Problem Solving
Problem 1.
In the mythical city of Oronto, citizens are facing a deadly pandemic called Ovid-19.
A. (4 points) The druids have figured out that the pandemic follows the model :
dN = 0.75t2 at
where N(t) is the number of affected people and t is measured in days.
What is the total number of people affected at the end of 15th day? Initially, how many days did it take for 100 people to get affected? How many days did it take for the number of effected people to grow from 100 to 200?
B. (6 points) It was found out that the pandemic is being spread by contact and the Ruler of Oronto decreed on strict social distancing protocols. The druids again computed that from day 20 the new model is :
—dN = 2t2C°5` as before N(t) is the number of affected people and t is measured in days. dt
According to this new model, after how many days, is the pandemic expected to end? What will be the total number of people affected through out this entire pandemic (that is, starting from day 0) ?
(The pandemic is considered to have ended when the number of effected people < 1. You can try plotting graphs to have an idea but for most of the questions in the above problem, you need to show your computations to support your answers. )
Problem 2. Escape velocity is defined as the minimum speed needed for an object to escape from the gravitational influence of a planet. In this problem we will see how improper integrals are used to compute them. The amount of work done to move an object from xo to x1 against a force F is computed via the integral
W =f Fdx xo
The gravitational force follows an inverse square law, F = k/r2. So as r co, F 0.
A. (5 points) Suppose we are on a planet of radius 105 km. We are trying to take a 1 kg object, from the surface of the planet to out of its gravitational reach ( that is to co) . Compute the work done in the process if k = 4 x 10′ kg•lcm3 Z.
B. (5 points) Another way to measure work done is the net change of kinetic energy : 1 w = _2(nivo2 nivi2)
where vo is the initial speed and v1 is the final speed. Find the minimum value of vo so that the 1 kg object in the previous problem never slows to a complete stop before reaching infinity. Express your answer in km/s.
Problem 3. A. (5 points)
(a) Try using Wolfram Alpha to solve the integral
1—x4 J ,c4(1+x2+x4)dx
Does it give you an elementary antiderivative? How do you know?
(b) Use a substitution of the form u = xk + X-k (for some positive integer k) to solve the integral
j..ilFx4(1+x2+x4) (HINT: After your substitution, you might still need some help integrating!)
B. (5 points) Consider the function g(x)=e-x(l+e-r)-2dt.
(a) Compute f g(x)dx by hand.
(b) What answer does WolframAlpha give?
(c) Are the two answers the same? Explain why or why not.
1-x4
dx
