1. For the following functions determine the following limits (if they exist): (i) lim =-5e+3 (iii) lim (ii) _lim (iv) lim +2
1. For the following functions determine the following limits (if they exist): (i) lim =-5e+3 (iii) lim (ii) _lim (iv) lim +2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![8. The logistic model is important in the modelling of population dynamics and is represented by
1. For the following functions determine the following limits (if they exist):
(i) lim
7³–5x+37
(ii) lim
r+1
(iii) lim 9-5
(iv) lim
N(t) =
1+ (* - 1).
e-akt
(v) lim 2r+1
(vi) lim r-2
(vii) lim +3||
z-322+z-6
(viii) lim v25-z-3
z4 -16
where
N(t) is the population at some time t (days)
2. For the function f(x) defined by
No = N(0) is the initial population
O if x is an integer
f(x) =
a and k are positive constants
1 if æ is not an integer
• At t =0 the population is 5
(a) Sketch the graph of y = f(æ) for x € [-4,4]
• At one week (t = 7) the population is 10
(b) Find lim f(æ) (if it does not exist, explain why it is the case).
• Eventually after a long time the population is 100
3. Given the following functions find the limits (if they exist).If the function is discontinuous state
(a) Given these three pieces of information provided about the model, find the exact values of No, a
the type of discontinuity:
and k.
(i) lim,교_ir+4
-4
r +1,
x 21
(b) What is the approximate value of N after two weeks?
(ii) lim f(x) where f(x) = «
-1,
I < 1
(c) Approximately after what time was N= 50?
(iii) lim
I--3
-32 (z²+6z+9)
(z+3)²
(d) Sketch the graph of N(t) (labelling all axes,asymptotes and intercepts).
3x + c,
T< -1
(iv) Find the value of c such that f(x) =
is continuous
5+ 4r3,
x > -1
4. For functions F(r) =
8z2+20r-12
4r²+4x–3
(a) Completely factorise the numerator
(b) Completely factorise the denominator
(c) Find the coordinates of any "holes" in the graph of the function (removable discontinuities)
(d) Find all æ- and y, intercepts fr the function
(e) By using the polynomial division algorithm re-express the function is asymptote form
(f) Find all asymptotes for F(r)
(g) State the domain and range of F
(h) Explain the asymptotic behaviour of F (by asymptotic behaviour this means something
like "as a → x, f(x) → 2")
(i) Carefully sketch the graph of y = F(x) including all the information found in parts (a)-(h).
5. (a) Simplify –5 (xy¬1)¯° in a form such that all exponents are positive integers.
(b) Solve z22" = x2*+1 for r
(c) Solve 13"-27L = 0.for r
(d) Solve 25* – 26(5*) + 25 = 0 for r
(e) For f(x) = V , find dom(f)
6. Solve for a with each of the following:
(i) (32)³z+1
(ii) log5(4x – 3) = log3 (9)
(a) (iii) 9 ()° = 4 + 2 (4)ª
(iv) 3 (8-ª) = } (72z+1)
(v) 2 log2r+1 (2x + 4) – log2r+1 (4) = 2
= 128
7. (a) Exactly calculate ?log4 8
log, 2
(b) Write In (V2x + 1) + ln(x) – In(|sin æ|) in one expression.
(c) Solve for r : log2 () = 4
(d) Beginning with f(x) = In(x), sketch the sequence of transformations such that f(x) =
3 In(4 – a) + 2
(e) Find dom(f) and ran(f) for f(x) =
VIn(2r+1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fa051c2-2cde-44b8-a544-2fd7e9e3e51b%2F21b1aa72-a8e7-4b7a-8c82-d9e1f0edbf63%2Fpv96pcf_processed.png&w=3840&q=75)
Transcribed Image Text:8. The logistic model is important in the modelling of population dynamics and is represented by
1. For the following functions determine the following limits (if they exist):
(i) lim
7³–5x+37
(ii) lim
r+1
(iii) lim 9-5
(iv) lim
N(t) =
1+ (* - 1).
e-akt
(v) lim 2r+1
(vi) lim r-2
(vii) lim +3||
z-322+z-6
(viii) lim v25-z-3
z4 -16
where
N(t) is the population at some time t (days)
2. For the function f(x) defined by
No = N(0) is the initial population
O if x is an integer
f(x) =
a and k are positive constants
1 if æ is not an integer
• At t =0 the population is 5
(a) Sketch the graph of y = f(æ) for x € [-4,4]
• At one week (t = 7) the population is 10
(b) Find lim f(æ) (if it does not exist, explain why it is the case).
• Eventually after a long time the population is 100
3. Given the following functions find the limits (if they exist).If the function is discontinuous state
(a) Given these three pieces of information provided about the model, find the exact values of No, a
the type of discontinuity:
and k.
(i) lim,교_ir+4
-4
r +1,
x 21
(b) What is the approximate value of N after two weeks?
(ii) lim f(x) where f(x) = «
-1,
I < 1
(c) Approximately after what time was N= 50?
(iii) lim
I--3
-32 (z²+6z+9)
(z+3)²
(d) Sketch the graph of N(t) (labelling all axes,asymptotes and intercepts).
3x + c,
T< -1
(iv) Find the value of c such that f(x) =
is continuous
5+ 4r3,
x > -1
4. For functions F(r) =
8z2+20r-12
4r²+4x–3
(a) Completely factorise the numerator
(b) Completely factorise the denominator
(c) Find the coordinates of any "holes" in the graph of the function (removable discontinuities)
(d) Find all æ- and y, intercepts fr the function
(e) By using the polynomial division algorithm re-express the function is asymptote form
(f) Find all asymptotes for F(r)
(g) State the domain and range of F
(h) Explain the asymptotic behaviour of F (by asymptotic behaviour this means something
like "as a → x, f(x) → 2")
(i) Carefully sketch the graph of y = F(x) including all the information found in parts (a)-(h).
5. (a) Simplify –5 (xy¬1)¯° in a form such that all exponents are positive integers.
(b) Solve z22" = x2*+1 for r
(c) Solve 13"-27L = 0.for r
(d) Solve 25* – 26(5*) + 25 = 0 for r
(e) For f(x) = V , find dom(f)
6. Solve for a with each of the following:
(i) (32)³z+1
(ii) log5(4x – 3) = log3 (9)
(a) (iii) 9 ()° = 4 + 2 (4)ª
(iv) 3 (8-ª) = } (72z+1)
(v) 2 log2r+1 (2x + 4) – log2r+1 (4) = 2
= 128
7. (a) Exactly calculate ?log4 8
log, 2
(b) Write In (V2x + 1) + ln(x) – In(|sin æ|) in one expression.
(c) Solve for r : log2 () = 4
(d) Beginning with f(x) = In(x), sketch the sequence of transformations such that f(x) =
3 In(4 – a) + 2
(e) Find dom(f) and ran(f) for f(x) =
VIn(2r+1)
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