Math 1410 - Test #3A Score: Fall 2009 Name Row 80 Q1: This is a calculator problem. If t, in minutes, is the time since a drug was administered, the concentration, C(t) in ng/ml, of a drug in a patient's bloodstream is given by the surge function C(t) = 30te- O. 04t A. How long does it take for the drug to reach its peak concentration? What is the peak concentration? :<_-:;'_5_,_q_J_fl-"':cJ'---/...:...n..L- _ L.L J. ( L d- Yl" ~~ ) r:. =- b ::= --:04 =.:l6" ~ t'rr g1a--f FI... r '-V-- 0 / B. If the minimum effective concentration is 15 ng/ml and ifthe second dose should be given when the first dose becomes ineffective, when should the second dose be administered. ~ (1'-11.()58 ) IS) (0vJ : Q2: A curve representing the total number of people, P, infected with a virus often has the shape of a logistic curve in the form shown, with time t in weeks. 5000 P=----:-=:-: 1 + 49ge- 1. 7t A. What is the significance of the numerator, 5000? ~ ~,hr.-r '3 ~~~ Oj~~ a- ;rncj.1k-~ v2-- StJoo~. B. How many people were initially infected? C. Find the time at when the rate at which people are becoming infected begins to decrease. What is the value of P at this point? 50DO /-/- /')1 e- I, '7 t; time= -- y(~) -,.1 Ii A. (</99) -~ -J. -q 3. t{; 51 -/. -:rt ;::;
Page 2A of 5 Q3: Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made every two weeks, show the rate, R'(t), at which pollutants are escaping (in tons/week) in the gas: 10 A. Approximate the value of f R'(t)dt o Time (months) 0 2 4 6 8 10 R'(t) 5 7 8 10 13 17 /...s = s?- +- 7-';;L. f-s Z +--/D';:;' -t-j3':2 t<s == ( 5 + T +- &-' +--- /D -1-13). ;;L ==.t/3'.j. -?& (r-r fl f'lp+ 1/ O f}-s =- :-.- 7-'/0 -t-j 3 +- J::" ). ;;J... /1 /9 f.p ~ :: 5S,:;2- ;;2 - I/O B. What does the value you found in A represent? (j Q4: In pharmacology, the bioavailability of a drug is given by the definite integral T Total bioavailability ;:::; 1C(t)dt o where C(t) is the concentration of the drug in the bloodstream measured in ng/ cm in hours. 3 at time t, measured A. What are the units of bioavailability. Ylg/C.Ih ~ - ~ B. Which drug (A or B) has the larger bioavailability? C(t) Briefly s: explain how you determined your answer. ~U/V1t-r~ C(.f) Co' J.-u~ J,-6 VJ {0LttL. L-UY1 u- ~ ~I. -t:o- CJ cvw:l v \.. ~ cur- fb.;t/y ~~ ~~~... JA/r' 4. ~~~y~. B t
Page 3A of 5 QS: A- C refer to the function fex) = 3x 2 + 5 A. Use calculus to find the value of the definite integral. For credit you must show your calculations: (5 J e3x 2 + 5)dx :::: 2 = (J~5+;;l.~:))- ('Is'+-Jt:i) ::=. lso-18 :=: J32.. B. The integral in part A represents the area of a region. Shade-in the region on the graph to the right. fex) C. What is the average value of the function fex) = 3x 2 + 5 on the interval [2,5]7 Q6: A. Shown is the graph of f'ex). Sketch a graph of an antiderivative fct) that satisfies feo) = 5. Label each " y = {(x) critical value offex) with its coordinates. x (5,- 5) B. Set up the integration calculation used to find the area bounded by y = f' ex) and y = 0 for 0 ~ x ~ 5
Page 4A of 5 Find the antiderivatives: 3 ~Ix\ + 8 Vi+c.. Use au - substitution B. Jx 2 (3x 3 - S)4dx =: S('3 )(.'3-'0 ')~, -X'--dx LA-::::: 3x "3-5 ~ S ll4- ~~ du.- ~ 9x'-~ ~ du- = X <.~ -.L. u.. 5""' q 5 "3 \5 =; 1- (3'1: -:)) +-- CJ 11~ Q7: If dy = 5 sin (2x) find the particular antiderivative that satisfies when x = 0, y = 1 dx ~ ::::. j 5 ;J.1/vl (:;2.)<) chl la -' - S Q,e:) (2)( I +u J - ;;;t. ;;;:. 5 SflJfi" (2X) d,y ~ '5 c.m CO} +-0 \ ::2 lj-~q)( s ~ \ 2 du.- = ;;l chl J..ck.--=-eW. 2.- l,. (. l.l' 2-~ ~ S J (.WY1 :: S (_ uy-> '1L-) 2 1-' -= G
Math 1410 - Test #3B Score: Fall 2009 Name Row 80 Q1: This is a calculator problem. If t, in minutes, is the time since a drug was administered, the concentration, C(t) in ngiml, of a drug in a patient's bloodstream is given by the surge function C(t) = 2Ste- O. 02t A. How long does it take for the drug to reach its peak concentration? 5'0 ~ What is the peak concentration? 4 [) q. cr Lf 9 119 / roy) I L I \.,. ) L. = b ~.02.::: :5 0 yy~ C6Y' ~h 1- (nl1gvyyu ~e B. If the minimum effective concentration is 60 ngiml and if the second dose should be given when the first dose becomes ineffective, when should the second dose be administered. Q2: A curve representing the total number of people, P, infected with a virus often has the shape of a logistic curve in the form shown, with time t in weeks. 4000 P=------".. 1 + 49ge-1.7t A. What is the significance of the numerator, 4000? JL- ~~ u3 ~~~ ~ ~;)jrj-j ~11J:vK~ ~ u'j- t-ftjo{). B. How many people were initially infected? f(o):= If ODD,7'OOO;=. &' ~ I +-L/-Cj ~ 50D C. Find the time at when the rate at which people are becoming infected begins to decrease. What is the value of P at this point? :< tj60 time= ---- ~ (-fqq) b -:::; ~/, 1 t.---: /.1 :::; fr 4CJQ 3.& 5'1- tv.qj.-1w t..",..3. f.p s-tf
Page 2B of 5 Q3: Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made every two weeks, show the rate, R'(t), at which pollutants are escaping (in tons/week) in the gas: Time (months) 0 2 4 6 8 10 R'(t) 5 7 10 12 15 18 10 A. Approximate the value of f R'(t)dt o --:L'" L + I"_~.J-IS-::L S'd. -/-,'0<. /10';2... <>< == (5.J.-:}- +-/D +-1"2- +-/5)"). == J../q. ~ :::: '1 '8' == (:j. -rid f' 0;Z, ;)... /,9..1 -Lz..;;Z C).. =1 J I B. What does the value you found in A represent? Q4: In pharmacology, the bioavailability of a drug is given by the definite integral Total bioavailability ::::: itc(t)dt where C(t) is the concentration of the drug in the bloodstream measured in ngjcm 3 at time t, measured in hours. A. What are the units of bioavailability. _-!lfl-.-f-..li---,,~~.:...~_---,~,---,-,,----- B. Which drug (A or B) has the larger bioavailability? Briefly explain how you determined your answer. C(t) (2 (i) de n9/c.rn 3 ~ B t
Page 3B of 5 05: A- C refer to the function [(x) = 3x 2 + 5 A. Use calculus to find the value of the definite integral. For credit you must show your calculations: 5 L(3x 2 + 5)dx ;:: x. 3 +-S"xf,S" ::: (;:;;.s-r;l..s;)-(jf- S ) )~ '-f B. The integral in part A represents the area of a region. Shade-in the region on the graph to the right. C. What is the average value of the function [(x) = 3x 2 + 5 on the interval [1,5]? 5 } _I ~[3X'+-5) ~:= -:q (JLfq) = ~ ~ 5-1 I [(x) ~~\\ 2 3 4 5 Q6: A. Shown is the graph of r (x). Sketch a graph of an y y = [ex) antiderivative J(t) that satisfies f(o) = 5. label each critical value of [(x) with its coordinates. l ~/;).) B. Set up the integration calculation used to find the area bounded by y = f'(x) and y = 0 for 0 :$ x :$ 5 1- ( "3 S5 f 'exl h - f en c0< J\ 3
Page 4B of 5 Q7: Find the antiderivatives: A. == 3j X- M + L\ ~ X- V;z. Jx ~ 3~ I/CI + '-t ~ Y.z.!I,2. Ii:? 3 ~ I )C \ +- ~ X- +u Use a u - substitution B. Jx 2 (3x 3-4)5dx _ SC 3x3 - Ll )S".x2-chc :3 IA.. ::: 3)( -Lf Su- 5.*b-. '2. c!u. = 1-x: 4-1-. ~.L du.- = )L 2. dx OJ 9 to Ie = J- l3 X :3_4) l.> +-G 5.4 Q8: If dy = 7 sin (2x) find the particular antiderivative that satisfies when x = 0, y = 2 dx /).AyL j ~ (--&xj Jx- ;; - ~ CJ:o (-k. x) +-G ()I<. u. ==- :l )( du-=c2-h ~ck.= ~
Math 1410 Test #3A Score: Fall 2009 Name Row 80 Q1: This is a calculator problem. If t, in minutes, is the time since a drug was administered, the concentration, C t in ng ml, of a drug in a patient's bloodstream is given by the surge function C t = 30te 0.04t A. How long does it take for the drug to reach its peak concentration? What is the peak concentration? 8 B. If the minimum effective concentration is 15 ng ml and if the second dose should be given when the first dose becomes ineffective, when should the second dose be administered. Q2: A curve representing the total number of people, P, infected with a virus often has the shape of a logistic curve in the form shown, with time t in weeks. 5000 P = 1 + 499e 1.7t A. What is the significance of the numerator, 5000? 12 B. How many people were initially infected? C. Find the time at when the rate at which people are becoming infected begins to decrease. What is the value of P at this point? time =
Page 2A of 5 Q3: Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made every two weeks, show the rate, R t, at which pollutants are escaping (in tons/week) in the gas: 10 A. Approximate the value of R t dt 0 Time (months) 0 2 4 6 8 10 R t 5 7 8 10 13 17 12 B. What does the value you found in A represent? Q4: In pharmacology, the bioavailability of a drug is given by the definite integral Total bioavailability T C t dt 0 4 where C t is the concentration of the drug in the bloodstream measured in ng cm 3 at time t, measured in hours. A. What are the units of bioavailability. B. Which drug (A or B) has the larger bioavailability? Briefly explain how you determined your answer. C t A B T t
Page 3A of 5 Q5: A C refer to the function f x = 3x 2 + 5 A. Use calculus to find the value of the definite integral. For credit you must show your calculations: 2 5 3x 2 + 5 dx 12 B. The integral in part A represents the area of a region. Shade-in the region on the graph to the right. C. What is the average value of the function f x = 3x 2 + 5 on the interval 2, 5? y f x x Q6: A. Shown is the graph of f (x). Sketch a graph of an antiderivative f(t) that satisfies f 0 = 5. Label each critical value of f x with its coordinates. y y = f(x) 12 y = f (x) x x Area = 6 Area = 7 Area = 9 B. Set up the integration calculation used to find the area bounded by y = f (x) and y = 0 for 0 x 5
Page 4A of 5 Find the antiderivatives: A. 3 x + 4 x dx 14 Use a u substitution B. x 2 3x 3 5 4 dx Q7: If dy dx = 5 sin (2x) find the particular antiderivative that satisfies when x = 0, y = 1 6
Math 1410 Test #3B Score: Fall 2009 Name Row 80 Q1: This is a calculator problem. If t, in minutes, is the time since a drug was administered, the concentration, C t in ng ml, of a drug in a patient's bloodstream is given by the surge function C t = 25te 0.02t A. How long does it take for the drug to reach its peak concentration? What is the peak concentration? 8 B. If the minimum effective concentration is 60 ng ml and if the second dose should be given when the first dose becomes ineffective, when should the second dose be administered. Q2: A curve representing the total number of people, P, infected with a virus often has the shape of a logistic curve in the form shown, with time t in weeks. 4000 P = 1 + 499e 1.7t A. What is the significance of the numerator, 4000? 12 B. How many people were initially infected? C. Find the time at when the rate at which people are becoming infected begins to decrease. What is the value of P at this point? time =
Page 2B of 5 Q3: Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made every two weeks, show the rate, R t, at which pollutants are escaping (in tons/week) in the gas: Time (months) 0 2 4 6 8 10 R t 5 7 10 12 15 18 12 10 A. Approximate the value of R t dt 0 B. What does the value you found in A represent? Q4: In pharmacology, the bioavailability of a drug is given by the definite integral Total bioavailability T C t dt 0 4 where C t is the concentration of the drug in the bloodstream measured in ng cm 3 at time t, measured in hours. A. What are the units of bioavailability. B. Which drug (A or B) has the larger bioavailability? Briefly explain how you determined your answer. C t A B T t
Page 3B of 5 Q5: A C refer to the function f x = 3x 2 + 5 A. Use calculus to find the value of the definite integral. For credit you must show your calculations: 1 5 3x 2 + 5 dx 10 y f x B. The integral in part A represents the area of a region. Shade-in the region on the graph to the right. C. What is the average value of the function f x = 3x 2 + 5 on the interval 1, 5? x Q6: A. Shown is the graph of f (x). Sketch a graph of an antiderivative f(t) that satisfies f 0 = 5. Label each critical value of f x with its coordinates. y y = f(x) 14 y = f (x) x x Area = 6 Area = 7 Area = 9 B. Set up the integration calculation used to find the area bounded by y = f (x) and y = 0 for 0 x 5
Page 4B of 5 Q7: Find the antiderivatives: A. 3 x + 4 x dx 14 Use a u substitution B. x 2 3x 3 4 5 dx Q8: If dy dx = 7 sin (2x) find the particular antiderivative that satisfies when x = 0, y = 2 6