Question

3. Consider the region R in the first quadrant enclosed by y = x, y =...

3. Consider the region R in the first quadrant enclosed by y = x, y = x/2, and y = 5.

(a) Sketch this region, making sure to identify and label all points of intersection.

(b) Find the area of R, using the method of your choice.

(c) Using the method of your choice, set up an integral for the volume of the solid resulting from rotating R around the y-axis. Do NOT evaluate the integral.

(d) Using the method of your choice, set up an integral for the volume of the solid resulting from rotating R around the x-axis. Do NOT evaluate the integral.

4. Suppose a spring with natural length 20 cm satisfies a modified Hooke’s Law: the force needed to stretch it is F (x) = x + x3 newtons, where x is the distance beyond its natural length, measured in meters, that the spring is stretched. How much work is needed to stretch the spring from 30 cm to 40 cm?

You do not need to simplify your answer (but do perform any integrals that may occur).

5.

5. (12 points, 6/6) Two short problems on differential equations.

(a) It is a fact that the general solution of the differential equation x2y′ + xy = 2 is
y = x2(lnx+C). Using this fact, what is the particular solution of this equation with y(1) = 1?

2′2 (b) Find the general solution g(t) of the equation g g = t .

6. Consider the differential equation y′ = y − x + 1. Sketch a direction field for this differential equation, with −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1, using a 3-by-3 grid (that is, you should draw nine line segments).

7.

A fruit fly population has 1000 individuals initially. Suppose time is measured in weeks.

(a) If it obeys natural growth with k = 0.5, how long will it take to get to 5000 individuals? Do not simplify your answer.

(b) If it obeys logistic growth with k = 0.5 and carrying capacity 10000, how long will it take to get to 5000 individuals? Do not simplify your answer.

8.

Your professor’s wife has a favorite plant that she puts on the windowsill. However, it is also Miranda’s favorite plant:

  1. (a) The plant’s volume V (t) is measured in cubic centimeters. It grows via the law of natural growth, at a rate equal to one-tenth of its current volume per week. Assuming Miranda is ignoring it, what differential equation models the growth of the plant?

  2. (b) Now assume that, additionally, Miranda eats 8 cubic centimeters of plant per week. What differential equation models the growth of the plant now? (Just state the equation, don’t solve it!)

  3. (c) Are there any equilibrium solutions V (t) = C for your differential equation in part b)?

  4. (d) Suppose the plant’s volume is 60 cm3 at time zero. What do you think will happen in the long run? Will the plant grow, die off, or stay at a stable volume?

(e) Prove it: solve the equation in part b), with the initial condition V (0) = 60. Was your hypothesis in part d) correct?

(If you did not solve part b, or if you prefer a different problem, instead solve the unrelated differential equation (3x − 8)yy′ = 2, with the initial condition y(3) = 1.)

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