n! 7. n1000 Math 1280-1 n3 5n 8. 3n−n5 − Peter Alfeld. 4.(1 pt) Determine whether the sequences are in- creasing, decreasing, or not monotonic. If increasing, enter 1 as your answer. If decreasing, enter 1as your WeBWorK Problem Set 1. answer. If not monotonic, enter 0 as your answer− . 1 1. an = 2n+6 Due 2/7/06 at 11:59 PM. n 2 2. an = n−+2 Get to work on this set right away and an- √ + 3. a = n 2 • swer these questions well before the deadline. n 6n+2 4. a = cosn Not only will this give you the chance to fig- n 2n ure out what’s wrong if an answer is not ac- 5.(1 pt) Consider the sequence cepted, you also will avoid the likely rush and ncos(nπ) congestion prior to the deadline. a = . n 2n 1 − Write the first five terms of an, and find limn ∞ an. Procrastination is hazardous! If the sequence diverges, enter ”divergent” in the→ an- 1.(1 pt) Determine whether the sequence is diver- swer box for its limit. gent or convergent. If it is convergent, evaluate its a) First five terms: , , , , . limit. If it diverges to infinity, state your answer as b) limn ∞ an = . → ”INF” (without the quotation marks). If it diverges 6.(1 pt) Consider the sequence to negative infinity, state your answer as ”MINF”. If it diverges without being infinity or negative infinity, ln(1/n) an = . state your answer as ”DIV”. √2n 4n7 + sin2(11n) lim Write the first five terms of an, and find limn ∞ an. n ∞ 6 + → → n 10 If the sequence diverges, enter ”divergent” in the an- swer box for its limit. 2.(1 pt) Find the limit of the sequence whose terms a) First five terms: , , , , . are given by b) limn ∞ an = . a = (n2)(1 cos( 5.5 )). → n − n 7.(1 pt) Suppose (1 pt) Match each sequence below to statement 3. 1 2 3 4 5 that BEST fits it. a1 = ,a2 = ,a3 = ,a4 = ,a5 = . 2 1 3 1 4 1 5 1 6 1 STATEMENTS − 2 − 3 − 4 − 5 − 6 Z. The sequence converges to zero; a) Find an explicit formula for an: I. The sequence diverges to infinity; . F. The sequence has a finite non-zero limit; b) Determine whether the sequence is convergent D. The sequence diverges. or divergent: . SEQUENCES (Enter ”convergent” or ”divergent” as appropriate.) ( 1 ) c) If it converges, find limn ∞ an = 1. nsin n → (ln(n)) . 2. n 8.(1 pt) Suppose n100 3. (1.01)n 4. ln(ln(ln(n))) 1 2 a1 = 2,an+1 = an + . 5. sin(n) 2  an  6. arctan(n + 1) Find limn ∞ an = . 1 → ∞ Hint: Let a = limn ∞. Then, since an+1 = 7 → 5. 1 2 1 2 ∑ n10 16 an + , we have a = a + . Now solve n=1 − 2  an  2  a for a. 11.(1 pt) Determine the sum of the following se- ∞ 7 9.(1 pt) Consider the series ∑n=1 . Let sn be the ries. n+6 ∞ n 1 n-th partial sum; that is, ( 4) − n ∑ − n 7 n= 6 sn = ∑ . 1 i=1 i + 6 (1 pt) Determine the sum of the following se- Find s4 and s8 12. ries. s4 = ∞ 4n + 6n s8 = ∑ ( n ) 10.(1 pt) Match each of the following with the cor- n=1 11 rect statement. C stands for Convergent, D stands for Divergent. 13.(1 pt) Express 8.846846846... as a rational p ∞ 1 number, in the form q 1. ∑ 5 where p and q have no common factors. 6 + √n3 n=1 p = and ∞ 7 2. q = ∑ n(n + 7) n=1 14.(1 pt) A ball drops from a height of 18 feet. ∞ 2 + 7n Each time it hits the ground, it bounces up 30 per- 3. ∑ n n=1 8 + 9 cents of the height it fall. Assume it goes on forever, ∞ ln(n) 4. ∑ find the total distance it travels. n=1 10n

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

2 A. convergent Math 1280-1 • B. divergent • 5.(1 pt) Determine whether the following series is Peter Alfeld. ∞ 1 ( )n+1 ∑ 1 1.1 WeBWorK Problem Set 2. n=1 − 5n A. conditionally convergent Due 2/7/06 at 11:59 PM. • B. absolutely convergent • Get to work on this set right away and an- C. divergent • • swer these questions well before the deadline. 6.(1 pt) Determine whether the following series is Not only will this give you the chance to fig- ure out what’s wrong if an answer is not ac- ∞ 1 ( 1)n cepted, you also will avoid the likely rush and ∑ − √n2 1 n=2 − congestion prior to the deadline. A. absolutely convergent • B. conditionally convergent Procrastination is hazardous! • C. divergent (1 pt) Consider the series: • 1. 7.(1 pt) Each of the following statements is an at- ∞ 3 3 tempt to show that a given series is convergent or not using the Comparison Test (NOT the Limit Compari- ∑ (k 1)2 − k2  k=10 − son Test.) For each statement, enter C (for ”correct”) a) Determine whether the series is convergent or if the argument is valid, or enter I (for ”incorrect”) divergent: . if any part of the argument is flawed. (Note: if the (Enter ”convergent” or ”divergent” as appropriate.) conclusion is true but the argument that led to it was b) If it converges, find its sum: . wrong, you must enter I.) If the series diverges, enter here ”divergent” again. ln(n) 1. For all n > 2, > 1 , and the series ∑ 1 (1 pt) Determine the convergence or divergence n2 n2 n2 2. converges, so by the Comparison Test, the se- of the following series. ln(n) ries ∑ n2 converges. ∞ √ 1 2 1 2n + 1 2. For all n > 1, nln(n) < n , and the series 2∑ n ∑ 2 n=1 n diverges, so by the Comparison Test, the se- 1 A. convergent ries ∑ nln(n) diverges. • B. divergent n > ln(n) < 1 1 • 3. For all 1, n2 n1.5 , and the series ∑ n1.5 3.(1 pt) Determine the convergence or divergence converges, so by the Comparison Test, the se- of the following series. ln(n) ries ∑ n2 converges. 4. For all n > 2, n < 2 , and the series 2∑ 1 ∞ 3k + k n3 7 n2 n2 ∑ converges, so by−the Comparison Test, the se- n=1 k! n ries ∑ n3 7 converges. A. convergent − • 5. For all n > 1, arctan(n) < π , and the series B. divergent n3 2n3 • π 1 2 ∑ n3 converges, so by the Comparison Test, 4.(1 pt) Determine the convergence or divergence arctan(n) of the following series. the series ∑ n3 converges. ln(n) 6. For all n > 2, > 1 , and the series ∑ 1 di- ∞ 1 n n n √n 1 cos verges, so by the Comparison Test, the series ∑ ln(n) n=1  − n ∑ n diverges. 1 ∞ 1 6. 8.(1 pt) The three series ∑An, ∑Bn, and ∑Cn have ∑ n=1 √n terms 1 1 1 10.(1 pt) Find the convergence set of the given A = , B = , C = . n n7 n n4 n n power series: Use the Limit Comparison Test to compare the fol- ∞ 2nxn lowing series to any of the above series. For each of ∑ the series below, you must enter two letters. The first n=1 n! is the letter (A,B, or C) of the series above that it can The above series converges for < x < . be legally compared to with the Limit Comparison Enter ”infinity” for ∞ and ”-infinity” for ∞. Test. The second is C if the given series converges, − 11.(1 pt) A famous sequence fn, called the Fi- or D if it diverges. So for instance, if you believe bonacci Sequence after Leonardo Fibonacci, who in- the series converges and can be compared with series troduced it around A.D. 1200, is defined by the recur- C above, you would enter CC; or if you believe it di- sion formula verges and can be compared with series A, you would enter AD. f1 = f2 = 1, fn+2 = fn+1 + fn. ∞ 5n2 + 5n6 1. Find the radius of convergence of ∑ 4n7 + 6n3 4 n=1 − ∞ 4n4 + n7 ∞ 2. f xn. ∑ 187n11 + 6n4 + 5 ∑ n n=1 n=1 ∞ 5n4 + n2 5n 3. − Radius of convergence: . ∑ 6n11 5n9 + 1 n=1 − 12.(1 pt) Find the interval of convergence for the given power series. 9.(1 pt) Select the FIRST correct reason why the ∞ n4(x + 7)n given series diverges. ∑ 14 ( n)(n 3 ) A. Diverges because the terms don’t have limit n=1 10 zero The series is convergent: B. Divergent geometric series from x = , left end included (Y,N): C. Divergent p series to x = , right end included (Y,N): D. Integral test 13.(1 pt) Match each of the power series with its E. Comparison with a divergent p series interval of convergence. F. Diverges by limit comparison test ∞ (x 5)n G. Cannot apply any test done so far in class 1. − ∑ (n!)5n ∞ n=1 cos(nπ) ∞ n!(5x 5)n 1. ∑ 2. − n=1 ln(5) ∑ 5n ∞ n=1 1 ∞ (5x)n 2. ∑ 3. n=1 nln(n) ∑ n5 ∞ 2 n n=1 (n + 1)(6 + 1) ∞ (x 5)n 3. ∑ 2n 4. − n=1 6 ∑ (5)n ∞ n=1 n (2n)! 1 1 4. ( 1) A. [ − , ] ∑ − (n!)2 5 5 n=1 B. 5/5 ∞ 7n + 4 C. {( ∞,}∞) 5. ∑ n − n=1 ( 1) D. (0,10) − 2 14.(1 pt) Find the power series representation for 17.(1 pt) Find the Taylor series in (x a) through − (x a)3 for 1 − f (x) = ( + x)2 1 f (x) = 2 x + 3x2 x3, a = 1 and specify the radius of convergence. − − − f (x) = + (x + 1)+ (x + 1)2+ ∞ (x + 1)3. en pn f (x) = ∑ ( 1) anx , n=1 − 18.(1 pt) Suppose that f (x) and g(x) are given by where en = the power series f (x) = 6 + 4x + 7x2 + 5x3 + A. n ··· • B. n - 1 and 2 3 • C. 0 g(x) = 6 + 4x + 4x + 3x + . • By multiplying power series,··· find the first few where a = , n terms of the series for the product and p = . n h(x) = f (x) g(x) = c + c x + c x2 + c x3 + . Radius of convergence: . · 0 1 2 3 ··· c0 = 15.(1 pt) Find the power series representation for c1 = 2 c = f (x) = xex . 2 c3 = ∞ ∞ 1 6x n pn 19.(1 pt) Suppose that = c x . f (x) = ∑ x , (13 + x) ∑ n n=0 an! n=0 Find the first few coefficients. where an = and pn = . c0 = 16.(1 pt) Find the Taylor series in (x a) through − c1 = (x a)3 for − c2 = π c3 = f (x) = tanx, a = 4 c4 = π π 2 Find the radius of convergence R of the power se- f (x) = + x 4 + x 4 + 3 4 − − ries. x π + O x π .

− 4 − 4 R = .



Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

3 4.(1 pt) The Taylor series for f (x) = ln(sec(x)) at ∞ Math 1280-1 n a = 0 is ∑ cn(x) . n=0 Peter Alfeld. Find the first few coefficients. c0 = c1 = WeBWorK Problem Set 4. c2 = c3 = Due 2/9/06 at 8:00 AM. c4 = Find the exact error in approximating ln(sec( 0.1)) Get to work on this set right away and an- − • by its fourth degree Taylor polynomial at a = 0. swer these questions well before the deadline. The error is Not only will this give you the chance to fig- ( ) ure out what’s wrong if an answer is not ac- 5.(1 pt) Let T5 x be the fifth degree Taylor poly- ( ) = ( . ) = cepted, you also will avoid the likely rush and nomial of the function f x cos 0 2x at a 0. T (x). congestion prior to the deadline. A. Find 5 (Enter a function.) T5(x) = B. Find the largest integer k such that for all x for Procrastination is hazardous! which x < 1 the Taylor| | polynomial T (x) approximates f (x) with 1.(1 pt) Represent the function x0.5 as a power se- 5 ∞ error n 1 ries cn(x 9) . less than . ∑ − 10k n=0 k = c0 = x c = 2 1 6.(1 pt) Let F(x) = sin(6t ) dt. c2 = Z0 c3 = Find the MacLaurin polynomial of degree 7 for F(x). Find the left endpoint of the interval of conver- gence. Use this polynomial to estimate the value of 0.79 left end = . sin(6x2) dx. Find the right endpoint of the interval of conver- Z0 gence. 7.(1 pt) Find Taylor series of function f (x) = ln(x) right end = . at a = 7. ∞ 2.(1 pt) Compute the 9th derivative of n ( f (x) = ∑ cn(x 7) ) x3 n=0 − f (x) = arctan c0 =  5  c1 = at x = 0. c2 = (9) f (0) = c3 = Hint: Use the MacLaurin series for f (x). c4 = Find the interval of convergence. 3.(1 pt) Compute the 9th derivative of The series is convergent: cos 6x2 1 from x = , left end included (Y,N): f (x) = − x3
to x = , right end included (Y,N): at x = 0. 8.(1 pt) Evaluate (9) f (0) = 2 ln(1 x) + x + x Hint: Use the MacLaurin series for f (x). lim − 2 x 0 4x3 1 → Hint: Use power series. 15.(1 pt) Let C be a semicircle of radius r > 0 cen- tered at the origin. 9.(1 pt) Evaluate Let P be a point on the x-axis whose coordinates are 3x3 3 9 6 P = (r + rt,0) where t > 0. e− 1 + 3x 2 x lim − − L P x 0 13x9 Let be a line through which is tangent to the semi- → Hint: Use power series. circle. Let A denote the triangular region between the circle 10.(1 pt) Assume that sin(x) equals its Maclaurin and the line and above the x-axis (see figure.) series for all x. Use the Maclaurin series for sin(7x2) to evaluate the integral 0.72 sin(7x2) dx Z0 . Your answer will be an infinite series. Use the first two terms to estimate its value. (Click on image for a larger view ) 11.(1 pt) Find T5(x): Taylor polynomial of degree Find the exact area of A in terms of r and t. 5 of the function f (x) = cos(x) at a = 0. Area(A) = . (You need to enter function.) Use a Maclaurin Polynomial to get a simple ap- T5(x) = proximation for the area of A for small t. : Find all values of x for which this approximation is Area(A) . within 0.000366 of the right answer. Assume for sim- ≈ plicity that we limit ourselves to x 1. x | | ≤ 16.(1 pt) Suppose that we use the bisection algo- | | ≤ rithm to approximate r = √56 which the greatest ( ) 12.(1 pt) Let T6 x : be the Taylor polynomial of zero of the function f (x) = x2 56. We begin by degree 6 of the function f (x) = cos(x) at a = 0. − finding two numbers, say, a1 = 7 and b1 = 8 which Suppose you approximate f (x) by T6(x), and if bracket the zero. This is because f (a1) < 0 and x 1, what is the bound for your error of your es- f (b ) > 0. Then we find m = (a + b )/2 = 7.5 and |timate?| ≤ (Hint: use the alternating series approxima- 1 1 1 1 h1 = (b1 a1)/2 = 0.5. tion.) We proceed− with the bisection algorithm. Suppose that an and bn bracket the zero. Then we compute 13.(1 pt) Let Tk(x): be the Taylor polynomial of mn = (an + bn)/2 and hn = bn an /2. If f (mn) = degree k of the function f (x) = sin(x) at a = 0. | − | 0 we stop because r = mn is the desired zero. If Suppose you approximate f (x) by Tk(x), and if f (mn) > 0 then mn becomes the new right endpoint, x 1, how many terms do you need (that is, what | | ≤ so we set an+1 := an and bn+1 := mn. If f (mn) < 0 is k) for you to have your error to be less than 1 ? 5040 then mn becomes the new left endpoint, so we set (Hint: use the alternating series approximation.) an+1 := mn and bn+1 := bn. Then mn is an approxi- mation to r with an error of hn 14.(1 pt) Let T8(x): be the Taylor polynomial of Complete the following table: degree 8 of the function f (x) = ln(1 + x) at a = 0. n an bn hn mn Suppose you approximate f (x) by T8(x), find all 1 7 8 0.5 7.5 positive values of x for which this approximation is 2 within 0.001 of the right answer. (Hint: use the alter- 3 nating series approximation.) 4 0 < x ≤ 5 2 Then is an approximation so far to r with (b) Midpoint Rule an error of . 17.(1 pt) (c) Simpson’s Rule Suppose that we use Newton’s Method to approxi- mate r = √3 504 which the zero of the function f (x) = 20.(1 pt) Use Simpson’s Rule and all the data in 3 x 504. We begin with a good guess, say x1 = 9. the following table to estimate the value of the inte- Then− Newton’s Method proceeds by the recursion 2 gral ydx. f (xn) Z xn+1 = xn . 4 − f 0(xn) − Compute the first few terms of the sequence xn ob- x -4 -3 -2 -1 0 1 2 tained from Newton’s method. y -1 5 -2 -5 -7 -1 9 x1 = 9 x2 = 21.(1 pt) x3 = Determine an n so that the trapezoidal rule will ap- x4 = proximate the integral x5 = 10 7dx x6 = Z x7 = 6 7 + x with an error E satisfying E 0.0005. 18.(1 pt) FIXED POINT ALGORITHM. n | n| ≤ If g is a continuous function taking the interval [a,b] The theoretical error bound for the Trapezoid rule to itself, then it has a fixed point r [a,b] so that is given by r = g(r). If in addition, g is differentiable∈ and satis- (b a)3 fies g0(x) M for all a x b where M < 1 is a E = − f 00(c) | | ≤ ≤ ≤ n − 12n2 constant, then the recursion xn+1 = g(xn), x1 [a,b] yields a sequence that converges x r as n ∈ ∞. where c is some point between a and b. It predicts n → → Consider the equation x = √12 + x. Using the that the desired accuracy will be achieved if the num- ber of terms n is at least . Fixed Point Algorithm starting with x1 = 3, find x2 to x . 22.(1 pt) Suppose that we use Euler’s method to 7 approximate the solution to the differential equation x1 = 3 x = dy x2 2 = ; y(0.3) = 9. x3 = dx y x = 4 Let f (x,y) = x2/y. x = 5 We let x = 0.3 and y = 9 and pick a step size x = 0 0 6 h = 0.2. x = 7 Euler’s method is the the following algorithm. Solve for the (positive) x in x = √12 + x. From x and y , our approximations to the solution x = n n of the differential equation at the nth stage, we find Evaluate 12 + 12 + √12 + . the next stage by computing q p ··· x = x + h, y = y + h f (x ,y ). 19.(1 pt) Given the following integral and value n+1 n n+1 n · n n of n, approximate the following integral using the Complete the following table: methods indicated (round your answers to six deci- n xn yn mal places): 0 0.3 9 1 4x2 1 e− dx,n = 4 Z0 2 (a) Trapezoidal Rule 3 4 3 5 Thus the actual value of the function at the point The exact solution can also be found using separa- x = 1.3 tion of variables. It is y(1.3) = . y(x) =

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

4 Peter Alfeld WW Prob Lib1 Math course-section, semester year WeBWorK problems. WeBWorK assignment 3 due 2/17/06 at 11:59 PM.

Find the distance of the point ( 4, 6) from the 1.(1 pt) line through (6, 8) which points in− the−direction of − A child walks due east on the deck of a ship at 1 2I + 4J. miles per hour. The ship is moving north at a speed of 17 miles per 7.(1 pt) hour. Let T be the triangle with vertices at Find the speed and direction of the child relative to (10,8),(6, 1),( 5,2). The area of T is the surface of the water. − − Speed = mph Hint: Use the projection formula to find the length The angle of the direction from the north = of an altitude orthogonal to any chosen base. (radians) 8.(1 pt) 2.(1 pt) Find the vector V which makes an angle of 70 de- Find a b if grees with the vector W = 2I 1J and which is of · − − a = 9, the same length as W and is counterclockwise to W. | | b = 5, I+ J | | π and the angle between a and b is 4 radians. 9.(1 pt) a b = While a planet P rotates in a circle about its sun, a · 3.(1 pt) moon M rotates in a circle about the planet, and both Gandalf the Grey started in the Forest of Mirk- motions are in a plane. Let’s call the distance be- wood at a point with coordinates (-1, -1) and arrived tween M and P one lunar unit. Suppose the distance of P from the sun is 3.2 103 lunar units; the planet in the Iron Hills at the point with coordinates (0, 3). × If he began walking in the direction of the vector makes one revolution about the sun every 3 years, and v = 4I + 2J and changes direction only once, when the moon makes one rotation about the planet every he turns at a right angle, what are the coordinates of 0.1 years. Choosing coordinates centered at the sun, 3 the point where he makes the turn. so that, at time t = 0 the planet is at (3.2 10 ,0), 3 × ( , ) and the moon is at (3.2 10 ,1), then the location of the moon at time t, where× t is measured in years, is (1 pt) 4. (x(t),y(t)), where An object is at rest on the plane. Three forces, x(t)= , , are acting on the object. If V W X y(t)= V = 3I 7J,W = 2I 7J, 10.(1 pt) The position of a particle in motion in the − − − plane at time t is then X must be X(t) = 9t I + sin( 7t) J. + I J At time−any t, determine− the following: 5.(1 pt) (a) the speed of the particle is: The distance between the two parallel lines L1 : (b) the unit tangent vector to X(t) is: 3x 2y = 6,L : 3x 2y = 1 is − − 2 − − − I+ J (Note that, the unit tangent vector to X(t) is defined 6.(1 pt) as V(t)/ V(t) .) | |

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

1 Peter Alfeld MATH 1280-1 Spring 2006 Homework Set 5 due 2/28/06 at 11:59 PM You may need to give 4 or 5 significant digits for some (floating point) numerical answers in order to have them accepted by the computer.

Distance = 1.(1 pt) 7.(1 pt) Enter T or F depending on whether the If a = (-6, 9, -9) and b = (1, -8, 1), statement is true or false. (You must enter T or F find a b = . · – True and False will not work.) 2.(1 pt) 1. Two planes parallel to a line are parallel. What is the angle in radians between the vectors 2. Two lines either intersect or are parallel. a = (1, 10, -8) and 3. Two lines perpendicular to a third line are par- b = (4, -2, 6)? allel. Angle: (radi- 4. Two planes either intersect or are parallel. ans) 5. Two planes perpendicular to a third plane are 3.(1 pt) parallel. Find a unit vector in the same direction as a = (6, 6. A plane and a line either intersect or are par- 10, 7). allel. ( , 7. Two lines parallel to a third line are parallel. , 8. Two planes perpendicular to a line are paral- ) lel. 4.(1 pt) 9. Two lines perpendicular to a plane are paral- The equation X (t) = A + tL is the parametric lel equation of a line through the point P : (2, 3,1). 10. Two planes parallel to a third plane are paral- The parameter t represents distance from the−point lel. P, directed so that the I component of L is positive. 11. Two lines parallel to a plane are parallel. We know that the line is orthogonal to the plane with 8.(1 pt) Consider the planes 2x + 3y + 1z = 1 and equation 8x + 3y 9z = 2. Then 2x + 1z = 0. − A= I+ J+ K (A) Find the unique point P on the y-axis which is L= I+ J+ K on both planes. ( , , ) 5.(1 pt) Given vectors u and v such that (B) Find a unit vector u with positive first coordi- u v = 0I 1J + 10K, nate that is parallel to both planes. find:× − I + J + K a) v u = (C) Use the vectors found in parts (A) and (B) to ( ×, find a vector equation for the line of intersection of , the two planes,r(t) = ). I + J + K b) (u v) (u v) = 9.(1 pt) The position of a particle in motion in the . × · × plane at time t is c) (u v) (u v) = X(t) = exp(5.1t)I + exp(1t)J. ( ,× × × At time t = 0, determine the following: , (a) The speed of the particle is: ). (b) Find the unit tangent vector to X(t): + 6.(1 pt) I J What is the distance from the point (0, 8, -3) to the (c) The tangential acceleration: xz-plane? (d) The normal acceleration: 1 10.(1 pt) The position of a particle in motion in the B. The acceleration vector a(t) = I+ plane at time t is J+ K X(t) = tI + ln(cos(t))J. Note: the coefficients in your answers must be en- At time t = ( 0.7π)/2, determine the following: tered in the form of expressions in the variable t; e.g. (a) the unit tangent− vector: “5 cos(2t)” I+ J 13.(1 pt) Consider the helix X(t) = (b) the unit normal vector to X(t): I+ π (cos(2t),sin(2t),4t). Compute, at t = 6 : J A. The unit tangent vector T = ( , , (c) the acceleration vector: ) I+ J B. The unit normal vector N = ( , , (d) the curvature: ) 11.(1 pt) 14.(1 pt) (A) Find the parametric equations for the Consider the vector functions line through the point P = (-4, -5, 0) that is perpen- X(t) = 3I + cos(10t)J, Y(t) = sin(9t)J + 8K. dicular to the plane 3x + 4y + 2z = 1. Use ”t” as your variable,− t = 0 should correspond to P, Let and the velocity vector of the line should be the same Z(t) = X(t) Y(t). × as the normal vector to the plane found directly from Then its equation. dZ (t)= I+ J+ K. x = dt y = 12.(1 pt) If X(t) = cos( 7t)I + sin( 7t)J + 7tK, z = compute: − − (B) At what point Q does this line intersect the yz- A. The velocity vector v(t) = I+ plane? J+ K Q = ( , , )

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

2 Peter Alfeld MATH 1280-1 Spring 2006 Homework Set 6 due 3/24/06 at 11:59 PM You may need to give 4 or 5 significant digits for some (floating point) numerical answers in order to have them accepted by the computer.

A. Half plane 1.(1 pt) B. Plane Match the function with the description of its level C. Cone sets (the sets z =constant). D. Sphere A. a collection of circles centered at the origin. E. Circular Cylinder B. a collection of ellipses. F. Elliptic or Circular Paraboloid C. a collection of hyperbolas. π 1. φ = 3 D. a collection of parallel lines. 2. ρ = 2cos(φ) E. a collection of parabolas. 3. r = 2cos(θ) 1. z = √1 + x + y 4. θ = π 1 3 2. z = (x + y 1)− 5. r = 4 − 3. xy + z2 = 1 6. ρcos(φ) = 4 4. z = x2 2xy + y2 + x + y 7. r2 + z2 = 16 − 5. z = 5x 3y 8. ρ = 4 − 2 2 6. z = x2 + 2y2 − 9. z = r 7. z = 15 x2
y2 5.(1 pt) Let P be the point ( 1,3, 4) in cartesian p − − − − 2.(1 pt) coordinates. Consider the equation xz2 + 4yz 2lnz = 1 as A. The cylindrical coordinates of P are r = defining z implicitly as a function of−x and y. The− val- , θ = , z = . ∂z ∂z B. The spherical coordinates of P are ρ = , ues of and at ( 5,1,1) are and . ∂x ∂y − θ = , φ = . 3.(1 pt) Match the surfaces with the appropriate de- 6.(1 pt) Let P be the point with the spherical coor- scriptions. dinates ρ = 5, φ = π/3, θ = π/4. 1. z = 2x2 + 3y2 A. The cylindrical coordinates of P are r = 2. z = y2 2x2 , θ = , z = . 3. z = 2x−+ 3y B. The cartesian coordinates of P are x = , 4. z = x2 y = , z = . 5. x2 + y2 = 5 7.(1 pt) The vectors U = 0.989992496600445I+ − 6. x2 + 2y2 + 3z2 = 1 0.141120008059867J, V = 0.141120008059867I − − 7. z = 4 0.989992496600445J form a base for the plane; that A. ellipsoid is, they are orthogonal vectors of length 1. Let B. horizontal plane X = 5I + 9J. Then we can write C. hyperbolic paraboloid X = uU + vV with u = , v = D. parabolic cylinder 8.(1 pt) Let L be the line y = 2, x = 7z. If we rotate E. nonhorizontal plane L around the x-axis, we get a surface whose equation F. elliptic paraboloid is Ax2 + By2 +Cz2 = 1, where G. circular cylinder A= , B = , C = . 4.(1 pt) 9.(1 pt) Match the surfaces with the appropriate de- Match the given equation with the verbal descrip- scriptions. tion of the surface: A. ellipse. 1 B. parabola. Note: Your answers should be numbers C. Two lines intersecting at the origin. C. Find the directional derivative of f at P in the D. hyperbola. direction of v. 1. x2 2xy + y2 = 16 Du f = 2. x2 − 2xy + 2y2 = 16 Note: Your answer should be a number 3. x2 − 2xy = 16 D. Find the maximum rate of change of f at P. 4. 2x2− 3xy 2y2 + x + y = 12 5. x2 −2xy +−y2 + x + y = 0 Note: Your answer should be a number − E. Find the (unit) direction vector in which the 10.(1 pt) maximum rate of change occurs at P. Find the first partial derivatives of f (x,y) = sin(x u = i+ j y) at the point (-8, -8). − Note: Your answers should be numbers A. fx( 8, 8) = B. f (−8, −8) = 14.(1 pt) The axis of a light in a lighthouse is tilted. y − − 2 2 When the light points east, it is inclined upward at 4 11.(1 pt) If f (x,y) = 4x + 4y , find the value of degree(s). When it points north, it is inclined upward the directional derivativ−e at the point (2, 1) in the 2π − at 7 degree(s). What is its maximum angle of eleva- direction given by the angle θ = 1 . tion? degrees 12.(1 pt) Suppose f (x,y) = 3x2 1xy 2y2, P = 8 6 − − − 15.(1 pt) (0, 3), and u = − , . − 10 10 Consider the equation xz2 4yz + 2logz = 5 as A. Compute the gradient
of f. defining z implicitly as a function− of x and y. The−val- f = + ∇ i j ∂z ∂z Note: Your answers should be expressions of x and ues of and at ( 1,1,1) are and . ∂x ∂y − y; e.g. “3x - 4y” Remember, in webwork ”log” means the natural log- B. Evaluate the gradient at the point P. arithm. (∇ f )(0, 3) = i+ j Note: Your− answers should be numbers 16.(1 pt) Find the equation of the tangent plane to the surface z = 9y2 16x2 at the point (2, 2, 28). C. Compute the directional derivative of f at P in − − − the direction u . z = Note: Your answer should be an expression of x (Du f )(P) = Note: Your answer should be a number and y; e.g. “3x - 4y + 6” 13.(1 pt) Suppose f (x,y) = x , P = ( 3,4) and 17.(1 pt) y − v = 1i + 4j. The intensity of light at a distance r from a source 2 A. Find the gradient of f. is given by L = Ir− , where I is the illumination at ∇ f = i+ j the source. Starting with the values I = 80, r = 80, Note: Your answers should be expressions of x and suppose we increase the distance by 1 and the illumi- y; e.g. “3x - 4y” nation by 2. By (approximately) how much does the B. Find the gradient of f at the point P. intensity of light change? (∇ f )(P) = i+ j dL =

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2 Peter Alfeld Math 1280-1, Spring 2006 (F) What is the minimum value of f on R2? If WeBWorK Assignment 7 due 4/6/06 at there is none, type N. 11:59 PM Double Integrals and Applications 5.(1 pt) Suppose f (x,y) = xy(1 10x 8y). − − 1.(1 pt) Find an equation of the tangent plane to the f (x,y) has 4 critical points. List them in increasing parametric surface x = 1r cosθ, y = 4r sinθ, z = r at lexographic order. By that we mean that (x, y) comes before (z, w) if x < z or if x = z and y < w. Also, the point 1√2,4√2,2 when r = 2, θ = π/4.   describe the type of critical point by typing MA if it z = is a local maximum, MI if it is a local minimim, and Note: Your answer should be an expression of x S if it is a saddle point. and y; e.g. “3x - 4y” First point ( , ) of type 2.(1 pt) Consider the surface 25x2 + 25y2 + 9z2 = Second point ( , ) of type 59 and the point P = (1,1,1) on this surface. Third point ( , ) of type a) The outward unit normal at the point P is Fourth point ( , ) of type I + J + K. 6.(1 pt) You are to manufacture a rectangular box b) The equation of the tangent plane at the point P with 3 dimensions x, y and z, and volume v = 27. is Find the dimensions which minimize the surface area z = x+ y+ . of this box. 3.(1 pt) Let w = 5xy + 3x 6y, x = r + s + t, y = x = r + s, and z = s + t. Find the−partial derivatives of w y = with respect to r, s and t at the point r = 1, s = 2, z = t = 3. 7.(1 pt) Find the coordinates of the point (x, y, z) ∂−w on the plane z = 2 x + 3 y + 3 which is closest to the = . ∂r origin. ∂w x = = . ∂s y = ∂w = . z = ∂t 8.(1 pt) Find the maximum and minimum values of 2 2 4.(1 pt) Suppose f (x,y) = x2 + y2 4x 6y + 1 f (x,y) = 4x + y on the ellipse x + 25y = 1 − − (A) How many critical points does f have in R2? maximum value: minimum value: (B) If there is a local minimum, what is the value 9.(1 pt) Evaluate the iterated integral 3 3 2 3 of the discriminant D at that point? If there is none, 0 0 12x y dxdy type N. R R 10.(1 pt) Calculate the double integral R(10x + (C) If there is a local maximum, what is the value 2y+20)dA where R is the region: 0 x R1R,0 y ≤ ≤ ≤ ≤ of the discriminant D at that point? If there is none, 5. type N. 11.(1 pt) Calculate the volume under the ellip- (D) If there is a saddle point, what is the value of tic paraboloid z = 1x2 + 4y2 and over the rectangle the discriminant D at that point? If there is none, type R = [ 1,1] [ 4,4]. N. − × − Please note, the notation [-1,1] x [-4,4] refers to the (E) What is the maximum value of f on R2? If Cartesian product of these two closed intervals, that there is none, type N. is, all pairs (x,y) with x in [-1,1] and y in [-4,4] which 1 is a rectangle with corners at (-1,-4) (1,-4) (-1,4) and C. What is the moment about the y-axis? (1,4). 12.(1 pt) Find the volume of the solid bounded by D. Where is the center of mass? ( , the planes x = 0, y = 0, z = 0, and x + y + z = 1. ) E. What is the moment of inertia about the origin? 13.(1 pt) Match the following integrals with the verbal descriptions of the solids whose volumes they 16.(1 pt) find R x√ydA, where R is the region give. Put the letter of the verbal description to the left in the first quadrantR R bounded above by the curve of the corresponding integral. y = 64 x2. − 1. 2 2 4 y2 dydx 0 2 − (1 pt) R R1 − 1 2 17. √ p√1 3y 2. 3 2 − 1 4x2 3y2 dxdy A sprinkler distributes water in a circular pattern, 0 0 − − r R 2 R4+√4 x2 p supplying water to a depth of e− feet per hour at a 3. 2 4 − 4x + 3y dydx − distance of r feet from the sprinkler. R 1 R√y 2 2 4. 0 y2 4x + 3y dxdy A. What is the total amount of water supplied per R R 1 √1 x2 2 2 hour inside of a circle of radius 13? 5. − 1 x y dydx 1 √1 x2 3 R− R− − − − ft perhour A. One half of a cylindrical rod. B. What is the total amount of water that goes B. Solid under an elliptic paraboloid and over a throught the sprinkler per hour? planar region bounded by two parabolas. ft3 perhour C. One eighth of an ellipsoid. 18.(1 pt) D. Solid under a plane and over one half of a cir- Electric charge is distributed over the disk cular disk. x2 + y2 4 so that the charge density at (x,y) is E. Solid bounded by a circular paraboloid and a σ(x,y) =≤15 + x2 + y2 coulombs per square meter. plane. Find the total charge on the disk. 14.(1 pt) Using polar coordinates, evaluate the integral 19.(1 pt) sin(x2 + y2)dA where R is the region 9 x2 + Using polar coordinates, evaluate the integral R ≤ yR2R 36. which gives the area which lies in the first quadrant ≤ between the circles x2 +y2 = 324 and x2 18x+y2 = − 15.(1 pt) 0. A lamina occupies the part of the disk x2 + y2 1 ≤ in the first quadrant and the density at each point is 20.(1 pt) given by the function ρ(x,y) = 4(x2 + y2). Find the area of the region in the first quadrant A. What is the total mass? bounded by the curves y2 = 2x,, y2 = 3x, x2 = 9y, B. What is the moment about the x-axis? x2 = 10y.

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2 Peter Alfeld Let F be the radial force field F = xI + yJ. Find Math 1280-1, Spring 2006 the work done by this force along the following two WeBWorK Assignment 8 due 4/13/06 at curves, both which go from (0, 0) to (3, 9). (Compare 11:59 PM your answers!) 2 Vector Fields and Line Integrals A. If C1 is the parabola: x = t, y = t , 0 t 3, ≤ ≤ then F dX = 1.(1 pt) C1 R · 2 Compute the gradient vector fields of the following B. If C2 is the straight line segment: x = 3t , y = 9t, 0 t 1, then d = functions: C2 F X 2 2 ≤ ≤ R · A. f (x,y) = 10x + 2y 4.(1 pt) ∇ f (x,y) = I+ J Let C be the counter-clockwise planar circle with B. f (x,y) = x3y2, center at the origin and radius r > 0. Without comput- ∇ f (x,y) = I+ J ing them, determine for the following vector fields F C. f (x,y) = 10x + 2y whether the line integrals C F dX are positive, neg- ∇ f (x,y) = I+ J ative, or zero and type P, N,R or·Z as appropriate. D. f (x,y,z) = 10x + 2y + 3z A. F = the radial vector field = xI + yJ: ∇ f (x,y) = I+ J+ K B. F = the circulating vector field = yI + xJ: E. f (x,y,z) = 10x2 + 2y2 + 3z2 − ∇ f (x,y,z) = I+ J+ K C. F = the circulating vector field = yI xJ: − 2.(1 pt) D. F = the constant vector field = I + J: Match the following vector fields with the verbal 5.(1 pt) If C is the curve given by X(t) = descriptions of the level curves or level surfaces to (1 + 1sint)I + 1 + 4sin2 t J + 1 + 4sin3 t K, 0 ≤ which they are perpendicular by putting the letter of t π and F is the radial
vector field F(
x,y,z) = ≤ 2 the verbal description to the left of the number of the xI + yJ + zK, compute the work done by F on a par- vector field. ticle moving along C. 1. F = 2I + J 2. F = 2xI + yJ + zK 6.(1 pt) 3. F = xI + yJ zK Let R be the rectangle with vertices (0,0), (9,0), (0,7), − 4. F = xI + yJ + zK (9,7), and let C be the boundary of R traversed coun- 5. F = xI + yJ K terclockwise. For the vector field − 6. F = xI + yJ F(x,y) = 3yI + xJ, 7. F = xI yJ − find 8. F = yI + xJ F dX. 9. F = 2I + J + K ZC · 10. F = yI + xJ − 11. F = 2xI + yJ 7.(1 pt) For each of the following vector fields F , decide whether it is conservative or not by comput- A. ellipsoids ing curl . Type in a potential function f (that is, B. paraboloids F ∇ f = ). If it is not conservative, type N. C. ellipses F A. (x,y) = (16x + 2y) + (2x + 14y) D. hyperboloids F I J f (x,y) = E. lines B. (x,y) = 8y + 9x F. circles F I J f (x,y) = G. spheres C. (x,y,z) = 8x + 9y + H. hyperbolas F I J K f (x,y,z) = I. planes D. F(x,y) = (8siny)I + (4y + 8xcosy)J 3.(1 pt) f (x,y) = 1 E. F(x,y,z) = 8x2I + 2y2J + 7z2K f (x,y,z) = Note: Your answers should be either expressions of x, y and z (e.g. “3xy + 2yz”), or the letter “N” 10.(1 pt) 8.(1 pt) Suppose C is any curve from (0,0,0) to Calculate the divergence and curl of these vector (1,1,1) and F(x,y,z) = (4z + 1y)I + (5z + 1x)J + fields: A. F (X) = x3 3xy2 I + 3x2y + y3 J (5y + 4x)K. Compute the line integral C F dX. − − R · curl (F)= I+
J+
K 9.(1 pt) div (F)= 3 2 3 Find the work done by the force field F(x,y,z) = B. G(X) = x yI + x zJ + yz K 2xI+2yJ+7K on a particle that moves along the he- curl (F)= I+ J+ K lix X(t) = 6cos(t)I + 6sin(t)J + 3tK,0 t 2π. div (G)= ≤ ≤

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2 2 Peter Alfeld A. If C1 is the parabola: x = t, y = t , 0 t 10, ≤ ≤ Math 1280-1, Spring 2006 then d = Z F X WeBWorK Assignment 9 due 4/21/06 at C1 · 2 11:59 PM B. If C2 is the straight line segment: x = 10t , y = Independence of Path, Green’s Theorem 100t2, 0 t 1, then d = Z F X ≤ ≤ C2 · 1.(1 pt) Let C be the positively oriented circle yI+xJ 2 2 ( , ) = x + y = 1. Use Green’s Theorem to evaluate the 5.(1 pt) Let F x y −x2+y2 and let C be the circle line integral 4ydx + 11xdy. X(t) = (cost)I + (sint)J, 0 t 2π. C ≤ ≤ R A. Compute C F dX R · 2.(1 pt) Let F = 1yI + 2xJ. Use the tangential Note: Your answer should be a number vector form of Green’−s Theorem to compute the cir- B. Is F conservative? Type Y if yes, type N if no. culation integral F dX where C is the positively C · oriented circle x2 R+ y2 = 25. 6.(1 pt) Let C be the positively oriented square with vertices (0,0), (1,0), (1,1), (0,1). Use Green’s The- 2 2 3.(1 pt) Let F = 1xI + 2yJ and let n be the out- orem to evaluate the line integral C 2y xdx+1x ydy. ward unit normal vector to the positively oriented cir- R cle x2 + y2 = 9. Compute the flux integral F nds. 2/3 C · 7.(1 pt) Find a parametrization of the curve x + R y2/3 = 1 and use it to compute the area of the interior. 4.(1 pt) Let F be the radial force field F = xI + yJ. Find 8.(1 pt) Let F = 3x3I + 5y3J and let n be the out- the work done by this force along the following two ward unit normal vector to the positively oriented cir- 2 2 curves, both which go from (0, 0) to (10, 100). (Com- cle x + y = 4. Compute the flux integral C F nds. pare your answers!) R ·

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1 Peter Alfeld Math 1280-1, Spring 2006 5.(1 pt) Use Stoke’s Theorem to evaluate F dr WeBWorK Assignment 10 due 4/27/06 at ZC · 11:59 PM where F(x,y,z) = xI + yJ + 5(x2 + y2)K and C is the Surface Integrals, Divergence and Stokes’ boundary of the part of the paraboloid where z = Theorems 64 x2 y2 which lies above the xy-plane and C is 1.(1 pt) Let F = 9xI + 7yJ + 9zK. Compute the oriented− −counterclockwise when viewed from above. divergence and the curl. = A. div F 6.(1 pt) Suppose F = F(x,y,z) is a gradient field = + + B. curl F I J K with F = ∇ f , S is a level surface of f, and C is a curve 2.(1 pt) Let F = (8yz)I + (9xz)J + (5xy)K. Com- on S. What is the value of the line integral C F dr? pute the following: R · A. div F = (1 pt) Evaluate 1 + x2 + y2 dS where S B. curl F = I+ J+ K 7. Z Z S p C. div curl F = is the helicoid: r(u,v) = ucos(v)I + usin(v)J + vK, Note: Your answers should be expressions of x, y with 0 u 5,0 v 5π and/or z; e.g. ”3xy” or ”z” or ”5” ≤ ≤ ≤ ≤ 3.(1 pt) A fluid has density 5 and velocity field (Note, see Exercise 5 on page 772 of the text.) v = yI + xJ + 2zK. 8.(1 pt) Find the surface area of the part of the Find−the rate of flow outward through the sphere sphere x2 + y2 + z2 = 4 that lies above the cone z = x2 + y2 + z2 = 4 x2 + y2 p 4.(1 pt) Use Stoke’s theorem to evaluate 9.(1 pt) Let S be the part of the plane 4x+3y+z = 1 curlF dS where F(x,y,z) = 17yzI + 17xzJ + Z ZS · − which lies in the first octant, oriented upward. Find 3(x2 + y2)zK and S is the part of the paraboloid the flux of the vector field F = 4I + 3J + 3K across z = x2 + y2 that lies inside the cylinder x2 + y2 = 1, the surface S. oriented upward.

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