Easy Calculus Homework. Need this done by 4/16/2021 at 8:00 AM

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Need all problems in attached document done by no later than 8:00 AM on 4/16/2021. Need the answers fully worked out and written out so I can review them.

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Question 1 (10 points) Consider the differential equation dy dt = t2 − y. (a) Show that y(t) = ce−t + t2 − 2t + 2 is a solution to this equation for any value of c. (b) Show that d 2y dt2 = 2t− t2 + y (Hint: differentiate the DE!). (c) Consider a solution of the equation that passes through the point (2, 2). Use fact that dy dt = t2 − y and the result in (b) to answer: At this point, is the solution increasing or decreasing? Is it concave up or concave down? (You may want to verify your answer on the Geogebra applet!) Question 2 (8 points) Find a value of n for which the function y = x−1 x+1 is a solution of the differential equation 1 2 dy dx = ( y x−1 )n . (Hint: plug in to both sides of the equation!) 1 Question 3 (8 points) (a) Compute the integral ∫ ∞ 1 1 x3/2 dx to show that it converges. Do not forget to write in limits correctly. (b) Does ∫ ∞ 1 sin2(x) 2 + x3/2 dx converge or diverge? Explain. (Hint: use part (a), and recall that 0 ≤ sin2(x) ≤ 1.) Question 4 (10 points) Recall that Hooke’s law says that force exerted by a spring extended to length x is F (x) = kx, where k is the spring constant. Consider a spring that is extended to 10cm by a mass of 2kg. How much work is done in extending the spring from this length to length 25cm? You may assume that acceleration due to gravity is 9.8 m/s2. Be sure to specify all units! 2 Question 5 (20 points) Consider the following differential equations: (A) dy dx = arccos(y)x2 (B) dy dx = y + e−x (C) dy dx = ln(xy) (D) dy dx = xe−y (E) dy dx = x x+y (F) dy dx = x2 + y2 (a) Of these differential equations, exactly two are seperable. Circle those two above. (b) Of the two seperable equations, one is very hard to solve, and the other much more straightforward. Solve the straightforward one with initial value y(1) = 0. (c) Of all the equations, only one has an equilibrium solution. Which is it? What is the equilibrium solution? Is the equilibrium stable or unstable? (If you use a Geogebra applet to help with the last part of this question, please paste an image from it here, or at least sketch a sufficient amount of the slopefield to justify your answer.) (d) One of the two slopefields below is for Equation B, and one is for Equation F. Label which is which, and write below one distinguishing feature that enabled you to tell the difference. −2 −1 1 2 −2 −1 1 2 −2 −1 1 2 −2 −1 1 2 3 (e) Use Euler’s Method with ∆t = 2 to estimate y(5) for equation (F) given that y(1) = 0. Is your answer an over or underestimate of the true value of y(5)? Explain briefly. Question 6 (12 points) The density of pollution over a city at height h meters is well mod- eled by p(h) = 0.0004e−0.0025h kg/m3 at altitude h meters. The city can be seen as a circle of radius 8km, and the pollution measured in a cylinder with the city as its base. (Note: this question is substantively the same as Q15 of the Varying Density lab. Feel free to use your work from there here.) (a) Using the function p(h) write a Riemann sum which approximates the amount of pollutant in the cylindrical column of air up to height 1,000 meters, assuming we have divided the cylinder into n slices of equal thickness. Label the volume of each segment, and the density of pollutant on it. (b) Write down an integral which yields the total amount of pollutant in the air over the city to an altitude of 1,000 meters. (No need to compute the integral, just write it down.) (c) Pollution exists above 1,000 meters, of course. We could calculate it up to 2,000 meters, or 3,000, but each of these would be arbitrary cutoffs. Write down and compute an improper integral giving the total mass of pollutants over the city. (Hint: this is similar to part (b). What should the new upper bound be?) 4 Question 7 (15 points) Consider the differential equation dy dt = y2 − y − 2. (a) By factoring the right hand side of the equation, find the equilibrium solutions of this equation. (b) Fill in the blank and cross out the incorrect answer: dy dt is a degree polynomial in y. Its top coefficient is positive/negative. (c) Draw a graph of dy dt vs. y below, labeling your axes and roots. (d) Fill in the blanks (note that the first two should be the same): The graph in part (c) has a critical point at y = . This implies that any solution that passes through y = has an there. (e) Use your graph to determine the direction of the ticks on a slopefield representing this differential equation away from its equilibria. Make a sketch of the slopefield below, indicating the equilibria, the direction of the ticks elsewhere, and the stability of each equilibrium. Label your axes. Finally, sketch a solution that has the feature mentioned in (d), and label that feature. 5 Question 8 (8 points) Let f(x) = { 1 3√x −1 ≤ x < 0 1√ x 0 ≤ x < 1 Compute ∫ 1 −1 f(x) dx. (Hint: you will need to split this up and compute two improper integrals. Be sure to write and compute limits correctly!) Question 9 (9 points) The following statement are all false. For each, explain why using either calculation, worded explanation, or a counterexample, as appropriate. (a) Any two solutions to a differential equation always differ by a constant (i.e. their graphs are vertical shifts of each other). (b) If F (x) is the force applied to an object (in Newtons) at distance x meters from its origin, then the work done in moving the object a distance of d meters is W = Fd Newton-meters. (c) The function y(x) = Cex is the general solution of d 2y dx2 = y. 6

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