5.3: Accumulation Function
2. When A is increasing, f is...
When f is increasing, A is...
When A is increasing, f is POSITIVE
When f is increasing, A is concave up
6. If F is concave up, what do we know about the graph of f?
graph of f decreases
1c. what is the antiderivative?
e^3x + C
6a: write a definite integral that represents the amount dumped in the first b years after 2009.
integral from 0 to b P'(t)dt = integral from 0 to b 300 + 200tdt
what does FTC1 stand for?
Fundamental theorem of calc I
1. find F(4)
F(x) = integral from 0 to 4 h(t)dt = -6
2a: what is K'(x)?
K'(x_ = 8x^3 - 2ln(x)
3a
(3x^3/2)/(3/2) - (5x^3)/3 + 3x + C
6c
t = 8.6
FTC1 shows the relationship between what?
4b. On what interval(s) is A(x) decreasing?
(1.5, 2.5)
4. find the following:
b) F(2)
d) F'(2)
4b: 0.818
4d: -0.166
3d
ln|x| + 1/2ln |x| + C
6b
P(t) = 300t + 100t^2 + 10,000 tons
considering antiderivative formulas, what should you ALWAYS have at the end of your final answers?
(+ C )
this is important to remember as professors will take point
4e. where does A(x) have a relative minimum?
where does A(x) have an inflection point?
relative min. at x = 2.5
3 IPs --> x = 0, 2, 4
3. compute F(6) and F'(6)
F(6): 1,996.35
F'(6): 1,548.91
4a: find specific antiderivative (show work)
general antiderivative: e^2x + x^3 +C
F(0) --> C = 4
FINAL answer (specific antiderivative):
e^2x+x^3+4
5. find specific antiderivative
H(t) = -5t^3 + 60t^2 + 80t +20
what is the formula for an accumulation function?
A(x) = integral from a to x of f(t)dt
5. consider all parts (a, b, c, d, e, f)
a. negative
b. zero
c. decreasing
d. concave up
e. increasing
f. IP
7. parts a - e
a. (1, 3.5)
b. (0, 2.5) and about (4.75, 6)
c. x = 3.5
d. x = 1
e. x = 0, x = 2.5, x = 4.75, x = 6 (4 IPs)
4b - find specific antiderivative
antiderivative: (2x^3)/3 + 3x^2/2 - 2x + C
C = -17
so, FINAL ANSWER specific antiderivative:(2x^3)/3 + (3x^2)/2 - 2x -17
5. find general antiderivative
-5t^3 + 60^2 + 80t + C
refer to #1 (FTC1 packet)
what is the formula for FTC1?
d/dx of the integral a to x f(t)dt = f(x)