Unformatted text preview: Diﬀerential Equations Practice
1. Suppose that y (x) is a function which is never zero and satisﬁes the diﬀerential equation −y = 2xy 2 . Suppose also that y (0) = 3. Find y . 2. Is it possible that y 2 + (y )2 = 1? If not, explain why. If so, give an example. 3. Revolve the region in the ﬁrst quadrant bounded by the graph of the function y = −x2 + h, h > 0, about the y axis. The 3dimensional solid obtained is the shape of a melting heap of snow when it is of height h. (a) Express the volume V of the heap of snow as a function of its height h. (b) The heap of snow melts at a rate equal to the area of its base. What is the rate at which its height is decreasing? (Hint: ‘Rate’ means derivative. Solve for dh using what you know about dV .) dt dt (c) Assuming that h = 60 when t = 0, what is the rate of melting when t = 50? 4. The thirdlife of a radioactive element is deﬁned to be the time required for two thirds of the radioactive nuclei present in a sample to decay (leaving one third remaining). (a) Prove that the thirdlife is a constant that does not depend on the number of radioactive nuclei intially present in the sample. (b) Suppose you are given a substance with a thirdlife of one month. What is its halflife? 5. A cup of hot cocoa (initial temperature: 90 ◦ C) is left outside on a winter day for 10 minutes, during which time it cools to a lukewarm 40 ◦ C. It is then taken back inside the house where it cools further to a tepid 25 ◦ C after another 20 minutes. The ambient temperature inside the house is 20 ◦ C. Assuming that the k value (in Newton’s law of cooling, section 7.2) is a property of the cocoa and therefore is constant throughout this scenario, how cold is it outside? ...
View
Full Document
 Fall '06
 MARTIN,C.
 Calculus, Equations, Radioactive Decay, 10 minutes, radioactive nuclei present, differential equations practice, diﬀerential equation −y

Click to edit the document details