Mathematics 102 Review Questions Poblem 1: Multiple Choice Questions 1: Conside the function y = f(x) = 3e 2x 5e 4x (a) The function has a local maximum at x = (1/2)ln(10/3) (b) The function has a local minimum at x = (1/2)ln(10/3) (c) The function has a local maximum at x = ( 1/2)ln(3/5) (d) The function has a local minimum at x = (1/2)ln(3/5) (e) The function has a local maximum at x = ( 1/2)ln(3/20) 2: Let m 1 be the slope of the function y = 3 x at the point x = 0 and let m 2 be the slope of the function y = log 3 x at x = 1 Then (a) m 1 = ln(3)m 2 (b) m 1 = m 2 (c) m 1 = m 2 (d) m 1 = 1/m 2 (e) m 1 = m 2 /ln(3) 3: Conside the cuve whose equation is x 4 +y 4 +3xy = 5. The slope of the tangent line, dy/dx, at the point (1,1) is (a) 1 (b) -1 (c) 0 (d) -4/7 (e) 1/7 4: Two kinds of bacteia ae found in a sample of tainted food. It is found that the population size of type 1, N 1 and of type 2, N 2 satisfy the equations dn 1 dt = 0.2N 1, N 1 (0) = 1000, dn 2 dt = 0.8N 2, N 2 (0) = 10. Then the population sizes ae equal N 1 = N 2 at the following time: (a) t = ln(40) (b) t = ln(60) (c) t = ln(80) (d) t = ln(90) (e) t = ln(100) 5: In a conical pile of sand the atio of the height to the base adius is always /h = 3. (Recall that the volume of a cone with height h and adius is V = (π/3) 2 h.) If the volume is inceasing at ate 3 m 3 /min, how fast (in m/min) is the height changing when h = 2m? (a) 1/(12π) (b) (1/π) 1/3 (c) 27/(4π) (d) 1/(4π) (e) 1/(36π) 6: Newton s Law of cooling leads to a diffeential equation that pedicts the tempeatue T(t) of an object whose initial tempeatue is T 0 in an envionment whose tempeatue is E. The pedicted tempeatue is given by T(t) = E + (T 0 E)e kt whee t is time and k is a constant. Shown in Fig 1 on the following page is some data points plotted as ln(t(t) E) vesus time in minutes. The ambient tempeatue was E = 22 C. Also shown on the gaph is the line that best fits those 11 points. Accoding to this gaph, the value of the constant k is appoximately (a) -1/27 (b) e 1/27 (c) 1/27 (d) 4/27 (e) ln(1/27) 1
ln ( T(t) - E ) 4.1 4.0 3.5 Bestfitline 3.0 10 20 2.9 time in minutes 0.0 30.0 Figue 1: Figue fo Multiple Choice poblem 6 2
Long Answe Poblems Poblem 2a: Fig. 2 shows a 1 km ace tack with cicula ends. Find the values of x and y that will maximize the aea of the ectangle. y x Figue 2: This shape is investigated in both poblems 1 and 2. Poblem 2b: Now suppose that Fig 2 shows the shape of a leaf of some plant. If the plant gows so that x inceases at the ate 2 cm/yea and y inceases at the ate 1 cm/yea, at what ate will the leaf s entie aea be inceasing? Poblem 3: Find the dimensions of the lagest ectangle that can fit exactly into a cicle whose adius is 10 cm. Poblem 4: A cell of the bacteium E.coli has the shape of a cylinde with two hemispheical h Figue 3: Shape of the object descibed in Poblem 4. Note: Useful volumes and suface aeas: Fo a hemisphee, V = (2/3)π 3,S = 2π 2. Fo a cylinde, V = π 2 h and S = 2πh (not including end caps) caps, as shown in Fig 3. Conside this shape, with h the height of the cylinde, and the adius of the cylinde and hemisphees. (a) Find the values of and h that lead to the lagest volume fo a fixed constant suface aea, S= constant. (b) Descibe o sketch the shape you found in (a). (c) A typical E. coli cell has h = 1µm and = 0.5µm. Based on you esults in (a) and (b), would you agee that E. coli has a shape that maximizes its volume fo a fixed suface aea? (Explain you answe). 3
Poblem 5: If the cell shown above in Fig 3 is gowing so that the height inceases twice as fast as the adius. If the adius is gowing at 1 µm pe day at what ate will the volume of the cell incease? (Leave you answe in tems of the height and adius of the cell.) Poblem 6:(a) It takes you 1 hs (total) to tavel to and fom UBC evey day to study Philosophy 101. The amount of new leaning (in abitay units) that you can get by spending t hous at the univesity is given appoximately by L P (t) = 10t 9+t. How long should you stay at UBC on a given day if you want to maximize you leaning pe time spent? (Time spent includes tavel time.) (b) If you take Math 10000 instead of Philosophy, you leaning at time t is L M (t) = t 2. How long should you stay at UBC to maximize you leaning in that case? Poblem 7: The atoms of some adioactive mateial ae known to have a pobability k of decaying pe unit time. We will use y(t) to denote the amount of adioactivity emaining at time t. Suppose that thee is 100 gm of this adioactive substance initially, at time t = 0. Conside what happens duing a small time inteval t. How much adioactive mateial is left at time t 1 = t? At time t s = 2 t? Wite down an equation that links y(t n+1 ) to y(t n ) whee t n = n t. Convet this equation to a diffeential equation. If the half life of this substance is 1 day, find out how much is left afte 5 days. What is the value of k in this case, and how much adioactivity is left at time t? Poblem 8: Given a population of 6 billion people on Planet Eath, and using the appoximate gowth ate of = 0.0125 pe yea, how long ago was this population only 1 million? Assume that the gowth has been the same thoughout histoy (which is not actually tue). Poblem 9: Find citical points fo the function y = e x (1 ln(x)) fo 0.1 x 2 and classify thei types. Poblem 10: The function y = ln(x) e x has a citical point in the inteval 0.1 x 2. It is not possible to solve fo the value of x at that point, but it is possible to find out what kind of citical point that is. Detemine whethe that point is a local maximum, minimum, o inflection point. Poblem 11: (a) Conside the polynomial y = 4x 5 15x 4. Find all local minima maxima, and inflection points fo this function. (b) Find the global minimum and maximum fo the function in poblem (1) on the inteval [-1,1]. Poblem 12: Conside the polynomial y = x 5 x 4 +3x 3. Use calculus to find all local minima maxima, and inflection points fo this function. (b) Find the global minimum and maximum fo the function in Poblem (3) on the inteval [-1,1]. Poblem 13: Find a polynomial of thid degee that has a local maximum at x = 1, a zeo and an inflection point at x = 0, and goes though the point (1,2). Hint: assume p(x) = ax 3 +bx 2 +cx+d and find the values of a, b, c, d. 4
Poblem 14: Find a linea appoximation to the function y = x 2 at the point whose x coodinate is x = 2. Use you esult to appoximate the value of (2.0001) 2. Poblem 15: The Lennad-Jones potential, V(x) is the potential enegy associated with two unchaged molecules a distance x apat, and is given by the fomula V(x) = a x 12 b x 6 Molecules would tend to adjust thei sepaation distance so as to minimize this potential. Find any local maxima o minima of this potential. Find the distance between the molecules, x, at which V(x) is minimized and use the second deivative test to veify that this is a local minimum. Poblem 16: Conside an object thown upwads with initial velocity v 0 > 0 and initial height h 0 > 0. Then the height of the object at time t is given by y = f(t) = 1 2 gt2 +v 0 t+h 0. Find citical points of f(t) and use both the second and fist deivative tests to establish that this is a local maximum. Poblem 17: The figue (not dawn to scale) shows a tumo mass containing a necotic (dead) coe (adius 2 ), suounded by a laye of actively dividing tumo cells. The entie tumo can be assumed to be spheical, and the coe is also spheical 1. (a) If the necotic coe inceases at the ate 3 cm 3 /yea and the volume of the active cells inceases by 4 cm 3 /yea, at what ate is the oute adius of the tumo ( 1 ) changing when 1 = 1 cm. (NOTE: Show all you wok, leave you answe as a faction in tems of π; indicate units with you answe.) necotic coe 2 1 active cells (b) At what ate (in cm 2 /y) does the oute suface aea of the tumo incease when 1 = 1cm? Poblem 18: Shown below is a majo atey, (adius R) and one of its banches (adius ). A labeledschematic diagamisalso shown (ight). Thelength0AisL, andthedistance between 0and P is d, whee 0P is pependicula to 0A. The location of the banch point (B) is to be detemined so that the total esistance to blood flow in the path ABP is as small as possible. (R,,d,L ae positive constants, and R >.) 1 Recall that the volume and suface aea of a sphee ae V = (4/3)π 3, S = 4π 2 5
R L d 0 B A P (a) Let the distance between 0 and B be x. What is the length of the segment BA and what is the length of the segment BP? (b) The esistance of any blood vessel is popotional to its length and invesely popotional to its adius to the fouth powe 2. Based on this fact, what is the esistance, T 1, of segment BA and what is the esistance, T 2, of the segment BP? (c) Find the value of the vaiable x fo which the total esistance, T(x) = T 1 +T 2 is a minimum. 2 z is invesely popotional to y means that z = k/y fo some constant k 6