## Marginal Analysis

### 1. The total weekly cost (in dollars) incurred by a recording company in
pressing \(x\) CDs is given by the following function \(C(x)=2000+2x-0.0001x^2, \quad 0
\le x \le 6000 \). (a) What is the actual cost incurred in producing the 821st
disc? (b) What is the marginal cost when \(x=820\)? solution

### 2. The cost function for manufacturing \( x \) units of a product is given
by \( C(x) = 0.0002x^3 − 0.07x^2 + 120x + 3000\), where \( C(x) \) is the
total cost in dollars. (a) Find the marginal cost function. (b) Evaluate
\(C' (100)\). (c) Interpret the result of part (b).
solution

### 3. The cost function for making \( x \) units of a product is given by \(
C(x) = 4000 + 3x \), where \(C(x)\) is the total cost in dollars. (a) Find the average cost function \(\bar{C(x)}\). (b) Find the marginal average cost function.
solution

### 4. The total cost function \(C(x)\) for producing some CDs is given by
\(C(x)=1,600+3x-0.0002x^2\). (a) Find the average cost function. (b) Find the marginal average cost function. solution

### 5. The demand function for a certain product is given by \( p = −0.025x +
900 (0 ≤ x ≤ 40000) \) where \( p\) is the unit price in
dollars and \(x\) is the quantity demanded. (a) Find the revenue function \(R\). (b) Find the marginal revenue function. (c) Find and interpret \( R' (5000)\). (d) If the total cost function for the production is given by \(C(x) =
200x + 280000\), then (i) find the profit function \(P\). (ii)
find the marginal profit function, (iii) find and interpret \(P'(5000)\).
solution

### 6. The management of Acrosonic plans to market the Electrostat, an
electrostatic speaker system. The marketing department has determined that the
revenue can be calculated from the unit price in dollars: \( p = −.04x + 800\)
and the cost of manufacturing \(x\) ElectroStat speaker systems is: \(C(x) =
200x + 300,000\). (a) Find the revenue function. (b) Find the profit function. (c) Find the marginal revenue function. (d) Find the marginal profit function. (e) Evaluate P’(5000). (f) ) Interpret your solution in part (e).
solution