1 <= y <= 1 }. Also mention the points at which the maximum and minimum will occur.

1) Let f(x) = e^(xyz) . Find: fxx, fyy, fzz, fxy, fxz, fyx, fyz, fzx, fzy.

2) Use the Chane Rule to calculate derivatives ∂z/∂s and ∂z/∂t

z = e^xy tan y, x = s+2t, y = s/t

3) Use the Chane Rule to calculate derivatives ∂z/∂s and ∂z/∂t

z = xy−2x+3y, x = cos s, y = sin t

Given the funtion g(x,y) = (e^x)(y)+sin(x/y).

1. Find the linearization of g(x,y) at the point (1,2).

2. Estimate the value of g(1.01, 1.99).

8. Determine the centroid, ?(?̅,?̅,?̅), of the solid
formed in the first octant bounded by ?+?−16=0
and 2?^2−2(16−?)=0.

Evaluate the surface integral

F · dS

for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.

F(x, y, z) = x i − z j + y k

S is the part of the sphere

x² + y² + z² = 1

in the first octant, with orientation toward the origin

Nutt’s Nut Company has 500 pounds of peanuts, 100 pounds of pecans, and 50 pounds of cashews on hand. they package three types of 5-pound cans of nuts: Can I contains 3 pounds of peanuts, 1 pound of pecans, and 1 pound of cashews; Can II contains 4 pounds of peanuts, 1/2 pound of pecans, and 1/2 pound of cashews; and Can III contains 5 pounds of peanuts. The selling price is $28 for Can I, $24 for Can II, and $21 for can III. How many cans of each kind should be made to maximize revenue? Set up an LP problem. Do not solve.

Using method of variation of parameters, solve the differential equation: y''+y'=e^(2x)

Find the general solution, and particular solution using this method.

Curve sketching: Choose two of the functions to sketch a graph. you should include the following parts for each.

a). domain b). x and y intercepts c). any asymptotes d). intervals of increase/decrease e)/ extreme values f). intervals of concavity and infection points

f(x)= 2x+9 / x+3

g(x)= -2 / x+1

h(x)= x^ - 6x^2