## Solving for 'c'

Aim: To determine the value of 'c' in an indefinite integral.

To do this, we usually will have to rely on extra information. Here are some examples.

1. Find function y given that dy/dx = 3x2 and the curve y passes through the point (2,13).
Solution:
y = ∫ 3x2 dx
y = x3+ c ; now to find the y we have to use the given point (2,13).
13 = 23+ c  ------------------[substitute 2 into x and 13 into y ].
13 = 8 + c
c = 5.
y = x3+ 5 is the answer.
2.

3. A toy-plane is moving in a straight line with a velocity of 10 m/s at t=3 seconds. It's acceleration is modelled by a = 5 - 6t2. Find the equation of its velocity.
Solution:
v = ∫ 5 - 6t2 dt
v = 5t - 2t3+ c ; now to find the y we have to use the given point (3,10).
10 = 5(3) - 2(3)3+ c
10 = 15 - 54 + c
c = 49.
v = 5t - 2t3+ 49 is the answer.
4.

5. The marginal cost of producing a table is dC/dq = 6q -30 where q is the number of table produced. If no table is produced then the producer will still have cost amounting to $250. Find the equation C. Solution: C = ∫ 6q-30 dq C = (6/2)q2- 30q + c ; now to find the c we have to use the given fact that C=250 when q=0. 250 = 3(0)2- 30(0) + c c = 250 C = 3q2- 30q + 250 is the answer. [In economics, the c=$250 is called the fixed costs.]