Keywords: calculus; Marginal analysis; Elasticity; Cost; Income; Profit; Maximum value; Minimum value?
The application of 1 derivative in economic analysis?
1. 1 The application of marginal analysis in economic analysis?
1. 1. 1 marginal demand and marginal supply?
Let the demand function Q=f(p) be derivable at point p (where q is demand and p is commodity price), what is its marginal function q? =f？ (p) It is called marginal demand function, which is called marginal demand for short. Similarly, if the supply function Q=Q(P) is differentiable (where q is the supply quantity and p is the commodity price), its marginal function Q=Q(p) is called marginal supply function, which is called marginal supply for short. ?
1. 1.2 marginal cost function?
Total cost function C=C(Q)=C? 0+C？ 1(Q)； Average cost function = (q) = c (q) q; Marginal cost function c? =C？ (ask). c？ (Q？ 0) when the output is q? 0, its economic significance is: when the output reaches Q? 0, if one unit product is added or reduced, the cost will increase or decrease by c? '？ (Q？ 0) units. ?
1. 1.3 marginal revenue function?
Total income function r = r (q); Average income function = (q); Marginal income function r' = r' (q).
R'(Q？ 0) Called when the sales volume of goods is Q? The marginal revenue is 0. Its economic significance is: when the sales volume reaches Q? 0, increase or decrease a unit of products, income will increase or decrease r?' (Q？ 0) units. ?
1. 1.4 marginal profit function?
Profit function l = l (q) = r (q)-c (q); Average profit function; =(Q) Marginal profit function L'=L'(Q)=R'(Q)-C'(Q). L'(Q？ 0) when the output is q? 0, its economic significance is: when the output reaches Q? 0, every increase or decrease of a unit product, the profit will increase or decrease L'(Q? 0) units. ?
Example 1 the total cost of a company producing q (tons) products per month c (thousand yuan) is a function of output q, C(Q)=Q? 2- 10Q+20. If the sales price of each ton of products is 20,000 yuan, the monthly output 10 ton, 15 ton, and the marginal profit of 20 tons are required. ?
Solution: the total income function of producing q tons of products per month is:?
R(Q)=20Q？
L(Q)=R(Q)-C(Q)=20Q-(Q？ 2- 1Q+20)？
=-Q？ 2+30Q-20？
L'(Q)=(-Q？ 2+30Q-20)'=-2Q+30？
What are the marginal profits of monthly output 10 ton, 15 ton and 20 ton respectively?
L' (10) =-2×10+30 =10 (thousand yuan/ton); ?
L' (15) =-2×15+30 = 0 (thousand yuan/ton); ?
L'(20)=-2×20+30=- 10 (thousand yuan/ton); ?
The above results show that when the monthly output is 65,438+00 tons, if the output increases by 65,438+00 tons, the profit will increase by 65,438+00,000 yuan; When the monthly output is 15 tons, the output will increase by 1 ton, but the profit will not increase; When the monthly output is 20 tons, if the output increases by 1 ton, the profit will decrease by 1 ten thousand yuan. ?
Obviously, enterprises can't increase profits by increasing output, so what kind of output should be maintained to maximize profits? ?
1.2 the application of elasticity in economic analysis?
1.2. 1 elastic function?
Let function y=f(x) be derivable at point X, the ratio of the relative change of function δYY = f(X+δX)-f(X)y to the relative change of independent variable δxx, and the limit of δX→0 is called the relative change rate of function y=f(x) at point X, or called elastic function. Remember EyEx? EyEx=？ lim？ δx→0
? Δ? yy？ Δ? xx=？ lim？ δx→0？ Δ? y？ Δ? x.xy=f'(x)xf(x)
At point x=x? 0, elastic function value Ef(x? 0)Ex=f'(x？ 0)xf(x？ 0) Is it called f(x) at point x=x? When the elasticity value is 0, it is called elasticity. EE？ xf(x？ 0)% means that at point x=x? 0, when x changes 1%, f(x) approximately changes EE? xf(x？ 0)%。 ?
1.2.2 demand elasticity?
In economics, the relative change rate of demand to price is called demand elasticity. ?
With regard to the demand function Q=f(P) (or P=P(Q)), since the demand function Q=f(p) (or P=P(Q)) of commodities is a monotonic decreasing function, and δ p and δ q are different symbols, the elasticity function of demand to price is specially defined as η (p) =-f' (p).
Example 2 Let the demand function of a commodity be Q=e-p5? , find (1) demand elasticity function; (2) elasticity of demand when 2)p = 3, p = 5 and p = 6. ?
Solution: (1) η (p) =-f' (p) pf (p) =-(-15) e-P5? . pe-p5？ = p5？
(2)η(3)=35=0.6; η(5)=55= 1; η(6)=65= 1.2?
η(3)= 0.6 & lt； 1, which means that when P=3, the price rises by 1%, and the demand only drops by 0.6%, and the range of demand change is smaller than that of price change. ?
η(5)= 1, indicating that when P=5, the price increases 1%, the demand decreases 1%, and the price and demand change in the same range. η(6)= 1.2 & gt； 1, which means that when P=6, the price will increase by 1% and the demand will decrease by 1.2%, and the range of demand change is greater than that of price change. ?
1.2.3 Income elasticity?
Income r is the product of commodity price p and sales volume q, that is?
R=PQ=Pf(p)？
r ' = f(p)+pf '(p)= f(p)( 1+f '(p)pf(p))= f(p)( 1-η)？
Therefore, the income elasticity is erep = r' (p). PR(p)= f(p)( 1-η)PPF(p)= 1-η。
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Thus, the relationship between income elasticity and demand elasticity is deduced: at any price level, the sum of income elasticity and demand elasticity is equal to 1. ?
(1) If η; 0 price increase (or decrease) 1%, income increase (or decrease)1-η)%; ?
(2) if η >; 1, and then Erep
(3) If η= 1, then EREP=0, the price changes 1%, and the income remains unchanged. ?
1.3 the application of maximum and minimum in economic problems?
Optimization is the core of economic management activities, and various optimization problems are also one of the most concerned problems in calculus, such as making the cost lowest, the income most, the profit most and the expense least under certain conditions, and so on. The following introduces some applications of function maximum in economic benefit optimization. ?
1.3. 1 lowest cost problem?
Example 3 suppose that the total cost function of X units of a certain product produced by a factory in each batch is c(x)=mx? 3-nx？ 2+px, (constants m>0, n>0, p>0), (1) How many units are produced in each batch to minimize the average cost? (2) Find the minimum average cost and the corresponding marginal cost. ?
Solution: (1) average cost (X)=C(x)x=mx? 2-nx+p，？ C'? =2mx-n？
Order? C'? , x=n2m, and? c ' '？ (x)= 2m & gt； 0。 Therefore, when n2m units are produced in each batch, the average cost is the smallest. ?
(2)(n2m)=m(n2m)？ 2-n(n2m)+p=(4mp-n？ 24m), and C'(x)=3mx? 2-2nx+p，C'(n2m)=3m(n2m)？ 2-2m(n2m)+p=4mp-n？ 24m So, the minimum average cost is equal to its corresponding marginal cost. ?
1.3.2 Maximum profit problem?
Example 4 assumes that the fixed cost of producing a product is 60,000 yuan, the variable cost is 20 yuan per piece, and the price function is p=60-Q 1000(Q is the sales volume). Assuming a balance between supply and demand, ask what the output is, and the profit will be the largest. What is the maximum profit? ?
Solution: The total cost function of the product C(Q)=60000+20Q?
Revenue function r (q) = pq = (60-q1000) q = 60q-q? 2 1000?
Then the payoff function L(Q)=R(Q)-C(Q)=-Q? 2 1000+40Q-60000？
L'(Q)=- 1500Q+40, let L'(Q)=0 to get Q=20000?
∫L ' '(Q)=- 1500 & lt； When 0∴Q=2000, L is the largest, and L (2000) = 340,000 yuan?
Therefore, when producing 20,000 products, the maximum profit is 340,000 yuan.
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2 The application of integral in economy?
In economic management, the total function (original function) is usually solved by indefinite integral or a definite integral with variable upper limit; If the change of the total function is found in a certain range, the definite integral is used to solve it. ?
Example 5 Let the marginal cost of producing X product be C= 100+2x, and its fixed cost be c? 0= 1000 yuan, and the unit price of the product is designated as 500 yuan. Assuming that the products produced can be sold completely, what is the maximum profit when asking the output? Find out the maximum profit. ?
Solution: What is the total cost function?
C(x)=∫x0( 100+2t)dt+C(0)= 100 x+x？ 2+ 1000?
The total revenue function is R(x)=500x?
Total profit L(x)=R(x)-C(x)=400x-x? 2- 1000, L'=400-2x, let L'=0, x=200, because l' (200)
In this paper, the definite integral is used to analyze that the maximization of profit does not mean that the increase of output will definitely increase profit. Only by reasonably arranging the production volume can the total profit be obtained. ?
To sum up, it is very necessary for business operators to make a quantitative analysis of their economic ties. Using mathematics as an analytical tool can not only provide accurate numerical values for business operators, but also provide new ideas and perspectives for business operators in the process of analysis, which is also a concrete embodiment of the application of mathematics. Therefore, as a qualified business operator, we should master the corresponding mathematical analysis methods, so as to provide a reliable basis for scientific business decisions.