4.1 Antiderivatives

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4.1 Antiderivatives
Invert the problem given the derivative of a
function, find the function itself. Notation
the derivative is denoted as f the
antiderivative as F so that Ff.
Definition The antiderivative of a function f is
a function F such that Ff. Note
Antiderivative is not unique! Example Show that
F1x31 and F2x32 are both antiderivatives of f
3x2. Solution Differentiate the functions
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Theorem If F(x) is an atiderivative if a
function f(x), then any function F(x)C, where C
is any constant, is also an antiderivative of the
same function. Note the constant C is called
the constant of integration. Antiderivative of
Power Function Remember differentiation
does not act on a constant multiplier. Thus,
antidifferentiation does not act on a constant
multiplier either! Exercise Find the
antiderivative of , where k is a
constant.
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Another very important rule Antiderivative of a
sum is a sum of antiderivatives. (based on the
analogous property of the derivative) Exercises
Find the antiderivatives of the following
functions
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Homework Section 4.1 1,3,5,7,9,11,13,15.
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4.2 The Area Problem
Agenda calculate the area under a curve bounded
by the x-axis, and vertical lines xa and
xb. Approach approximate this region by many
vertical rectangles.
Then, the area is approximated by the sum of
areas of these rectangles
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To make the approximation better, we need to
increase the number of the rectangles (and
simultaneously decrease their bases). So, the
area is the limit of the above sum as n
approaches infinity The middle expression is
called the definite integral of f from a to b f
is called the integrand a and b are limits of
integration dx shows that x is the variable of
integration.
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