Micro24fallAssn2

1) in R print and calculate them: Differentiate the following functions:

a) y = 4x^3

dy/dx = 12x^2 ## b) y = 4x^3 − x^2 + 3x + 7 dy/dx = 12x^2 - 2x + 3 ## c) y = 2x^5 + 3x^3 + 7x^2 − x − 2 dy/dx = 10 x^4 +9 x^2 +14x - 1 ## d) y = 4x^(1/2) − (1/3)x^2 + 3/(x^2) + 5/x dy/dx = -2x/3 + 2/ (x^(1/2)) - 5/(x^2) - 6/ (x^3) ## e) y = ln (4x^3− x^2+ 3) dy/dx = (12x^2 - 2x)/ (4x^3 - x^2 + 3)

2) Differentiate y with respect to x using implicit differentiation

f) 3𝑥^2 − 𝑦^2 = 18

6x - 2y (dy/dx) = 0 dy/dx = 3x/y

g) 5y/(x+1) = 25

[5(dy/dx) (x+1) - 5y]/(x+1)^2 = 0 5(dy/dx)/(x+1) - 5y/(x+1)^2 = 0 dy/dx = y/(x+1)

#3) Differentiate y with respect to x using chain rule and/or the quotient rule.

h) 𝑦 = (2x^3 + x)^4

u = 2x^3 + x dy/dx = u^4 = 4 (6x^2 + 1) (2x^3 + x)^3

i) y = x^3 (2x + 5)^4

u=2x+5 dy/dx=x^3 u^4 dy/dx = 3x^2 (2x +5 )^4 + 8x^3 (2x+5)^3 = x^2 (2x+5)^3 (8x + 3(2x+5))

j) 𝑦 = 5x/(x+1)

dy/dx = [5(x+1)-5x]/(x+1)^2 = 5/(x+1)^2

k) y^2 + 2y = x

2y(dy/dx) + 2(dy/dx) = 1 dy/dx = 1 / 2(y + 1)

l) 100 = 5y/[x^3 (2x + 5)^4 ] *

100[x^3 (2x + 5)^4 ] = 5y u=(2x+5)^4 20[x^3 (2x + 5)^4 ] = y dy/dx = 20[4(2x + 5)^3 x^3 + 3x^2 (2x + 5)^4]

m) 𝑦 = (sqrt(x^2 + x - 1))/ (x^2 -1) *

u = x^2 - 1 sqrt(u+x) / u = [(1/2) (u+x)^(-1/2) u - du/dx sqrt(x^2 + x - 1)]/u^2 dy/dx = - (2x^3 + 3x^2 - 2x + 1)/[2(X^2 + x -1)^(1/2) (x+1)^2 (x-1)^2)

n) y = 𝑥 / (𝑒^(4x)) + ln (4x^2 ) *

dy/dx = [(𝑒^(4x)) - 4(𝑒^(4x)) x]/ (𝑒^(8x)) + 2 (4/4x) = (1-4x)/(𝑒^(4x) + 2/x

4) Let’s look at a utility function that satisfies the assumptions that more is better and that marginal utilities are diminishing. Suppose a consumer’s preference between food and clothing can be represented by the utility function U = sqrt(xy), where x measures the number of units of food and y the number of units of Clothing.

a) Show that a consumer with this utility function believes that more is better for each good.

MUx = ∂x/∂U = sqrt(y) MUy = ∂y/∂U = sqrt(0.5xy^(-0.5)) MUx (Marginal Utility of Food) represents the extra utility gained from consuming one more unit of food, while keeping clothing constant. It depends on the square root of the number of clothing units, meaning as clothing (y) increases, the marginal utility of food rises but at a decreasing rate.

MUy (Marginal Utility of Clothing) represents the extra utility from consuming one more unit of clothing, while keeping food constant. It is proportional to the square root of the food units (x), meaning as more clothing is consumed, its marginal utility decreases, and it’s also influenced by the amount of food consumed. To show that a consumer believes more is better for each good, we need to confirm that both MUx and MUy are positive.

MUx is always positive since the square root of a positive number is positive. MUy is also positive as long as both x and y are positive. Thus, more of both food and clothing leads to higher marginal utilities, indicating that the consumer believes more is better for each good.

b) Show that the marginal utility of food is diminishing and that the marginal utility of clothing is diminishing.

To show diminishing marginal utility, we check the second derivatives of the utility function with respect to x and y.

∂^2 U / ∂ x^2 = 0 meaning the marginal utility of food is constant, not diminishing ∂^2 U / ∂ y^2 = -0.25 xy^(-1.5), which is negative for positive x and y, confirming that the marginal utility of clothing is diminishing as more clothing is consumed

Thus, the marginal utility of clothing diminishes, while the marginal utility of food remains constant.