Math

h.w13.docx

1. Check that = 1/2 x^2+x+3 satisfies the differential equation y/dx = x+1 .

2. Check that = sin t satisfies the differential equation d^2y]/[dt^2] =-y .

3. Sketch a slope field corresponding to the equation y/dt = 2t-y. Then sketch the solution that passes through the point 1,2) .

4. Sketch a curve = f(x) that satisfies: (0) = 3 , and y/dx =-2 . What is (x) ?

5. A curve = f(x) has the property that the slope of tangent to the curve at every point is reciprocal to the value of the curve. Write down a differential equation whose solution is = f(x) . No need to solve this equation.

6. Determine the time it takes an amount of money to triple if compounded continuously at an annual rate of $5 \%$ .

7. I got a Guinness can whose temperature is 1^(circ)C . To enjoy it cold at the perfect temperature of ^(circ) C , I set my refrigerator to . The temperature (t) of the can at time measured in seconds satisfies Newton's Cooling Law equation:

dT]/[dt] =-.0001 T.

How long will it take for the can to cool off to ^(circ) C ?

8. A cup of tea with initial temperature of = 85C^(circ) is left in the room with a constant temperature = 21^(circ) . In one minute the tea cools down to 4C^(circ) . What will be the temperature of the tea after an hour? When will the temperature of the tea be 2^(circ) ?

9. Share on the forum, but no need to submit: Describe in a short paragraph a phenomenon around you that manifests exponential growth or decay (other than a Guinness beer). When you share your example, ensure that no student has discribed an identical example earlier on the forum.

10. Share on the forum, but no need to submit: Suppose that we can drill a tunnel connecting San Francisco and Paris. Traveling though it will be Gravistar, a frictionless train moving only due to the pull of gravity. How long will travel by Gravistar take? Will it be faster than the direct flight between San Franisco and Paris?

ttp://calculus.sfsu.edu/latexrender/pictures/cd865d7147b0e905098d6212f8f95309.png

Let denote the position of an object of mass on the axis joining San Francisco and Paris. We set zero to be the midpoint between the two cities. From physics one can show that such an object is subjected to the effective gravitational force

=-kx

where = m*1.55*10^(-6) [meter]/[s^2] and = m[d^2x]/[dt^2] according to the Newton's second Law of motion. Write down the differential equation that relates the position and time of an object in the tunnel. Then assume that at time = 0 the object is let go in the tunnel from San Francisco. Use the the theorem from last section to calculate the time it will take Gravistar to reach Paris.

11. Share on the forum but do not submit. Could two solutions to first order differential equation y/dt = F(t,y) colide? Would there be a need for speed limit if we always drove along a solution to the same differential equation?