1) The document contains examples of using calculus to find equations and functions based on given marginal, cost, profit, or revenue functions. 2) Example problems include finding equations that satisfy given slope and point conditions, finding antiderivatives that satisfy given initial conditions, determining revenue and demand functions from a marginal revenue function, and finding cost and profit functions from marginal cost and profit information. 3) One example is to find the total cost function and cost of producing 50 units given the marginal cost function is c’(x)=0.3x^3 + 2x and fixed cost is 3000.

1. Problem
Application

2. Example 1
Find the equation of the curve that passes
through
a. (0,1) b. (0, -1) c. (0,0)
If its slope is given by = 2x at any point x.

3. SOLUTION:
Finding a function y=f(x) such that = 2x and
y=1 @ x=0
If = 2x, then y= ∫ 2xdx
y= ∫2xdx
y= x2 + c
Since y= 1 when x 0, then we can terminate
the particular value of C

4. a.) 1= 02 + C
C = 1
y = x 2 + 1
C = 1
1

5. C = -1
0
-1
b.) y = -1, x = 0
-1 = (0) 2 + C
y = x 2 -1

6. C = 0
c.) y = 0, x = 0
0 = (0) 2 + C
C = 0

7. Example no. 2
 Find the particular antiderivative that satisfies
the given condition.
= 200t -2 ; A(1) = 300
Solution:
dA = 200t -2 dt
A = -200t -1 + C
A = -200/t + C
A(1) = -200/t, @ A(1) = 300
300 = -200 + C
C = 500 thus,
A = 200/t + 500

8. Example no. 3 (Marginal Analysis)
 If the marginal revenue function is R’ = 12 – 8x + x
2 , determine the
(a)Revenue (b) demand functions
Solution:
a) = (12-8x+x 2)
dR = (12-8x+x 2)dx
R = 12x-8/2x 2 + 1/3x 3 + C = 12x-4x 2 + 1/3x 3 + C
when R = 0, x = 0, C = 0.
Thus, revenue function: R = 12x-4x 2 + 1/3x 3
b) R= xy, y=R/x = 1/x (12x-4x 2 + 1/3x 3 )
demand function = 12x-4x + 8/2x2

9. Example no. 4
 If the marginal cost of producing x units is given
by c’(x) = 0.3x 3 + 2x and the fixed cost is 3000,
find the cost function c(x) and the cost of
producing 50 units.
Solution:
• Marginal cost is the derivative of the cost
function and fixed cost
Is cost at 0 production level, therefore, the
mathematical problem is to find c(x). Given
c(x)= 0.3x3 + 2x, c(0)= 3,000, by indefinite
integral and determining the arbitrary
integration constant using c(0)= 3,000

10. c’(x) = 0.3x3 + 2x
C(x)= ∫ (0.33 + 2x) dx
= 0.075x4 + x2 + k
Since c represents the cost, we use k for the
constant of integration.
But c (0)= 0.075(0)4 + 02 + k= 3000, k=3000
The particular cost function is c(x)= = 0.075x4 +
x2 + 3000
The cost of producing 50 units:
C(50)= 0.075(50)4 + (50)2 + 3000
= 474, 250

11.  Example no. 5
If the marginal cost function for a week for a
certain product is MC= 4x + 100
The fixed cost associated to the product
amounting to P500 per week, find the total cost
for week.
SOLUTION:
Total cost function is C(x)= ∫ c’ (x)dx= ∫ (4x
+100)dx
= 4/2 x2 + 100x + k
= 2x2 + 100x + k
Since c represents the cost, we use k for the
constant integration but 1c(0)= 2(0)2 + 100(0) +
k=500, k= 500

12. Example no. 6
The Unilever Philippines has found out that its
marginal profit in tens of pesos is given by:
dP/ds 30s2 + 200
Determine the profir for 100 salesmen.
*Where s represents the number of salesman.
SOLUTION:
P= ∫ 30 s2 + ∫ 200ds
= 30/3 + 200s + C
= 10s3 + 200s + C, @ s=0, P=0, thus C=0

13. The profit function is P= 10s3 + 200s
@s= 100, P=?
P= 10 (100)3 + 200 (100)
P= 10,000,000 + 20, 000
P= 10, 020, 000 * 10
P= 100, 200, 000