Math 142 Lecture Notes for Section 6.1
Section 6.1 -
Antiderivatives
Example 6.1.1:
List any function whose derivative is x.
Example 6.1.2:
List any function whose derivative is x3 + 5.
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Math 142 Lecture Notes for Section 6.1
2
Definition 6.1.3:
The General antiderivative of f (x) on an interval I is the function F (x) + C where
C stands for any real number, if
Also written as
which is known as the indefinite integral of f (x), where F 0 (x) = f (x).
General Rules of Integration:
Z
(1)
xn dx =
Z
(2)
kdx =
Z
(3)
Z
(4)
ex dx =
1
dx =
x
Z
(5)
[f (x) + g(x)]dx =
Z
(6)
[k · f (x)]dx =
Math 142 Lecture Notes for Section 6.1
Example 6.1.4:
Find the following:
Z
(a)
3dx =
Z
(b)
Z
1 4
x dx =
3
Z
2
dt =
t7
(c)
(d)
Z
(e)
Z
(f)
x5 dx =
(5x3 + 2x2 + 3)dx =
4
dx =
x
3
Math 142 Lecture Notes for Section 6.1
Z
(g)
(ex + x2 )dx =
Z
√
1
(2 x − 3 + x4/7 )dx =
x
Z
4x3 + 6x2
dx =
2x
(h)
(i)
Z
(j)
(x + 4)(x − 3)dx =
Example 6.1.5:
Find y if y(1) = 3 and
dy
6
1
= − 3
dx
x x
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Math 142 Lecture Notes for Section 6.1
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Example 6.1.6:
The daily marginal cost function for the Stark Iron Buddy prototype is given by M R(x) =
2
10 − x3 for 0 ≤ x ≤ 1000, where x represents the number of Iron Buddies produced.
(a) Suppose that R(3) = $120, find the revenue function for Iron Buddies.
(b) Using the information given and the fact that R(x) = p(x)·x give the price function
for Iron Buddies.
(c) What will the price be when the demand is for 300 Iron Buddies?
Section 6.1 Suggested Homework: 1-47 (odd)