# Interpretations and Applications of the Derivative and the Definite Integral. d dx [R(i)] = R '(t)

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1 Interpretations and Applications of the Derivative and the Definite Integral [A.MOUNT]= The rate at which that amount is changing dx For example. if water is being drained from a swimming pool and R(t) represents the amount of water. measured in cubic feet, that is in a swimming pool at any given time. measured in hours. then R'(t) would represent the rate at which the amount of water is changing. d dx [R(i)] = R '(t) What would the units of R'(r)be?,c:... er ht)ij(" b f RATE= A.MOUNT OF CHANGE a b ln the context of the ex le situation above, explain what this value represents: J R'(t)dt = R(b)- R(a). (. - Rc:o..") = S o- R 1 C ')J.*- o.w.ov "\) 4 I,>'\. wo.:*:-e.v- l ia,,_,.-.l. * : 4,- =- b. The table gi, en below represents the velocity of a particle at gi, en values of 1. where I is measure in minutes. I.,.... I 5 1 I. 2 _)?- 3o' minutes,1,l, '(t) ft/minute 3 a. Approximate the l'alue of J r:;: ;;; ;' s.,. v(t)dt using a midpoint Riemann Sum. Using correct units of measure, 11,(,. i) -.u.1-l + \ ( 1.") \" 3 rjo \I(. IIA4P t) f>os.1t1<21'. { o.rlnc.k --\;=- 'Tl> t= 3 l,.)te.s.. p -,-_:, b. What is the value of J a(1)d1. and using correct units. explain what this value represents. s o\!)' l )ci.. : 5 \J('l.5)-"(s) = \. - \.G:. =\ -ttlm,u"wt e- \ sfs l-t)clt --4 t"e.\c -\'t...t p c..l. tr'::' t -:. 5-1:: ':l.s" WM.U.

2 The temperature of water in a tub ru timer is modeled by a strictly increasing.. twice differentiable function, II'. where TJ'(r) is measured in degrees Fahrenheit and r is measured in minutes. t (minutes} W(t) (degrees Fahrenheit) l Using the data in the table. estimate the value of Qr'( Using correct units. interpret the meaning of - this, alue in the context of this problem. ""'(, c.,s)- wc.c() - \5- ' <.7.'f-,.g_,.. l'lce.vj 1 (1i) >, +, -t '\ -t -:.. I 5 YM>'\l.ltes, o - I. o I 7 b 1, lt\.c.r ij,. ', I,.,, Use the data in the table to evaluate f v (t)dt. Using correct units. interpret the meaning of this integral : in the context of this problem. s: 1(. J.:b = vj (,o)-v..i (o') = 71 - SS : \ I (o DP:_\ J ; -..i'(:t")6--t "':"'f '- Cl.>MovJt 'b =t' 1...1"\.. I - - t -:..C +t> t::. :). '"' e.t. I r f.t. For 2 Sr< 25. the function ll'that models the water temperature has a first derivative gi, en by the functio o.+ji cos(.61). Based on this model. what is the temperature of the water at time 1=15? 'i j,4j b (O. Ob-t.)it " '-""' D43 (2.5)-wC ) - \tj ('J.S) - 71 \""('ls):: '73.43 OF)...

3 A pan of bisc is removed from an oven at which point in time. r =, the temperature of the biscuits is I ()()(;D The rate at which the temperature of the biscuits is changing is modeled by the function 8'(1) 13.84e --. r ) Find the value of 8'(3). Using correct units. explain the meaning of this value in the conte.xt of the problem. 'B \ Cf) - 8. '2.3 C.O C... pv- v+4!.. S"'CA. 2> 1 (-;)c:. o, --re.,.., f 1:>.OL i "C 3 M(l"\vtt. bt.'.t '5 Q..._. Sketch the graph of 8'(1)on the axes below. Explain in the context oftbe problem why the graph makes At time t = I. what is the temperature of the biscuits? Show your work.. J'l) -,,.att- o. \i3-t d..-4: = B(1) - '6(')... o ; _ _, <a(,o)- \

4 A cylindrical can of radius IO millime1ers is used io measure rainfall in Stormville. During the first 5 days of a 6-dav period. 3 millimeters of rainfall bad been collected. The height of water in the can is modeled by!he function. S. where S(t) 1s measured in millime1ers and I is measured in days for 5:;: 1 6. The rate at which the height of the wa1er is rising is given by the function S '(1) = 2sin(.3t) Find the value of f S'(1)dt. Using correc1 units. explain the meaning of this value in the comext of this J1 problem. \It, S S,t'\(.3 +1.Sdwt: \\.ISO\ M\ me- <.<:. 1 S. " s 1 )clt rd o..-(:i.v.lc.=- lo ht ::..ls,. b'o J "b cj,...,tf-- i..io..t At the end of the 6-day period. wha1 is the volume of water lbat had accumulated in the can? Show your work. S St\"\L,!>-4::) \.S c:l:e : t \b 1o.s 5 s \J-:. ii<" : ii(1o) i (11o<o.sb -=llb J t-s<o.srr w i<-wu:'ffl e,! The rate at which people enter an auditorium for a concert is modeled by the function R given by R(t) = 138, 2-675i 3 for O 1 2 hours: R(t) is measured in people per hour. V.I.P. tickets were sold to 1 people who are already in the auditorium when the when the doors open at 1 = for general admission ticket holders to emer. The doors close and the conce r t begins at t = 2. lf all of the V.I.P. ticket holders s1ayed for the start oflbe concert, how many people are in the auditorium "hen concert begins? Jo 1 8-t '). - '15.\:. '\ d;t; = go f \c t\

5 Date Day #53 Homework - Class - - At time 1 =. there are 12 pounds of sand in a conical tank. Sand is being added 1 the tank at the rate of 2e sin 2 ' + 2 pounds per hour. Sand from the tank is used at a rate 5 sin 2 c + 3./i per hour. The tank can hold a maicimum of2 pounds of sand. ').. I. Find the value of J:s(1)d1. Using correct units. what does this value represent? ) S l )c- -:= ;l. \. l '1 po 1) e. cu:i -to '1.f'\ \fn...\ Find the value of J: R(c)dc. Using correct units. what does this value represent? j Rl )d..-t = 14. "1 povvia.o 1 \$ \ \,+,, t ;;3. 3. Find the value oft J:s(i)dt. Using correct units, what does this value represent?..l.. It S(.t) :t : S. 2,q 3 po. UJ.A.r...J.. ci.urio '\, &(_ _J \) A \) \ :';:)-4 \ - \51,r,.,-t:.l-tbi.,- -1. Write a function. A(t), containing an integral expression that represents the amoun1 of sand in the tank at any given time. l. t :t Ali') = l ic + j S( <l-t -j Rlt>d:t I:) 5. Ho Al):: un :\. arejj : r : = ( d. 1 : : \I. 4U) p,orj.s\ 6. After time c = 7. sand is not used any more. Sand is. however. added until the tank is full. lf k represents the, alue of c at which the tank is at malcimum capacity. write. but do not solve. an equation using an integral e>..-pression 1 find how many hours it will take before the tank is completely full of sand. K \ c S, Sl o..t = oo

6 25 AP CALCULUS AB Problem #2 The tide re sand from Sandy Point Beach at a rate modeled by the function R, given by 2 +5sin( )- A pumping station sand 1 the beach at a rate modeled by the function S, gi\ en by 'j 2. 15t S(r) = 31 I + Both R(t) and S(r) have units of cubic yards per hour and I is measured in hours for O \$; t S 6. At lime t =. the beach contains 25 cubic yards of sand. (a) How much sand will the Lide remove from the beach during this 6-hour period? Indicate units of measure. (b) Write an expression for Y(r). the total number of cubic yards of sand on the beach at time t. (c) Pind the rate at which the total amount of sand on the beach is changing at time t = 4. (d) For K- at what time t is the amount of sand on the beach a roi'lli!!!!? What is the minimum,'aloe? Justify your 5: RC. 6..t l \. \ Co eu.t>ic. '(( ) =- soo -t s: \$t tk.t -s: ( 't 1 t St-t) - C:1&) '{ '( 4\-J -:.. SC. 4U - ( 4) \- \. C\ o q c.u'bic. E '('(. S(t')- R( )::.le: 5. I \8 s"" Y(o') 2.soc cs. 11, "' '{ t S. l\ i") -: 2. sec + J S( *' - '(("): 2.'t93..(.77 ( :{-; 2'+ t. 3L. 'l E:.\J."T t""' v",\l)vv\ t,t soj.., :2.'i-q.3"9 PourJ..s I v.>iu b b t = S hovrs.

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