Kernel: SageMath 9.6

C02 CoCalc Introduction

Arithmetic, Algebra, and Calculus

Exercise 1. Calculate 615+cos(π)+12e3\frac{6}{15} + \cos(\pi) + \sqrt{12e^3}.

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Exercise 2. Solve the equation ax2+bx+c=0ax^2 + bx + c = 0 for the variable xx in terms of the parameters aa, bb, and cc.

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Exercise 3. Graph the equation y=3cos(x2)xy = 3\cos(x^2) - x for 1x3-1 \leq x \leq 3. what is yy when x=3x = 3?

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Exercise 4. Graph the function f(x)=3cos(x2)xf(x) = 3\cos(x^2) - x on the domain [1,3][-1, 3]. what is f(3)f(3)?

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Exercise 5. Find the zeros of the function ff defined in the previous exercise on the domain [1,3][-1, 3].

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Exercise 6. Find the area between the graph of ff, the horizontal axis, and the domain [1,3][-1, 3].

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Exercise 7. Find the equation of the tangent line to the graph of ff at 11. Graph ff and its tangent line at 11 on the domain [1,3][-1, 3].

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Atlantic Cod Harvesting Model

Exercise 8. Equation (1.12) in the text is u(t)=ru(t)(1u(t)/K)hu(t)u'(t) = ru(t)(1-u(t)/K) - hu(t). Describe the meaning and possible units for each symbol.

Exercise 9. The solution to the harvesting differential equation and initial condition u(0)=u0u(0) = u_0 is claimed to be equation (1.13): u(t)=(1hr)K1+e(rh)t(Ku0(1hr)1).u(t) = \frac{\left(1-\frac{h}{r}\right)K}{1 + e^{-(r-h)t}\left(\frac{K}{u_0}\left(1-\frac{h}{r}\right) - 1\right)}. Verify the claim.

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Exercise 10. Graph the solution on the domain [0,10][0, 10] when K=10K = 10, r=1r = 1, u0=2u_0 = 2, and h=0.1h = 0.1

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Exercise 11. According to the harvesting model, what happens to the population as time goes on?

Exercise 12. the code below stores Atlantic cod population and harvesting data from Table 1.2 in the text and then shows part of the table. Find the mean of the harvesting data. Graph the population data versus years after January 1, 1978, and the model solution using reasonable parameter values.

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ydata = list(range(1978,2009)) udata = [72148, 73793, 74082, 92912, 82323, 59073, 59920, 48789, 70638, 67462, 68702, 61191, 49599, 46266, 34877, 28827, 21980, 17463, 18057, 22681, 20196, 25776, 23796, 19240, 16495, 12167, 21104, 18871, 21241, 22962, 21848] hdata = [0.18847, 0.14974, 0.21921, 0.17678, 0.28203, 0.34528, 0.20655, 0.33819, 0.14724, 0.19757, 0.23154, 0.2086, 0.33565, 0.29534, 0.33185, 0.35039, 0.2827, 0.19928, 0.18781, 0.19357, 0.18953, 0.17011, 0.1566, 0.28179, 0.25287, 0.25542, 0.08103, 0.0874, 0.08195, 0.10518, NaN] table(columns = [ydata[0:32:6], udata[0:32:6], hdata[0:32:6]], header_row = ["year", "$u_t$", "$h_t$"])
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