Curriculum Module: Calculus: Fundamental Theorem
Worksheet 2, Applying the Second Fundamental Theorem of Calculus: Finding Derivatives of Functions Defined bY an Integral Find F'(x)when:
1.
F(x) =
F(x): tisat
f,(i-- t J.
F'(r)=
F(x) = [rzt at
4.
F' ( x)-- 2x
s.
F(x)= [ifz-2,>a,
u.
,@)= Iie-2t)dt ?'&)-- 2'Lx
F(x)= JUllsrr,rdt
g.
,r.
F '(x)= s' ^
= J0lt* zt dt ('(t) ')itx)
F(x)
F(r) =
f
= Jrf* sint dt
F(x)
F'(Y)' 5'^x g.
3
F(x)= [iztat €'(Y\'LY
F'(x) 'L-vY 7.
[itat
l-
10.
F(x)
2r dt
F'(.x)= /5irr^x co.:x
l8/
f{r)a,
n.
f'(r). I (r\
rsinx
= JUl^
'
F(*): ft'' f@dt F'(x)=
| (gtr)) 5'(r)
function? ''l"i'r 'te{; ol h1 d-cr,,61h,6 (rrn^-it,ri^,.i
between the integrand function and the integral What is the relationship'{w,^Ch,"^
tnk3,rt,,c4
"ti)i-l'
;;"''J))')'"o+'!o
'
a,Aa ;'*ri^',p,'l)^
%
{r)
Maria pfirezRandle, Bishop Kenny High school, ]acksonville, Florida
13
,
**'''}"vc'"6'{ {wn(h :n :{ r^lrOr= Change
in Ffrom t = a to t -- b = F(b)- F(a)'
Exercises questions' Write a sentence to answer each of the following
l.
Tt
h(t) is the
hour, what
pile of sand measured in feet per r-aIe-qf.change of the height of a conical
do".
Jt
'li,s 6lt'e*E: 2.
If v(f)
h(t)
dt
represent? Answer in correct units'
* i*-;
'
;''
p'L\' l'"
'{'{* f t':i 'l
t-r{p':
the x-axis' measured in feet per is the velocity of a particle moving along
'"'"fi*Tl]:i;. {i";lfl.'"i'"i"ii.}"rH'-,'L:"i'i'?I'i' r"*' 3,
r 3 :'ei {"
lfb(t)istherylgg!ryglhofthenumberofbacteriainadish,measuredinnrrmber of bacteria per hour, what
#vt-r
rJ'"'O-"*-\r
does Iru:,rr^*;::;.ese1t?
ui\ Srtw+i'I 15
correct units.
l"T.'i"
lr'' (('(
Theorem Curriculum Module: Calculus: Fundamental
. " alongthe'-Tt attime t i:1]T:"",:l::: moving of a' particle YqL.-v'- -"reDresents the position of 'itt**itn that represe expression an integral a) write . .r . ,!-r ""----t'hatgivesthetotal W\4'l at time t lO , an{ *":." the particle Q) *,time / : 1 0 ' ov tr" particle from time :, ' ^2. is the
i
il;#;';;;i;a
o) 5* J* v(c)at J.' o ,U'r'rs
* ]"]:-*.t:it:l::
L)
,
'' I'tr)lJ+ _f 2
the rate of growth of a rabbit p*11tt::::1':-1T,'3:il.l'[1i"T;
integral expression that in the year zaal (r = 0), write an in2007 ' represents the rabbit population
ffy;il'i"#ffi, nT
loo r J rttle+ o
High School, Jac}sonville, Florida Marra p6rez Randre, Bishop Kenny
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t/
Worksheet 4. Example Multiple-Choice Ouestions #23. (zoa3AB Exam, Section I, Part A, non-calculator section).
(a)
(c) sin(x6)
(b) sin(x3)
-cos(x6)
*(t:
(d)
r*(')r) = s,. (g')') " 2,
2xsin(x3)
6;'*Cl
The following questions are from the 2003 AB Exam, Section I, Part B, calculator section. #82. The rate of change of the altitude of a hot air balloon is given by:
,U):t'
-4tz +6
l'l-.1 t1 I o < t
