Abstract
The document resolves a series of 9 questions about algebraic functions. They include inverse demand functions, profit functions, concave functions, etc.
Extract
[...] Algebra - Studying Functions with Questions 1. p q The downward slope of the demand curve reflects the law of demand: as the price of the washing machine decreases, the quantity demanded increases. When the price is higher, fewer units are demanded. Elasticity of demand is a measure of how responsive the quantity demanded is to a change in price. The formula for price elasticity of demand is: First, solve for q when p=4 using the demand function: 4=?0.2q+6 ? [...]
[...] Thus, the function is concave up on the interval Réponse C 8. To find dQ/dK?, we differentiate the production function Q=50K3 /4? with respect to dQ/dK=50?(3/4)K3/4?1=37,5K-1/4 Now, substitute K=10: dQ/dK=21.09 To find the capital K when Q=100, use the equation: Q=50K3/4 Substitute Q=100: 100=50K3/4 Divide by 50: 2=K3/4 Now, solve for K by raising both sides to the power of K=24/3?2.52 Thus, the capital invested when the quantity produced is 100 items is approximately 2.52 thousand euros 9. We need to find the first derivative, and then compute for further interpretation. [...]
[...] The system of equations has no solution. 3. The profit equation is the difference between the total revenue TR(q) and total cost Substituting the given expressions for TR(q) and P(q)=(?q²+70q)?(300+30q) Simplifying: P(q)=?q²+70q?300?30q P(q)=?q²+40q?300 So, the profit equation is: P(q)=?q²+40q?300 At break-even, profit is zero. Set P(q)=0and solve for ?q²+40q?300=0 This is a quadratic equation. We can solve using the quadratic formula where b=40 and c=?300 Therefore, the break-even quantities are q=10 and q=30. To find the quantity that maximizes profit, take the derivative of the profit function with respect to and set it equal to zero to find the critical points. [...]
[...] 10p=30 At a price of the elasticity of demand is meaning demand is unit elastic. When elasticity is any change in price does not affect total revenue. At this point, the company maximizes its revenue. Consumer surplus is the difference between what consumers are willing to pay and what they actually pay. It is the area between the demand curve and the price level, up to the quantity demanded. The consumer surplus is calculated as the area of the triangle between the price level and the demand curve: Consumer Surplus=0,5×base×height So, the consumer surplus is: Consumer Surplus=0,5×10×(6?4)=10 2. [...]
[...] Now, substitute P=10 into either the demand or supply function to find the equilibrium quantity. Using the supply function: Qs=3(10)+30=30+30=60 Thus, the equilibrium quantity is Q=60 When the price is P=8 we can calculate both the quantity demanded and the quantity supplied. Qd=120?6(8)=120?48=72 Thus, the quantity demanded is Qd=72 Qs=3(8)+30=24+30=54 Thus, the quantity supplied is Qs=54 The quantity exchanged in the market is the smaller of Qd and which is Qs=54 Since Qd=72and Qs=54, there is a shortage of 72?54=18 units because the quantity demanded exceeds the quantity supplied If the quantity demanded falls by 5 units at every price, the new demand function (Qd2) becomes: Qd2=Qd?5=120?6P?5=115?6P Now, find the new equilibrium price and quantity by equating the new demand function to the supply function: 115?6P=3P+30 Solve for 115?30=6P+3P P=85/9?9.44 Thus, the new equilibrium price is approximately P=9.44. [...]
