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**Q1:**Question 2. (4 points) Intensive harvesting of a population of a fish species can cause population extinction. We consider the following two models for harvesting of a given fish species and analyze how the extinction depends on the nature of the harvesting. The population size P (measured in thousands) is a function of harvesting effort h. (The harvesting effort is a mathematical measure of "fishing effort", which you are not expected to know in details.) \begin{aligned} &\text { Model }\\ &\text { 1: } \quad P(h)=\left\{\begin{array}{ll} 3(1-h) & \text { if } 0 \leq h \leq 1 \\ 0 & \text { if } h>1 \end{array}\right. \end{aligned} \begin{aligned} &\text { Model }\\ &\text { 2: } \quad P(h)=\left\{\begin{array}{ll} 1+\sqrt{4-3 h} & \text { if } 0 \leq h \leq \frac{4}{3} \\ 0 & \text { if } h>\frac{4}{3} \end{array}\right. \end{aligned} (1) (1 point) What is the initial population of this species when no harvesting efforts were applied at all? (2) (1 point) Draw the graph of each model. (3) (2 points) Here you will analyze each model in terms of the Intermediate Value Theorem by answering the following questions. Which model has a situation where a small change in harvesting effort causes a sudden extinction? In Model 2, is there a harvesting effort to obtain the population of 500?See Answer**Q2:**a.Prove \tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right) \text { where }-\pi / 2<\tan ^{-1} x+\tan ^{-1} y<\pi / 2 \text { . } \tan ^{-1}(1 / 2)+\tan ^{-1}(1 / 3)=\frac{\pi}{4} Hint: Use an identity for tan(x + y). b. Use part (a) to show that c. Use the first four terms of the Maclaurin series of tan^-x and part (b) to approximate the value of n.See Answer**Q3:**\text { 4. Let } C([-3,3]) \text { be the vector space of continuous functions } f:[-3,3] \rightarrow \mathbb{R} \text { with the norm of uniform convergence }\|f\|_{\infty}:=\max _{x \in[-3,3]}|f(x)| \text {. } \text { (i) Consider the linear mapping } L: C([-3,3]) \rightarrow \mathbb{R} \text {, } L f:=\int_{-3}^{3} x f(x) d x \text { Prove that the mapping } L: C([-3,3]) \rightarrow \mathbb{R} \text { is continuous. } \text { (ii) Consider the mapping } F: C([-3,3]) \rightarrow \mathbb{R} \text {, } F(f):=\int_{-3}^{3}|x \| f(x)| d x Why you can not use the same strategy as in part (i) to prove that F is[2 Marks]continuous?See Answer**Q4:**\lim _{x \rightarrow 0} \frac{\ln \sqrt{x+1}-\tan ^{-1} x}{x} 11. (7 points) Use Taylor series to evaluate the limitSee Answer**Q5:**5. Find the Taylor series for the following functions with the given centres [4 marks each]. (a) f_{1}(x)=\ln (x-1), \quad a=2 \text { (b) } f_{2}(x)=\cos (3 x), \quad a=0 \text { (c) } f_{3}(x)=\sin (x), \quad a=\piSee Answer**Q6:**Instructions 1. Create your own example of an alternating series that is (conditionally) convergent, but not absolutely convergent. Your series should be different than any of those in the notes or text examples. 2. Post your infinite series on Discussion Board on Canvas. Give a brief explanation of how you created your series. There are several different approaches you might take. 3. Peer response: look at the post from at least one other classmate and critique the method used to create the series. Do you think the method should work? Alternatively, if you believe your classmate's method works, describe another way the series might have been created.See Answer**Q7:**6. 7. 8. · Σ (5(3)" + (3)") n=1 Ans. 13 n=l 8 η n+√n Σ()" n=0 Σ(3) n=0 Need handwritten step wise full solutionSee Answer**Q8:**When determining whether a series is convergent or not, state the test you use to make the determination and show that the conditions of the test are met. 1. Provide an example of an alternating series that does not converge even though it meets all the conditions of the Alternating Series Test except the one that anan+1 for all natural numbers n. When you come up with this example, you'll have demonstrated why this condition is needed. 2. Determine whether the series (-1)"- 3. Determine whether the series (-1)" Vn converges. Explain why. converges. Explain why.See Answer**Q9:**4. Determine whether the series 6. Determine whether the series n²+1 72³ 5. Determine whether the series (-1)" converges. Explain why. n=1 converges. Explain why. n! 72" converges. Explain why. 7. Determine whether the series (-)" converges. Explain why. 8. Determine whether the series Σn converges. Explain why.See Answer**Q10:**Determine for which the power series converges. (x − 2)" n 1. Σ n=1 Ans. All a in the interval [1, 3) mm 2. Σπ n=1 3. En!" n=] 4. Σ n=1 (-2) n νη 5. Σn!(x+1)n 2n n=1 6. Σ n=l n(x + 1)" 2nSee Answer**Q11:**If the tenth term of an arithmetic sequence is 15 and the common difference is 8, find the sum of the first 20 terms. The sum of the first 20 terms of the arithmetic sequence is XSee Answer**Q12:**For each sequence, determine whether it appears to be arithmetic, geometric, or neither. 5, 15, 25, 35, 500, 100, 20, 4. 15.45, 135. O Arithmetic O Geometric O Neither O Arithmetic O Geometric O Neither O Arithmetic O Geometric ONeitherSee Answer**Q13:**Sequence The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric, and write the formula for the term a, of that sequence. (a) 2,-4,8,... (b) -7,-5, -3,... Type O Arithmetic O Geometric O Arithmetic O Geometric term formula 9-0 4-0 808 0+0 0-0 0.0 X 10 $ 12 E GSee Answer**Q14:**2 Find the 8th term of the geometric sequence whose common ratio is 3 0 X 8 5 and whose first term is 3. 9 10See Answer**Q15:**Find and simplify a formule for a the term of the given sequence. (Hint: Decide what kind of sequence it is, then use information from within the section.) 15. 19, 23, 27, 31. The simplified formula for the term is a- X Ana Reported BORHOSee Answer**Q16:**Find the 13th, 0 term of the arithmetic sequence whose common difference is d=-7 and whose first term is a =4. d 8 X 5See Answer**Q17:**5 f the fifth term of a geometric sequence is 625, and the common ratio is find the sum of the first 20 terms to two decimal places. The sum of the first 20 terms of the geometric sequence is 0. X 5See Answer**Q18:**Find the 12th term of the following geometric sequence. 4, 8, 16, 32, 0 8 08 2 X 5See Answer**Q19:**Find and simplify a formula for a,, the term of the given sequence. (Hint: Decide what kind of sequence it is then use information from within the section.) 4, 20, 100, 500, 2500.... The simplified formula for the term is a,- 8 X D.O G Nokia c ESee Answer**Q20:**Find the 58th term of the following arithmetic sequence. 13, 18, 23, 28, 0 X 唱吧 5See Answer

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