1. Does ? ?

2. 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:

3. Does ?

4. Yes!

5. Because
&
Differ by a constant!

6. Review connection to Derivatives
Derivative & Integral Formulas:

7. Examples:
Given v(t) = 9.8t + 5 & s(0) = 10, Find the distance function.

8. Examples
Find the antiderivative F(t) of
f(t)=2t-3sint when F(0)=5

9. Examples
Find the antiderivative F(u) of
when F(1) = 3.

10. Given the graph of f’(x): Sketch the graph of f
if f is continuous and f(-3) = -6

11. Distance: How do we measure distance traveled?
Distance = velocity x time

12. Estimate distance traveled
Time
(sec) 1 2 3 4 5
Velocity (ft/sec) 20 30 38 44 48 50

13. More Data Points Time (sec) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Vel (ft/s)
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
Vel (ft/s) 20 26 30 35 38 42 44 46 48 49 50

14. Joanna was riding her bike when she suddenly hit her brakes (and stopped in 12 seconds)

15. Can use this idea to approximate area between function and x -axis y = x^2 on [1,3]

16. 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.

17. Summation Notation

18. Summation Notation
Here are a couple of formulas. Note that they all start with one, but could start anywhere.
where c is any number. 2.

19. Summation Notation