. Find the area between: MATH 4 Practice Problems for Exam (a) x =, y = / + x, y = x/ Answer: ln( + ) 4 (b) y = e x, y = xe x, x = Answer: e6 4 7 4 (c) y = x and the x axis, for x 4. x Answer: ln 5. Calculate the volume obtained by rotating: (a) The region in problem a around the x-axis Answer: π 4 π 6 (b) The region in problem a around the y-axis ( ) 5 Answer: π 6 (c) The region ( in problem b around the x-axis e Answer: π 7 ) (d) The region in problem b around the y-axis Answer: π(e 6 + ) (e) The region in problem c around the x-axis Answer: π( ln 4 ln 5 + 5 ) (f) The region in problem c around the y-axis Answer: π( ln ln 5 + 4) (g) The region in problem c around the line x = Answer: π(ln ln 5 + ) (h) The region in problem c around the line y = Answer: π( ln + 4 ln 5 + 5 ). Integrate: (straightforward) (a) x 4 e x Answer: 4 ex (x 4 4x + 6x 6x + ) + C
(b) x tan (x) Answer: x arctan x 6 x + ln( + 6 x ) + C x (c) x 5x + 4 Answer: 4 ln(x 4) ln(x ) + C (d) + 4x Answer: x + 4x + ln(x + + 4x 4 ) + C (e) + x Answer: x ln( + x) + C cos x (f) x Answer: x + sin( x) + C sec(ln x) tan(ln x) (g) x Answer: sec(ln x) + C 4. Integrate: (trickier) (a) sin 4 (x) Answer: x cos(x) sin(x) cos(x) 8 6 8 sin (x) + C x (b) 5 x Answer: x 5 + 5 arcsin(5/x) + C (Note that arcsin(5/x) = π arcsec(x/5) [Why?]) e t (c) e t 4 dt Answer: 4 (ln(et ) ln(e t + )) + C (d) + ex Answer: + e x + ln( + e x ) ln( + e x + ) + C (e) e x Answer: xe x e x + C 5. Evaluate: (a) e x(ln x) Answer: /
(b) x(x + 4) Answer: π/ y (c) dy y Answer: π/ 6. Find the general solution to each of the following differential equations: (a) x dy = y Answer: y = /(C ln(x)) (b) (x + ) dy = y Answer: y = Ce arctan x 7. Find the specific solution of each equation that satisfies the given condition: (a) dy = xy, y() = Answer: y = e (x )/ (b) dy = xy + x, y() = Answer y = e x / 8. In a second-order chemical reaction, the reactant A is used up in such a way that the amount of it present decreases at a rate proportional to the square of the amount present. Suppose this reaction begins with 5 grams of A present, and after seconds there are only 5 grams left. How long after the beginning of the reaction will there be only grams left? Will all of the A disappear in a finite time, or will there always be a little bit present? Answer: 4 seconds, and there will always be a little bit present. 9. According to Newton s law of heating and cooling, if the temperature of an object is different from the temperature of its environment, then the temperature of the object will change so that the difference between the object s temperature and the ambient temperature decreases at a rate proportional to this difference. On a hot day, a thermometer was brought outdoors from an air-conditioned building. The temperature inside the building was C, and so this is what the thermometer read at the moment it was brought outside. One minute later the thermometer read 7 C, and a minute after that it read C. What was the temperature outside? (Impress us and express the answer without using logarithms or the number e.) Answer: 9 C
. A super-fast-growing bacteria reproduces so quickly that the rate of production of new bacteria is proportional to the square of the number already present. If a sample starts with bacteria, and after hours there are bacteria, how long (after the starting time) will it take until there are (theoretically) an infinite number of bacteria? Answer: 6 hours... x x = (a) π 5 Answer: B x e x = (b) π 4 (c) π 5 (d) π (e) π 5 (f) π (a) /4 (b) 4/ (c) (d) /8 (e) e / (f) diverges 6 x 6x 4 4 x x 6 = (a) + ln() (b) 4/ (c) + ln() (d) 47/4 (e) + ln(4/) (f) + ln() 4. What is the volume of the solid obtained by rotating the region between the graph of y = and the x-axis for x around the y-axis? x + 4x + (a) π(ln + ln ) (b) π(4 ln 5 ln ) (c) π ln 5. (d) π( ln + ln ) (e) π ln 8 (f) π(5 ln ln ) Answer: F x + x + = (a) (b) π (c) π (d) π 4 (e) π (f) diverges 6. Find the surface area of the surface obtained by revolving the part of the graph of y = x /9 where x around the x-axis. (a) 8π (b) π (c) 76π (d) 77π (e) 98π (f) 86π 7 7 9 48 8 7 Answer: E
7. Solve the initial-value problem: dy = ey sin x, y() =. (a) y = ln(sec x) (b) ln(cos x) (c) π 4 + ln(cos x) (d) π 4 ln(sec x) (e) ln(sec x) (f) ln(cos x) 8. The function kx x f(x) = otherwise is a probability density function for a certain value of k. Find the mean of that probability density function. (a) 4 (b) (c) 4 (d) (e) (f) 9. If water leaks out of a small hole in a cylindrical bucket, then the height of the water level above the bottom of the bucket decreases at a rate proportional to the square root of the height. If the water level starts out at a height of 5 cm, and if after minutes it is down to 6 cm, how long after the start will the bucket be empty? (a) min (b) 4 min (c) 5 min (d) 6 min (e) 7 min (f) the bucket will never be completely empty. What is the length of the part of the curve y = x / x/ between x = and x =? (a) 5 7 7 5 (b) (c) (d) (e) (f) 6 6 6.. e x 4 ln x = (a) e4 6 Answer: B (b) 4e5 5 + x x 6 = (c) 5e6 6 (d) 6e7 49 (e) 7e8 64 (f) 8e9 8 (a) 5 ( + ) (b) 5 (4 + ) (c) ( )
(d) 5 ( ) (e) 5 (4 ) (f) diverges 4. x x 6x + 5 = (a) ln ln (b) ln ln 4 (c) ln ln 4 (d) ln ln 6 Answer: E (e) ln ln 4 (f) ln ln 4. What is the surface area of the surface obtained by rotating the part of the curve y = x for x around the x-axis? (a) π 8 (5 5 ) (b) π (7 7 ) (c) π 4 (8 8 ) (d) π 7 ( ) (e) π 6 (7 7 ) (f) π 45 ( ) 5. Let y(x) be the solution of initial-value problem y + 4xy =, y() =. Then y() = (a) e (b) e 4 (c) e 6 (d) e 8 (e) e (f) e 6. Some enterprising Penn scientists have created a sample of Unobtanium in their lab. One of the remarkable properties of this material is that when it is heated, contrary to Newton s law of cooling, its temperature decreases to room temperature at a rate proportional to the square root of the difference between its temperature and the ambient temperature. In a laboratory kept at degrees C, the sample is heated to a temperature of 6 degrees C. After minutes have passed, the temperature of the sample is 9 degrees C. How long after the initial heating will the sample s temperature be equal to the room temperature? 7. (a) 6 minutes (b) 8 minutes (c) minutes (d) minutes (e) 4 minutes (f) 6 minutes Answer B π/8 tan 4t dt = (a) (b) ln (c) ln (d) ln (e) ln (f) diverges Answer: F
8. The function kxe 4x x f(x) = otherwise is a probability density function for a certain value of k. Find the mean of that probability density function. (a) (b) (c) (d) 5 (e) (f) 8 9. The functions y (t) and y (t) are both solutions of the autonomous differential equation dy ( ) y dt = sin but satisfy different initial conditions: y () = and y () =. Either by solving the differential equation or, better, by thinking about its geometry (slope field), calculate lim (y (t) y (t)). t (a) (b) π (c) 4π (d) 6π (e) 8π (f) MATH 4 Second Midterm Exam - Fall 4. π sin x cos x (a) 4 6 (b) 4 5 (c) 5 (d) 4 5 (e) (f) 6. π/ x sin(x) (a) π 4 (b) π (c) π (d) π 4 (e) π (f) π. x 9 x (a) ln 7 (b) ln 9 5 (c) ln 5 (d) 4 ln 5 (e) 5 ln 4 (f) ln
4. 5. / arcsin x π (a) π (d) 6 ( ) 4 ln Answer A e x ( + e x ) / (b) π ln (c) π ln 4 (e) π + (f) π 6 (a) 8 (b) 5 (c) (d) 5 4 (e) 5 6 (f) 4 6. The function k x f(x) = x 6 otherwise is a probability density function for a certain value of k. Find the mean of that probability density function. (a) 5 4 (b) 5 (c) 4 (d) 5 (e) (f) 7. The solution of the initial-value problem satisfies y() = x dy + 5y = 6x y() = (a) (b) 4 (c) 5 (d) 6 (e) 8 (f) 9 8. On a cold winter day, when the temperature outside is degrees, Bart finds his skateboard on the roof and brings it indoors, where the temperature is 7 degrees. After being indoors for minutes, the temperature of the skateboard rises to 4 degrees. What will the temperature of the skateboard be after another minutes (i.e., 4 minutes after being brought indoors)? Assume Newton s law of cooling (and heating) applies.
(a).6 degrees (b) 4. degrees (c) 48.4 degrees (d) 56.8 degrees (e) 6.4 degrees (f) 67.6 degrees 9. The region between the x-axis and the graph of y = sin x for x π is rotated around the x-axis to generate a solid. What is the volume of the solid? (a) π Answer: B (b) π (c) π (d) 4π (e) 8π (f) 6π. Tank number holds liters of water in which 5 kg of salt is initially dissolved. At time t =, pure water begins to flow into tank number at a rate of liters per minute and the well-stirred mixture flows out at the same rate, into a second tank, which initially contains liters of pure water. The well-stirred mixture in the second tank also flows out at the same rate. If S(t) is the amount of salt (in kg) in the second tank at time t (minutes), what is the differential equation satisfied by S? (a) S = e t/5 5 (b) S = e t/5 (c) S = e t/ (d) S = e t/5 (e) S = e t/5 5 (f) S = e t/ 5 MATH 4 Second Midterm Exam - Fall 5. π/4 tan x sec 4 x (a) 4 (b) 7 6 4 (c) 5 (d) 8 5 (e) 5 (f) 6. ln x e x (a) 6 8 ln (b) 8 + ln (c) 5 4 6 ln (d) 7 7 4 ln (e) 7 9 + 8 ln (f) 4 + ln
. 6x + x + (x + )(6x + ) (a) ln + arctan 4 (b) ln + 4 arctan 4 (c) ln + arctan (d) 4 ln arctan (e) ln + arctan 4 (f) ln arctan Answer: E 4. e 9 x ln x (a) + 7e 8 (b) + 9e (c) + 5e 8 (d) + 5e 6 (e) 4 + 8e (f) + 7e 6 5. x(9 + (ln x) ) / (a) (b) (c) 4 (d) 4 (e) 5 4 (f) 6. The function k < x f(x) = x / otherwise is a probability density function for a certain value of k. Find the mean of that probability density function. (a) 5 (b) 4 (c) 4 (d) (e) 5 (f) 7. The solution of the initial-value problem dy = 4x y y() = 4 satisfies y() = (a) (b) (c) (d) 4 (e) 9 (f) Answer: B 8. In a second-order chemical reaction, the reactant R is used up in such a way that the amount of it present decreases at a rate proportional to the square of the amount present. Suppose this reaction begins with grams of R present, and after seconds
there are only grams left. How long after the beginning of the reaction will there be only 4 grams left? (a) 4 seconds (b) 45 seconds (c) 48 seconds (d) 5 seconds (e) 64 seconds (f) 8 seconds 9. The region between the x-axis and the graph of y = cos x for 4π x 4π is 8 rotated around the x-axis to generate a solid. What is the volume of the solid? (a) π (b) π (c) π (d) 4π (e) 8π (f) 6π. Consider the initial-value problem: y y = 5 sin x y() = A. For which value of the constant A will the solution be periodic (with period π)? (a) A = 4 (b) A = 4 (c) A = (d) A = (e) A = (f) A = Answer: E