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Does
x2
 x sin xdx  2 sin x  C
?
 x sin xdx  x cos x  sin x  C
?
Definition of Antiderivative


A function F(x) is an antiderivative of f(x)
if F’(x)=f(x) for all x in the domain of f.
Symbolically:
 f (x)dx  F(x)  C 
F '(x)  f (x)
Does
2
tan

sec
 d 

sec 3 
(a)
C
3
1 2
(b)
tan   C
2
1 2
(c)
sec   C
2
?
Yes!
2
tan

sec
 d 

1 2
(b)
tan   C
2
1 2
(c)
sec   C
2
Because
1 2
tan   C
2
&
Differ by a constant!
1 2
sec   C
2
Review connection to
Derivatives


Derivative & Integral Formulas:
http://www.math.vt.edu/people/hagen
Examples:

Given v(t) = 9.8t + 5 & s(0) = 10, Find
the distance function.
Examples

Find the antiderivative F(t) of
f(t)=2t-3sint when F(0)=5
Examples

Find the antiderivative F(u) of
u  u
f (u) 
u
2
when F(1) = 3.


Given the graph of f’(x): Sketch the graph of f
if f is continuous and f(-3) = -6
Distance: How do we measure distance
traveled?

Distance = velocity x time
Estimate distance traveled
Time
(sec)
Velocity
(ft/sec)
0
20
1
2
3
4
5
30 38
44
48
50
More Data Points
Time
(sec)
0
Vel
(ft/s)
20
0.5
1 1.5
26 30
2 2.5
35 38
3 3.5
4 4.5
5
42 44 46 48 49 50

Joanna was riding her bike when she
suddenly hit her brakes (and stopped in
12 seconds)
Time
(sec)
Velocity
(ft/sec)
0
3
4
7
10
12
88
60
40
25
10
0
Can use this idea to approximate area between function
and x -axis
y = x^2 on [1,3]
Summation Notation


Let x1, x2, x3,…, xn denote a set of n numbers. x1 is
the first number in the set. xi is represents the ith
number in the set.
The summation sign, the symbol,∑, which is the
Greek upper case letter, S. The summation sign, ∑,
instructs us to sum the elements of a sequence.
Summation Notation
Summation Notation

Here are a couple of formulas. Note that
they all start with one, but could start
anywhere.
1.
where c is any number.

2.

Summation Notation
1
1
1
1
1
1
 1  i 2  1  12  1  22  1  32  1  4 2  1  52
i 1
5
3

i 0
2i  1  1 3  5  7