Solution manual for calculus for the life sciences 2nd edition by adler

Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at Chapter Introduction to Models and Functions 1.3 Variables, Parameters, and Functions 1.3.1 The variables are the altitude and the wombat density, which we can call respectively The parameter is the rainfall, which we can call R 1.3.2 The variables are the altitude and the bandicoot density, which we can call respectively The parameter is the wombat density which we can call W 1.3.3 The graph is the horizontal line crossing the nor decreasing -axis at -4.2 and and b, is neither increasing y x -4.2 f (x) = -4.2 1.3.4 The graph is the line of slope whose numbers y -intercept is -6 is increasing for all real f (x) = 3x-6 x -6 1.3.5 The graph is a hyperbola; we draw it by plotting points and joining them with a smooth curve (in the next section we will learn how to transform the graph of to obtain this graph) The function is decreasing on and on y y = 2/x +4 01 x 1.3.6 The graph is a cube root parabola; we draw it by plotting points and joining them with a is increasing for all real numbers smooth curve The function Full file at Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Ins.aanual nkDtoiraccompany ect.eu/ Calculus for the Life Sciences, Second Canadian Edition y f (x) = x -1 x 01 -2 1.3.7 The graph is the hyperbola and decreasing on shifted one unit upward is increasing on y f (x) = 1/x2 +1 -1 x 1.3.8 The graph is V-shaped; it is the graph of the absolute value function moved units to the right is increasing on and decreasing on y f (x) = | x - | 1.3.9 Full file at x Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at Chapter 1.3.10 15 y 10 0 x 1.3.11 20 y 15 10 0 x 1.3.12 30 25 y 20 15 10 0 x 1.3.13 1.3.14 ⎛ c ⎞ ⎛ 5⎞ c 1.3.15 h ⎜ ⎟ = ,h ⎜ ⎟ = ,h(c + 1) = 5c + ⎝ ⎠ c ⎝ c ⎠ 25 Copyright © 2015 Nelson Education Ltd Full file at Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Ins.aanual nkDtoiraccompany ect.eu/ Calculus for the Life Sciences, Second Canadian Edition 1.3.16 1.3.17 is a polynomial, so its domain consists of all real numbers In symbols, the domain is
= (−∞,∞) 1.3.18 From we get { The domain consists of all real numbers that are not } equal to The domain is x ∈
x ≠ ; or, we can say that it consists of two intervals, and 1.3.19 The equation has no real number solutions, and so the denominator of is never zero The domain of consists of all real numbers In symbols, the domain is
= (−∞,∞) 1.3.20 From { } The domain is x ∈
x ≥ ; using intervals, we get 1.3.21 is defined for all , and is defined when all non-zero real numbers Using set notation, and Thus, the domain consists of ; using interval notation, 1.3.22 From we get In short, we write and Thus, the domain of is 1.3.23 The requirements (because of the fraction) and (because of the square root) imply Thus, the domain consists of all real numbers such that Using interval notation, we write the domain as 1.3.24 There are no restrictions coming from the linear function The square root is defined when , and therefore the domain of consists of all real numbers y x 2x+1 Full file at 01 x Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at Chapter 1.3.25 There are no restrictions on the domain coming from the square root, since it is computed of the numbers that are greater than or equal to The only restriction is the denominator in , and thus the domain is , or, in short, y 1/x 1/4 x x 1/x 1.3.26 There are no restrictions on the domain coming from the square root, since it is computed of the numbers that are greater than or equal to The remaining pieces are linear functions, which are always defined The domain is the set of all real numbers y x x -1 x -1 -2 1.3.27 Both pieces are quadratic functions, so the domain of is the set of all real numbers The graph of is the parabola for positive values of , and its reflection across the -axis for negative values of and for y x2 x -x2 1.3.28 The graph of is the horizontal line that crosses the -axis at range consists of the single real number Using set notation, we write Thus, the , or for short 1.3.29 The graph is the line of slope going through the origin Thus, the range is the set of all real numbers To confirm this fact algebraically, we take any real number and find an such that From we get Copyright © 2015 Nelson Education Ltd Full file at Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Ins.aanual nkDtoiraccompany ect.eu/ Calculus for the Life Sciences, Second Canadian Edition 1.3.30 Squaring a number we obtain a positive number, or zero Thus, we believe that the range of consists of all numbers To confirm this fact algebraically, we take any real number and find a number and such that From we get (note that the quantity under the square root is positive or zero) 1.3.31 The graph suggests that the range is and find a number such that To prove it we take any real number From we get and (note that the quantity under the square root is positive or zero) Thus, any number greater than or equal to can be obtained by applying the function to two values of and f (x) = x2+3 y , x 1.3.32 The graph suggests that the range is Alternatively, we ask ourselves: what numbers can we get as a result of subtracting from 3? Since expression cannot be larger than is positive or zero, the To prove this fact algebraically we take any real number and find a number such that From we get that and (note that the quantity under the square root is positive or zero) Thus, any number less than or equal to can be obtained by applying the function to two values of , and y f (x) = -x2+3 x 1.3.33 Because the range of take any From is , the range of we compute is the same Formally: 1.3.34 The range of the square root function consists of zero and all positive numbers, and that’s our guess To prove it, we take any real number and find a number such that Full file at Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at Chapter From we get that and Thus, any number larger than or equal to zero can be obtained by applying the function to ; in other words, the range of is 1.3.35 Because and the cube root of a positive number is positive (and the cube root of zero is zero), the range of consists of zero and positive numbers To confirm: we take any real number and find a number such that From we get and 1.3.36 1.3.37 80 Length 60 40 20 Cooper’s Goshawk Sharp-shinned Bird 
.38 The cell volume is generally increasing but decreases during part of its cycle The cell might get smaller when it gets ready to divide or during the night 
.39 The fish population is steadily declining between 1950 and 1990 
.40 Initially, the height is about metre, then increases until about age 30 when the trees reach the approximate height of metres; after that, the height decreases 
.41 The stock increases sharply, then crashes (falls to a bit below its original value), then increases sharply again, crashes to an even lower value than before; around day 12 the stock increases again (bit less sharply than previously), crashes to about its initial value, and levels out Copyright © 2015 Nelson Education Ltd Full file at Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Ins.aanual nkDtoiraccompany ect.eu/ Calculus for the Life Sciences, Second Canadian Edition .42 Population initial

intermediate tiny start
tiny levels out Time
.43 DNA maximum initial start plateau decline crash Time
.44 Temperature maximum average temp minimum day night day Time night
.45 Wetness wet dry dawn noon evening midnight dawn Time
.46 If if if plant even if there are no flowers Full file at Perhaps one bee will check out the Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at Chapter Number of bees 50 40 30 20 10 0 10 15 20 Number of flowers 25 Number of cancerous cells
.47 If r = 0,c = ; if r = 4,c = ; if r = 6,c = ; if It looks like these cells can tolerate up to rad before they begin to become cancerous 0 Radiation 10
.48 If if if if The insect develops most quickly (in the shortest time) at the highest temperatures Development time 40 30 20 10 10 20 30 Temperature 40
.49 If a = 0,h = ; if a = 100,h = 50 ; if a = 500,h = 83.3 ; if a = 1000,h = 90.9 ; if It looks like it would reach 100 m 100 Height 80 60 40 20 0 200 400 600 Age 800 Copyright © 2015 Nelson Education Ltd Full file at 1000 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Ins.aanual nkDtoiraccompany ect.eu/ Calculus for the Life Sciences, Second Canadian Edition 1.3.50 The function increases rapidly until , where its value raises above 15 Then it decreases until , reaching the value slightly below After remaining around for some time, it starts decreasing sharply at It reaches its lowest value of about when and then starts increasing rapidly f (x) 15 10 −5 −10 −15 0.5 1.5 2.5 3.5 4.5 x 1.3.51 The function reaches its lowest value of approximately when As grows larger and larger, or smaller and smaller, the function approaches Near the origin it drops sharply from values between and to below 0, and then rises sharply back to the values above The graph is decreasing for negative and increasing for positive It seems to be symmetric with respect to the -axis g (x) −1 −6 −4 −2 x 1.3.52 The graph of the function consists of four curves (although they might look like it, they are not straight lines) The function starts at when , then decreases until , reaching its lowest point Between and the function is increasing, reaching its initial value of again when The graph seems to be symmetric with respect to the vertical line h(x) 3.2 2.8 2.6 2.4 2.2 1.8 10 Full file at x Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition Thus, to graph move the resulting graph y -1/2 , we shift the graph of units up ½ units to the left and then f (x) = x2+x+6 23/4 x 1.4.58 Completing the square, we get Thus, to graph the right and then y -5/4 , we start with the graph of units down , shift it units to f (x) = x2-3x+1 3/2 x 1.4.59 Start with the graph of and move it units to the right (thus getting the graph of ) Move the resulting graph units down, to get the graph of y y=|x| y=|x-2| -1 x f (x) = | x - | -3 -3 28 Full file at https://TestbankDirect.eu/ Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ 1.4.60 Chapter Length as a function of age 10 Length 0 1.4.61 0.5 1.5 Age 2.5 Tail length as a function of age Tail length 0.8 0.6 0.4 0.2 0 1.4.62.

0.5 1.5 Age 2.5 Tail length as a function of length Tail length 0.8 0.6 0.4 0.2



1.4.63.
Length 10 Length as a function of mass 10 40 30 Length Mass Mass as a function of length 50 20 10 4 Length 10 0 10 20 30 40 50 Mass The graph of length as a function of mass looks like the graph of mass as a function of length turned on its side 1.4.64.
Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ Plugging in gives 26 bites 29 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition 1.4.65.
Plugging in gives 6.4 hours of sleep The fever is 1.4.66.
( ( )) ( ) if 1.4.67 M V L (T ) = 1.3V L (T ) = 2.6L (T ) = 2.6 (10 + T 10 ) At 3 , That’s a large insect, but I would not be frightened 1.4.68.
The formula is 0.5 1.25 1.5 3.25 2.5 7.25 10 0.25 0.25 2.25 1.5 3.5 9.5 14 P(t) 14 Population 12 a(t) 10 b(t) 0 time The population is increasing, whereas decreases to at time and then increases; the sum decreases slightly and then increases 1.4.69 The formula is t 0.5 1.5 2.5 20 21.625 23.5 25.625 28 30.625 33.5 40 39.5 39 38.5 38 37.5 37 60 61.125 62.5 64.125 66 68.125 70.5 30 Full file at https://TestbankDirect.eu/ Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ Chapter 80 V(t) Volume 70 60 50 Vb (t) Va (t) 40 30 20 Time increases, decreases, and the total increases 1.4.70.
Because the mass is the same at ages 2.5 days and days, the function relating a and M has no inverse Knowing the mass does not give enough information to estimate the age Mass 0 Age 1.4.71.
Volume never has the same value twice, and therefore V has an inverse that contains sufficient information to find the age 10 Volume 0 Age 1.4.72.
Glucose production is 8.2 mg at ages days and days We cannot figure out the age from this measurement Glucose production 10 0 Age Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ 31 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition 1.4.73.
There are two different values of (9.4 and 8.2) when is 5.6 This isn’t a function, because it fails the vertical line test Furthermore, we couldn’t tell if was 5.1 or 5.6 when = 8.2 Glucose production 10 0 Mass 1.4.74.
Denote the total mass by Then 80 70 60 50 40 30 Population 0 20 40 60 80 100 Year 160 168 168 160 144 120 Mass per individual Mass in millions of kilograms 20 40 60 80 100 Population (millions) 2.4 2.8 3.2 3.6 Mass per individual (kilograms) t 100 80 60 40 20 0 20 40 60 80 100 Year Total mass 200 160 120 80 40 0 20 40 60 80 100 Year The population increases, the mass per individual decreases, and the total mass increases and then decreases ( ) Then T ( t ) = P ( t )W (T ) = × 106 − × 104 t (80 + 0.5t ) 1.4.75.
Denote the total mass by t 20 40 60 80 100 1.6 1.2 0.8 0.4 80 90 100 110 120 130 32 Full file at https://TestbankDirect.eu/ 160 144 120 88 48 Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ Mass per individual (kilograms) Population (millions) Population 0 20 40 60 80 100 Year Mass per individual 140 120 100 80 60 40 20 0 20 40 60 80 100 Mass in millions of kilograms Chapter Total mass 200 160 120 80 40 0 20 40 Year 60 80 100 Year The population decreases, the mass per individual increases, and the total mass decreases ( P ( t ) [million] 2.4 3.6 5.6 8.4 12 80 70 60 50 40 30 Population 12 10 20 40 60 80 100 Year 160 168 216 280 336 360 Mass per individual 80 75 70 65 60 55 50 45 40 35 30 20 40 60 Year 80 100 Mass in millions of kilograms 20 40 60 80 100 Mass per individual (kilograms) t Population (millions) ) Then T ( t ) = P ( t )W (T ) = × 106 + 1000t (80 − 0.5t ) 1.4.76.
Denote the total mass by Total mass 400 350 300 250 200 150 20 40 60 80 100 Year The population decreases, the mass per individual increases, and the total mass increases 1.4.77.
We denote the total mass by Then t 20 40 60 80 100 2.4 2.8 3.2 3.6 80 78 72 62 48 30 160 187.2 201.6 198.4 172.8 120 The population decreases, the mass per individual increases, and the total mass increases and then decreases Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ 33 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Population (millions) Population 3.5 2.5 20 40 60 80 100 Year Mass per individual 80 75 70 65 60 55 50 45 40 35 30 20 40 60 Year 80 100 Mass in millions of kilograms Mass per individual (kilograms) Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition Total mass 210 200 190 180 170 160 150 140 130 120 20 40 60 80 100 Year 1.4.78 Using long division we write Alternatively, we add and subtract in the numerator: 2x 2x + − 2x + 2 f (x) = = = − = 2− x +1 x +1 x +1 x +1 x +1 So, move the graph of left by unit (thus getting vertically by a factor of (to obtain ), then stretch it ), then reflect across the -axis (to obtain ), and finally move up units The final graph is shown below 1.5 0.5 0 10 15 20 25 1.4.79 Using long division we write 34 Full file at https://TestbankDirect.eu/ Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ Chapter Alternatively, we add and subtract in the numerator: , move units (thus getting To graph by a factor of left by (to obtain ), then scale vertically ), then reflect across the ), and finally move up -axis (to obtain units See below r A c 1.4.80 is a linear function, so it does have an inverse horizontal line test, so have no inverse functions and and not pass the seem to be increasing, in which case they have inverse functions Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ 35 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition Depending on what software is used, we can ask it to find given , , , and so on (we used Maple) In the case of a linear function, Maple returns the correct inverse function For the functions involving quadratic and cubic terms, Maple does not have an returns two and three solutions respectively, ignoring the fact that inverse and has a (unique) inverse Maple is unable to solve for the inverses and of 1.4.81 The function increases rapidly until just after then it starts decreasing, and levels off , where it reaches its highest value; 1.4.82 The function oscillates, with the amplitude of its oscillations decreasing First, it climbs to a maximum value of about 2.4, then decreases to a bit above 0.8, then climbs again to about 2, and so on 1.4.83 The function consists of numerous oscillations, with a rapid interchange of highs and lows The complicated oscillating pattern is actually periodic, with period of about 36 Full file at https://TestbankDirect.eu/ Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ Chapter 1.4.84 As in Exercise 1.4.82, the function oscillates, but this time the amplitudes are affected by the function , as can be seen well in the initial behaviour: the first peak of is higher than in Exercise 1.4.82 1.4.85 The function increases rapidly until it reaches a bit about 1.2 and then decreases rapidly to just after ; after further slower decrease, it levels off at zero 1.4.86 The function seems to be periodic, with the period of a bit over units It consists of a number of sharp peaks, some with high values and some with low values Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ 37 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition Chapter Summary and Review: True/False Quiz 1.1 FALSE 1.2 FALSE For example, all vertical shifts inverse functions As well, all functions of the form real number have inverse functions 1.3 TRUE Reflecting with respect to (where is a real number) have , where is a non-zero -axis we obtain by a factor of corresponds to replacing by Horizontal compression , so the resulting function is 1.4 TRUE An increasing function cannot assume repeated values If , the function ; as well, if is equal to ) (so for no is not equal to “cancel each other”) Alternatively, we show that 1.6 TRUE, since 1.7 TRUE Pick value of , , then for all then 1.5 FALSE (so other than and or not From we get 1.8 FALSE and For that 1.9 TRUE, by the definition of the inverse function 1.10 FALSE Both and its inverse are increasing functions Chapter Summary and Review: Supplementary Problems 1.1 Divide both the numerator and the denominator by , to get is a vertical compression, or expansion, or a copy of The graph of the function the graph (if ) of the hyperbola From the graph, we see that the values become smaller and smaller as increases Likewise, the graph of is a scaled 38 Full file at https://TestbankDirect.eu/ Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ Chapter version of the graph of Thus, as increases, the values of approach zero approach We conclude that, as grows larger and larger, the values of 1.2 Using long division we obtain An alternative to long division goes like this (add and subtract 4): So, starting with the graph of , expand it vertically by a factor of 4, then reflect across the -axis and finally move the graph up units; see the figure below 1.3 Move the graph of two units down to obtain Keep the positive parts of the graph, and reflect the negative parts across the -axis, thus obtaining the graph of (figure a below) Now move it three units down, to generate the graph of , and finally repeat the absolute value of a graph construction: keep the positive parts and reflect the negative parts across the -axis to obtain (figure b below) y y −2 a Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ x −5 −2 x b 39 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition 1.4 The volume of the Earth is it follows that Since and so Thus, the mass of Earth is or 1.5 A number in the domain of must satisfy and From the former, Solving we obtain , then (by squaring) and Thus, the domain of consists of all real numbers smaller than or equal to and different from In interval notation, the domain consists of two sets and 1.6 Given the density (of a soil sample), what depth did it come from? Or: given density, what depth? To compute the inverse function, we solve for 40 Full file at https://TestbankDirect.eu/ : Copyright © 2015 Nelson Education Ltd Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file at https://TestbankDirect.eu/ 1.7 The area of the rectangular region is Chapter , so the density of the deer is The area of the circular region is , so there are approximately 187 ≈ 1879.9 45 deer (i.e., 1879 or 1880 deer) [Depending on how the answers to intermediate calculations are rounded off, the answer could be slightly different.] area ⋅ density = 144π ⋅ 1.8 We compute 1.9 We compute 1.10 By definition of the absolute value Thus, to obtain we keep the parts of which are above the -axis (or touch it), and replace the parts below the -axis with the horizontal line For an illustration, see below Copyright © 2015 Nelson Education Ltd Full file at https://TestbankDirect.eu/ 41 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, Second Canadian Edition y y g(x) f(x) 1.11 x x is approximately equal to , which is i.e., approximately 466.7 watts 1.12 (a) The mass of the colony is (b) From density = mass/volume, it follows that (c) From Since we obtain , it follows that i.e., about 2.1 mm 42 Full file at https://TestbankDirect.eu/ Copyright © 2015 Nelson Education Ltd ... 1000 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Ins.aanual nkDtoiraccompany ect.eu/ Calculus for the Life Sciences, Second Canadian Edition 1.3.50 The function... https://TestbankDirect.eu/ 23 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences, ... https://TestbankDirect.eu/ 25 Solution Manual for Calculus for the Life Sciences 2nd Edition by Adler Full file Instructor’s at https://TestbankDirect.eu/ Solutions Manual to accompany Calculus for the Life Sciences,