Class notes APPM 1350 Unit 3.7 and Appendix B

3.7 Antiderivatives
De a function I is is an antiderivative off on an intercal
I if Flas) = f(s) for cell x of I.
Geumplei
fad = 2x has untiderivatives Fras=x²
2
F₂(x) = x²+1
52885=821,008736.2
Thevarm. If I is an antiderivative of for on interved I, then the
most general untiderivatie of f on I is fast c where
C is an arbitrary constant.

t
Examples: Find most general contideriventive
st curi
FastC
FC85862
a) fexs = (x)
F(x) = sin(x)+(
6) tas = x²
of fas=
Fas = 3x² +C
6
f(x) = x"
‫اہمی‬
=
y(x) = x² sincx Fus= = x² + cos(x) + C
Bash Antiderivative Rodesi
Function
Particular Antiderivative
if(x)
iFしょう
f(x)=(x) Fuse Gue
X", nxt xntl
・
ин
-caxes
corcas sincxs etc....




हिरे fi

,3.7 Cont
Grumple
1
Find hxxs if h'(x) = 7cos(x) +√x and h(0) = 4
W(x) = 75(x)+ 2x² +C
2
W(x) = Zinc+ 33 x ² ² +4.
4 = 7 sin (0) + 3 = √(6)² = 4/6
4=C
Ex A store is thrown straight up over the edge of a building that
is 320 ft tall at
2
gravity is -32 ft/sec²
un
initial velocity of lofther. Earth's
2) Determine the acceleraton, velouty, and position funders of
this store at t secondly offer it has been thrown
u(t) = -32 ft/x²
2
sc+)=-16+=+=+16++c,
320=-16(0) +16(0)+C
✓(4)=-32++C
16=-3260)+(
16 =c vct)=-32+ #16 ft/sec
s(+) = -16+3 +16+ + 320 ft athabt tc
b) What is the maximum height of the store
sch
320
-16
Verter: += = = = =½ seconds
2(-16)2
Max height s($) = -16(=)²+16($) +320
-4 +8 + 320
= 324 fect