Math Peter Alfeld. WeBWorK Problem Set 1. Due 2/7/06 at 11:59 PM. Procrastination is hazardous!

Transcription

1 Math 80- Peter Alfeld. WeBWorK Problem Set. Due /7/06 at :59 PM. Get to work on this set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. Procrastination is hazardous!.( pt) Determine whether the sequence is divergent or convergent. If it is convergent, evaluate its limit. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. lim n 4n 7 + sin (n) n ( pt) Find the limit of the sequence whose terms are given by a n = (n )( cos( 5.5 n )). 3.( pt) Match each sequence below to statement that BEST fits it. STATEMENTS Z. The sequence converges to zero; I. The sequence diverges to infinity; F. The sequence has a finite non-zero limit; D. The sequence diverges. SEQUENCES. nsin( n ). (ln(n)) n 3. n 00 (.0) n 4. ln(ln(ln(n))) 5. sin(n) 6. arctan(n + ) 7. n! n n 3 5n 3n n 5 4.( pt) Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter as your answer. If decreasing, enter as your answer. If not monotonic, enter 0 as your answer.. a n = n+6. a n = n n+ n+ 6n+ 3. a n = 4. a n = cosn n 5.( pt) Consider the sequence a n = ncos(nπ) n. Write the first five terms of a n, and find lim n a n. If the sequence diverges, enter divergent in the answer box for its limit. a) First five terms:,,,,. b) lim n a n =. 6.( pt) Consider the sequence a n = ln(/n) n. Write the first five terms of a n, and find lim n a n. If the sequence diverges, enter divergent in the answer box for its limit. a) First five terms:,,,, b) lim n a n =. 7.( pt) Suppose a =,a = 3 3,a 3 = 3 4,a 4 = ,a 5 = a) Find an explicit formula for a n :. b) Determine whether the sequence is convergent or divergent:. (Enter convergent or divergent as appropriate.) c) If it converges, find lim n a n =. 8.( pt) Suppose a =,a n+ = ) (a n + an. Find lim n a n =.

2 Hint: Let ) a = lim n. Then, since a n+ = (a n + an, we have a = ( a + ). Now solve a for a. 9.( pt) Consider the series 7 n+6. Let s n be the n-th partial sum; that is, 7 s n = i + 6. n i= Find s 4 and s 8 s 4 = s 8 = 0.( pt) Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent n 3 7 n(n + 7) + 7 n n ln(n) 0n 5. 7 n 0 6.( pt) Determine the sum of the following series. ( 4) n 6 n.( pt) Determine the sum of the following series. ( 4n + 6 n n ) 3.( pt) Express as a rational number, in the form p q where p and q have no common factors. p = and q = 4.( pt) A ball drops from a height of 8 feet. Each time it hits the ground, it bounces up 30 percents of the height it fall. Assume it goes on forever, find the total distance it travels. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

3 Math 80- Peter Alfeld. WeBWorK Problem Set. Due /7/06 at :59 PM. Get to work on this set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. Procrastination is hazardous!.( pt) Consider the series: ( k=0 3 (k ) 3 k a) Determine whether the series is convergent or divergent:. (Enter convergent or divergent as appropriate.) b) If it converges, find its sum:. If the series diverges, enter here divergent again..( pt) Determine the convergence or divergence of the following series. A. convergent B. divergent n + 3.( pt) Determine the convergence or divergence of the following series. n 3 k + k k! A. convergent B. divergent 4.( pt) Determine the convergence or divergence of the following series. [ ( )] n cos n ) A. convergent B. divergent 5.( pt) Determine whether the following series is ( ) n+ 5n. A. conditionally convergent B. absolutely convergent C. divergent 6.( pt) Determine whether the following series is n= ( ) n n A. absolutely convergent B. conditionally convergent C. divergent 7.( pt) Each of the following statements is an attempt to show that a given series is convergent or not using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for correct ) if the argument is valid, or enter I (for incorrect ) if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) n. For all n >, ln(n) >, and the series n n n converges, so by the Comparison Test, the series ln(n) converges. n. For all n >, nln(n) < n, and the series n diverges, so by the Comparison Test, the series nln(n) diverges. 3. For all n >, ln(n) <, and the series n n.5 n converges, so by the Comparison Test, the series ln(n) converges..5 n n 4. For all n >, n 3 7 <, and the series n n converges, so by the Comparison Test, the series n n 3 7 converges. arctan(n) 5. For all n >, < π, and the series n 3 n 3 π converges, so by the Comparison Test, n 3 the series arctan(n) converges. n 3 6. For all n >, ln(n) > n, and the series n diverges, so by the Comparison Test, the series diverges. ln(n) n

4 8.( pt) The three series A n, B n, and C n have terms A n = n 7, B n = n 4, C n = n. Use the Limit Comparison Test to compare the following series to any of the above series. For each of the series below, you must enter two letters. The first is the letter (A,B, or C) of the series above that it can be legally compared to with the Limit Comparison Test. The second is C if the given series converges, or D if it diverges. So for instance, if you believe the series converges and can be compared with series C above, you would enter CC; or if you believe it diverges and can be compared with series A, you would enter AD n + 5n 6 4n 7 + 6n 3 4 4n 4 + n 7 87n + 6n n 4 + n 5n 6n 5n ( pt) Select the FIRST correct reason why the given series diverges. A. Diverges because the terms don t have limit zero B. Divergent geometric series C. Divergent p series D. Integral test E. Comparison with a divergent p series F. Diverges by limit comparison test G. Cannot apply any test done so far in class cos(nπ) ln(5) nln(n) (n + )(6 + ) n 6 n ( ) n (n)! (n!) 7n + 4 ( ) n 6. n 0.( pt) Find the convergence set of the given power series: x n n! The above series converges for < x <. Enter infinity for and -infinity for..( pt) A famous sequence f n, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 00, is defined by the recursion formula f = f =, f n+ = f n+ + f n. Find the radius of convergence of f n x n. Radius of convergence:..( pt) Find the interval of convergence for the given power series. n 4 (x + 7) n (0 n )(n 4 3 ) The series is convergent: from x =, left end included (Y,N): to x =, right end included (Y,N): 3.( pt) Match each of the power series with its interval of convergence. (x 5). n (n!)5 n n!(5x 5) n (5x) n n 5 5 n (x 5) n (5) n A. [ 5, 5 ] B. {5/5} C. (,) D. (0,0)

5 4.( pt) Find the power series representation for f (x) = ( + x) and specify the radius of convergence. f (x) = ( ) e n a n x p n, where e n = A. n B. n - C. 0 where a n =, and p n =. Radius of convergence:. 5.( pt) Find the power series representation for f (x) = xe x. f (x) = n=0 a n! xp n, where a n = and p n =. 6.( pt) Find the Taylor series in (x a) through (x a) 3 for f (x) = tanx, a = π 4 ( ) ( ) f (x) = + x π 4 + x π 4 + ( ) ( x π 3 (x ) ) 4 + O π ( pt) Find the Taylor series in (x a) through (x a) 3 for f (x) = x + 3x x 3, a = f (x) = + (x + )+ (x + ) + (x + ) 3. 8.( pt) Suppose that f (x) and g(x) are given by the power series f (x) = 6 + 4x + 7x + 5x 3 + and g(x) = 6 + 4x + 4x + 3x 3 +. By multiplying power series, find the first few terms of the series for the product h(x) = f (x) g(x) = c 0 + c x + c x + c 3 x 3 +. c 0 = c = c = c 3 = 6x 9.( pt) Suppose that (3 + x) = c n x n. n=0 Find the first few coefficients. c 0 = c = c = c 3 = c 4 = Find the radius of convergence R of the power series. R =. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 3

6 .( pt) Represent the function x 0.5 as a power series Math 80- Peter Alfeld. WeBWorK Problem Set 4. Due /9/06 at 8:00 AM. Get to work on this set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. Procrastination is hazardous! n=0 c n (x 9) n. c 0 = c = c = c 3 = Find the left endpoint of the interval of convergence. left end =. Find the right endpoint of the interval of convergence. right end =..( pt) Compute the 9th derivative of ( x 3 ) f (x) = arctan 5 at x = 0. f (9) (0) = Hint: Use the MacLaurin series for f (x). 3.( pt) Compute the 9th derivative of f (x) = cos( 6x ) x 3 at x = 0. f (9) (0) = Hint: Use the MacLaurin series for f (x). 4.( pt) The Taylor series for f (x) = ln(sec(x)) at a = 0 is n=0 c n (x) n. Find the first few coefficients. c 0 = c = c = c 3 = c 4 = Find the exact error in approximating ln(sec( 0.)) by its fourth degree Taylor polynomial at a = 0. The error is 5.( pt) Let T 5 (x) be the fifth degree Taylor polynomial of the function f (x) = cos(0.x) at a = 0. A. Find T 5 (x). (Enter a function.) T 5 (x) = B. Find the largest integer k such that for all x for which x < the Taylor polynomial T 5 (x) approximates f (x) with error less than 0 k. k = 6.( pt) Let F(x) = Z x 0 sin(6t ) dt. Find the MacLaurin polynomial of degree 7 for F(x). Use this polynomial to estimate the value of Z 0.79 sin(6x ) dx. 0 7.( pt) Find Taylor series of function f (x) = ln(x) at a = 7. ( f (x) = n=0 c n (x 7) n ) c 0 = c = c = c 3 = c 4 = Find the interval of convergence. The series is convergent: from x =, left end included (Y,N): to x =, right end included (Y,N): 8.( pt) Evaluate ln( x) + x + x lim x 0 4x 3

7 Hint: Use power series. 9.( pt) Evaluate e 3x3 + 3x 3 9x6 lim x 0 Hint: Use power series. 3x 9 0.( pt) Assume that sin(x) equals its Maclaurin series for all x. Use the Maclaurin series for sin(7x ) to evaluate the integral Z 0.7 sin(7x ) dx 0. Your answer will be an infinite series. Use the first two terms to estimate its value..( pt) Find T 5 (x): Taylor polynomial of degree 5 of the function f (x) = cos(x) at a = 0. (You need to enter function.) T 5 (x) = Find all values of x for which this approximation is within of the right answer. Assume for simplicity that we limit ourselves to x. x.( pt) Let T 6 (x): be the Taylor polynomial of degree 6 of the function f (x) = cos(x) at a = 0. Suppose you approximate f (x) by T 6 (x), and if x, what is the bound for your error of your estimate? (Hint: use the alternating series approximation.) 3.( pt) Let T k (x): be the Taylor polynomial of degree k of the function f (x) = sin(x) at a = 0. Suppose you approximate f (x) by T k (x), and if x, how many terms do you need (that is, what is k) for you to have your error to be less than 5040? (Hint: use the alternating series approximation.) 4.( pt) Let T 8 (x): be the Taylor polynomial of degree 8 of the function f (x) = ln( + x) at a = 0. Suppose you approximate f (x) by T 8 (x), find all positive values of x for which this approximation is within 0.00 of the right answer. (Hint: use the alternating series approximation.) 0 < x 5.( pt) Let C be a semicircle of radius r > 0 centered at the origin. Let P be a point on the x-axis whose coordinates are P = (r + rt,0) where t > 0. Let L be a line through P which is tangent to the semicircle. Let A denote the triangular region between the circle and the line and above the x-axis (see figure.) (Click on image for a larger view ) Find the exact area of A in terms of r and t. Area(A) =. Use a Maclaurin Polynomial to get a simple approximation for the area of A for small t. : Area(A). 6.( pt) Suppose that we use the bisection algorithm to approximate r = 56 which the greatest zero of the function f (x) = x 56. We begin by finding two numbers, say, a = 7 and b = 8 which bracket the zero. This is because f (a ) < 0 and f (b ) > 0. Then we find m = (a + b )/ = 7.5 and h = (b a )/ = 0.5. We proceed with the bisection algorithm. Suppose that a n and b n bracket the zero. Then we compute m n = (a n + b n )/ and h n = b n a n /. If f (m n ) = 0 we stop because r = m n is the desired zero. If f (m n ) > 0 then m n becomes the new right endpoint, so we set a n+ := a n and b n+ := m n. If f (m n ) < 0 then m n becomes the new left endpoint, so we set a n+ := m n and b n+ := b n. Then m n is an approximation to r with an error of h n Complete the following table: n a n b n h n m n

8 Then is an approximation so far to r with an error of. 7.( pt) Suppose that we use Newton s Method to approximate r = which the zero of the function f (x) = x We begin with a good guess, say x = 9. Then Newton s Method proceeds by the recursion x n+ = x n f (x n) f (x n ). Compute the first few terms of the sequence x n obtained from Newton s method. x = 9 x = x 3 = x 4 = x 5 = x 6 = x 7 = 8.( pt) FIXED POINT ALGORITHM. If g is a continuous function taking the interval [a, b] to itself, then it has a fixed point r [a,b] so that r = g(r). If in addition, g is differentiable and satisfies g (x) M for all a x b where M < is a constant, then the recursion x n+ = g(x n ), x [a,b] yields a sequence that converges x n r as n. Consider the equation x = + x. Using the Fixed Point Algorithm starting with x = 3, find x to x 7. x = 3 x = x 3 = x 4 = x 5 = x 6 = x 7 = Solve for the (positive) x in x = + x. x = Evaluate ( pt) Given the following integral and value of n, approximate the following integral using the methods indicated (round your answers to six decimal places): Z e 4x dx,n = 4 (a) Trapezoidal Rule 0 3 (b) Midpoint Rule (c) Simpson s Rule 0.( pt) Use Simpson s Rule and all the data in the following table to estimate the value of the integral Z 4 ydx. x y ( pt) Determine an n so that the trapezoidal rule will approximate the integral Z 0 7dx x with an error E n satisfying E n The theoretical error bound for the Trapezoid rule is given by (b a)3 E n = n f (c) where c is some point between a and b. It predicts that the desired accuracy will be achieved if the number of terms n is at least..( pt) Suppose that we use Euler s method to approximate the solution to the differential equation dy dx = x ; y(0.3) = 9. y Let f (x,y) = x /y. We let x 0 = 0.3 and y 0 = 9 and pick a step size h = 0.. Euler s method is the the following algorithm. From x n and y n, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing x n+ = x n + h, y n+ = y n + h f (x n,y n ). Complete the following table: n x n y n

9 5 The exact solution can also be found using separation of variables. It is y(x) = Thus the actual value of the function at the point x =.3 y(.3) =. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 4

10 Peter Alfeld WeBWorK problems. WW Prob Lib Math course-section, semester year WeBWorK assignment 3 due /7/06 at :59 PM..( pt) A child walks due east on the deck of a ship at miles per hour. The ship is moving north at a speed of 7 miles per hour. Find the speed and direction of the child relative to the surface of the water. Speed = mph The angle of the direction from the north = (radians).( pt) Find a b if a = 9, b = 5, and the angle between a and b is π 4 radians. a b = 3.( pt) Gandalf the Grey started in the Forest of Mirkwood at a point with coordinates (-, -) and arrived in the Iron Hills at the point with coordinates (0, 3). If he began walking in the direction of the vector v = 4I + J and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn. (, ) 4.( pt) An object is at rest on the plane. V,W,X are acting on the object. If V = 3I 7J,W = I 7J, then X must be I+ J Three forces, 5.( pt) The distance between the two parallel lines L : 3x y = 6,L : 3x y = is 6.( pt) Find the distance of the point ( 4, 6) from the line through (6, 8) which points in the direction of I + 4J. 7.( pt) Let T be the triangle with vertices at (0,8),(6, ),( 5,). The area of T is Hint: Use the projection formula to find the length of an altitude orthogonal to any chosen base. 8.( pt) Find the vector V which makes an angle of 70 degrees with the vector W = I J and which is of the same length as W and is counterclockwise to W. I+ J 9.( pt) While a planet P rotates in a circle about its sun, a moon M rotates in a circle about the planet, and both motions are in a plane. Let s call the distance between M and P one lunar unit. Suppose the distance of P from the sun is lunar units; the planet makes one revolution about the sun every 3 years, and the moon makes one rotation about the planet every 0. years. Choosing coordinates centered at the sun, so that, at time t = 0 the planet is at (3. 0 3,0), and the moon is at (3. 0 3,), then the location of the moon at time t, where t is measured in years, is (x(t),y(t)), where x(t)= y(t)= 0.( pt) The position of a particle in motion in the plane at time t is X(t) = 9t I + sin( 7t) J. At time any t, determine the following: (a) the speed of the particle is: (b) the unit tangent vector to X(t) is: I+ J (Note that, the unit tangent vector to X(t) is defined as V(t)/ V(t).) Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

11 Peter Alfeld MATH 80- Spring 006 Homework Set 5 due /8/06 at :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..( pt) If a = (-6, 9, -9) and b = (, -8, ), find a b =..( pt) What is the angle in radians between the vectors a = (, 0, -8) and b = (4, -, 6)? Angle: (radians) 3.( pt) Find a unit vector in the same direction as a = (6, 0, 7). (,, ) 4.( pt) The equation X (t) = A + tl is the parametric equation of a line through the point P : (, 3,). The parameter t represents distance from the point P, directed so that the I component of L is positive. We know that the line is orthogonal to the plane with equation 8x + 3y 9z =. Then A= I+ J+ K L= I+ J+ K 5.( pt) Given vectors u and v such that u v = 0I J + 0K, find: a) v u = (,, ). b) (u v) (u v) =. c) (u v) (u v) = (,, ). 6.( pt) What is the distance from the point (0, 8, -3) to the xz-plane? Distance = 7.( pt) Enter T or F depending on whether the statement is true or false. (You must enter T or F True and False will not work.). Two planes parallel to a line are parallel.. Two lines either intersect or are parallel. 3. Two lines perpendicular to a third line are parallel. 4. Two planes either intersect or are parallel. 5. Two planes perpendicular to a third plane are parallel. 6. A plane and a line either intersect or are parallel. 7. Two lines parallel to a third line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two lines perpendicular to a plane are parallel 0. Two planes parallel to a third plane are parallel.. Two lines parallel to a plane are parallel. 8.( pt) Consider the planes x + 3y + z = and x + z = 0. (A) Find the unique point P on the y-axis which is on both planes. (,, ) (B) Find a unit vector u with positive first coordinate that is parallel to both planes. I + J + K (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 9.( pt) The position of a particle in motion in the plane at time t is X(t) = exp(5.t)i + exp(t)j. At time t = 0, determine the following: (a) The speed of the particle is: (b) Find the unit tangent vector to X(t): I+ J (c) The tangential acceleration: (d) The normal acceleration:

12 0.( pt) The position of a particle in motion in the plane at time t is X(t) = ti + ln(cos(t))j. At time t = ( 0.7π)/, determine the following: (a) the unit tangent vector: I+ J (b) the unit normal vector to X(t): I+ J (c) the acceleration vector: I+ J (d) the curvature:.( pt) Consider the vector functions Let X(t) = 3I + cos(0t)j, Y(t) = sin(9t)j + 8K. Z(t) = X(t) Y(t). Then dz (t)= dt I+ J+ K..( pt) If X(t) = cos( 7t)I + sin( 7t)J + 7tK, compute: A. The velocity vector v(t) = I+ J+ K B. The acceleration vector a(t) = I+ J+ K Note: the coefficients in your answers must be entered in the form of expressions in the variable t; e.g. 5 cos(t) 3.( pt) Consider the helix X(t) = (cos(t),sin(t),4t). Compute, at t = π 6 : A. The unit tangent vector T = (,, ) B. The unit normal vector N = (,, ) 4.( pt) (A) Find the parametric equations for the line through the point P = (-4, -5, 0) that is perpendicular to the plane 3x + 4y + z =. Use t as your variable, t = 0 should correspond to P, and the velocity vector of the line should be the same as the normal vector to the plane found directly from its equation. x = y = z = (B) At what point Q does this line intersect the yzplane? Q = (,, ) Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

13 Peter Alfeld MATH 80- Spring 006 Homework Set 6 due 3/4/06 at :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..( pt) Match the function with the description of its level sets (the sets z =constant). A. a collection of circles centered at the origin. B. a collection of ellipses. C. a collection of hyperbolas. D. a collection of parallel lines. E. a collection of parabolas.. z = + x + y. z = (x + y ) 3. xy + z = 4. z = x xy + y + x + y 5. z = 5x 3y 6. z = ( x + y ) 7. z = 5 x y.( pt) Consider the equation xz + 4yz lnz = as defining z implicitly as a function of x and y. The values of z z and at ( 5,,) are and. x y 3.( pt) Match the surfaces with the appropriate descriptions.. z = x + 3y. z = y x 3. z = x + 3y 4. z = x 5. x + y = 5 6. x + y + 3z = 7. z = 4 A. ellipsoid B. horizontal plane C. hyperbolic paraboloid D. parabolic cylinder E. nonhorizontal plane F. elliptic paraboloid G. circular cylinder 4.( pt) Match the given equation with the verbal description of the surface: A. Half plane B. Plane C. Cone D. Sphere E. Circular Cylinder F. Elliptic or Circular Paraboloid. φ = π 3. ρ = cos(φ) 3. r = cos(θ) 4. θ = π 3 5. r = 4 6. ρcos(φ) = 4 7. r + z = 6 8. ρ = 4 9. z = r 5.( pt) Let P be the point (,3, 4) in cartesian coordinates. A. The cylindrical coordinates of P are r =, θ =, z =. B. The spherical coordinates of P are ρ =, θ =, φ =. 6.( pt) Let P be the point with the spherical coordinates ρ = 5, φ = π/3, θ = π/4. A. The cylindrical coordinates of P are r =, θ =, z =. B. The cartesian coordinates of P are x =, y =, z =. 7.( pt) The vectors U = I J, V = I J form a base for the plane; that is, they are orthogonal vectors of length. Let X = 5I + 9J. Then we can write X = uu + vv with u =, v = 8.( pt) Let L be the line y =, x = 7z. If we rotate L around the x-axis, we get a surface whose equation is Ax + By +Cz =, where A=, B =, C =. 9.( pt) Match the surfaces with the appropriate descriptions. A. ellipse.

14 B. parabola. C. Two lines intersecting at the origin. D. hyperbola.. x xy + y = 6. x xy + y = 6 3. x xy = 6 4. x 3xy y + x + y = 5. x xy + y + x + y = 0 0.( pt) Find the first partial derivatives of f (x,y) = sin(x y) at the point (-8, -8). A. f x ( 8, 8) = B. f y ( 8, 8) =.( pt) If f (x,y) = 4x + 4y, find the value of the directional derivative at the point (, ) in the direction given by the angle θ = π..( pt) Suppose f (x,y) = 3x xy y, P = (0, 3), and u = ( 8 0, 0) 6. A. Compute the gradient of f. f = i+ j Note: Your answers should be expressions of x and y; e.g. 3x - 4y B. Evaluate the gradient at the point P. ( f )(0, 3) = i+ j Note: Your answers should be numbers C. Compute the directional derivative of f at P in the direction u. (D u f )(P) = Note: Your answer should be a number 3.( pt) Suppose f (x,y) = y x, P = ( 3,4) and v = i + 4j. A. Find the gradient of f. f = i+ j Note: Your answers should be expressions of x and y; e.g. 3x - 4y B. Find the gradient of f at the point P. ( f )(P) = i+ j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. D u f = Note: Your answer should be a number D. Find the maximum rate of change of f at P. Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P. u = i+ j Note: Your answers should be numbers 4.( pt) The axis of a light in a lighthouse is tilted. When the light points east, it is inclined upward at 4 degree(s). When it points north, it is inclined upward at 7 degree(s). What is its maximum angle of elevation? degrees 5.( pt) Consider the equation xz 4yz + logz = 5 as defining z implicitly as a function of x and y. The values of z z and at (,,) are and. x y Remember, in webwork log means the natural logarithm. 6.( pt) Find the equation of the tangent plane to the surface z = 9y 6x at the point (,, 8). z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y ( pt) The intensity of light at a distance r from a source is given by L = Ir, where I is the illumination at the source. Starting with the values I = 80, r = 80, suppose we increase the distance by and the illumination by. By (approximately) how much does the intensity of light change? dl = Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

15 Peter Alfeld Math 80-, Spring 006 WeBWorK Assignment 7 due 4/6/06 at :59 PM Double Integrals and Applications.( pt) Find an equation of the tangent plane to the parametric ( surface x = r cosθ, y = 4r sinθ, z = r at the point,4 ), when r =, θ = π/4. z = Note: Your answer should be an expression of x and y; e.g. 3x - 4y.( pt) Consider the surface 5x + 5y + 9z = 59 and the point P = (,,) on this surface. a) The outward unit normal at the point P is I + J + K. b) The equation of the tangent plane at the point P is z = x+ y+. 3.( pt) Let w = 5xy + 3x 6y, x = r + s + t, y = r + s, and z = s + t. Find the partial derivatives of w with respect to r, s and t at the point r =, s =, t = 3. w r =. w s =. w =. t 4.( pt) Suppose f (x,y) = x + y 4x 6y + (A) How many critical points does f have in R? (B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N. (C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N. (D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. (E) What is the maximum value of f on R? If there is none, type N. (F) What is the minimum value of f on R? If there is none, type N. 5.( pt) Suppose f (x,y) = xy( 0x 8y). f (x,y) has 4 critical points. List them in increasing lexographic order. By that we mean that (x, y) comes before (z, w) if x < z or if x = z and y < w. Also, describe the type of critical point by typing MA if it is a local maximum, MI if it is a local minimim, and S if it is a saddle point. First point (, ) of type Second point (, ) of type Third point (, ) of type Fourth point (, ) of type 6.( pt) You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume v = 7. Find the dimensions which minimize the surface area of this box. x = y = z = 7.( pt) Find the coordinates of the point (x, y, z) on the plane z = x + 3 y + 3 which is closest to the origin. x = y = z = 8.( pt) Find the maximum and minimum values of f (x,y) = 4x + y on the ellipse x + 5y = maximum value: minimum value: 9.( pt) Evaluate the iterated integral R 3 0 x y 3 dxdy R ( pt) Calculate the double integral R R R (0x + y+0)da where R is the region: 0 x,0 y 5..( pt) Calculate the volume under the elliptic paraboloid z = x + 4y and over the rectangle R = [,] [ 4,4]. Please note, the notation [-,] x [-4,4] refers to the Cartesian product of these two closed intervals, that is, all pairs (x,y) with x in [-,] and y in [-4,4] which

16 is a rectangle with corners at (-,-4) (,-4) (-,4) and (,4)..( pt) Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z =. 3.( pt) Match the following integrals with the verbal descriptions of the solids whose volumes they give. Put the letter of the verbal description to the left of the corresponding integral.. R 0R 4 y dydx. R 3 R 3y 0 0 4x 3y dxdy 3. R R 4+ 4 x 4 4x + 3y dydx 4. R R y 0 4x + 3y dxdy 5. R y R x x x y dydx A. One half of a cylindrical rod. B. Solid under an elliptic paraboloid and over a planar region bounded by two parabolas. C. One eighth of an ellipsoid. D. Solid under a plane and over one half of a circular disk. E. Solid bounded by a circular paraboloid and a plane. 4.( pt) Using polar coordinates, evaluate the integral RR R sin(x + y )da where R is the region 9 x + y ( pt) A lamina occupies the part of the disk x + y in the first quadrant and the density at each point is given by the function ρ(x,y) = 4(x + y ). A. What is the total mass? B. What is the moment about the x-axis? C. What is the moment about the y-axis? D. Where is the center of mass? (, ) E. What is the moment of inertia about the origin? 6.( pt) find R R R x yda, where R is the region in the first quadrant bounded above by the curve y = 64 x. 7.( pt) A sprinkler distributes water in a circular pattern, supplying water to a depth of e r feet per hour at a distance of r feet from the sprinkler. A. What is the total amount of water supplied per hour inside of a circle of radius 3? ft 3 perhour B. What is the total amount of water that goes throught the sprinkler per hour? ft 3 perhour 8.( pt) Electric charge is distributed over the disk x + y 4 so that the charge density at (x,y) is σ(x,y) = 5 + x + y coulombs per square meter. Find the total charge on the disk. 9.( pt) Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x +y = 34 and x 8x+y = 0. 0.( pt) Find the area of the region in the first quadrant bounded by the curves y = x,, y = 3x, x = 9y, x = 0y. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

17 Peter Alfeld Math 80-, Spring 006 WeBWorK Assignment 8 due 4/3/06 at :59 PM Vector Fields and Line Integrals.( pt) Compute the gradient vector fields of the following functions: A. f (x,y) = 0x + y f (x,y) = I+ J B. f (x,y) = x 3 y, f (x,y) = I+ J C. f (x,y) = 0x + y f (x,y) = I+ J D. f (x,y,z) = 0x + y + 3z f (x,y) = I+ J+ K E. f (x,y,z) = 0x + y + 3z f (x,y,z) = I+ J+ K.( pt) Match the following vector fields with the verbal descriptions of the level curves or level surfaces to which they are perpendicular by putting the letter of the verbal description to the left of the number of the vector field.. F = I + J. F = xi + yj + zk 3. F = xi + yj zk 4. F = xi + yj + zk 5. F = xi + yj K 6. F = xi + yj 7. F = xi yj 8. F = yi + xj 9. F = I + J + K 0. F = yi + xj. F = xi + yj A. ellipsoids B. paraboloids C. ellipses D. hyperboloids E. lines F. circles G. spheres H. hyperbolas I. planes 3.( pt) Let F be the radial force field F = xi + yj. Find the work done by this force along the following two curves, both which go from (0, 0) to (3, 9). (Compare your answers!) A. If C is the parabola: x = t, y = t, 0 t 3, then R C F dx = B. If C is the straight line segment: x = 3t, y = 9t, 0 t, then R C F dx = 4.( pt) Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine for the following vector fields F whether the line integrals R C F dx are positive, negative, or zero and type P, N, or Z as appropriate. A. F = the radial vector field = xi + yj: B. F = the circulating vector field = yi + xj: C. F = the circulating vector field = yi xj: D. F = the constant vector field = I + J: 5.( pt) If C is the curve given by X(t) = ( + sint)i + ( + 4sin t ) J + ( + 4sin 3 t ) K, 0 t π and F is the radial vector field F(x,y,z) = xi + yj + zk, compute the work done by F on a particle moving along C. 6.( pt) Let R be the rectangle with vertices (0,0), (9,0), (0,7), (9,7), and let C be the boundary of R traversed counterclockwise. For the vector field find F(x,y) = 3yI + xj, Z C F dx. 7.( pt) For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, f = F). If it is not conservative, type N. A. F(x,y) = (6x + y)i + (x + 4y)J f (x,y) = B. F(x,y) = 8yI + 9xJ f (x,y) = C. F(x,y,z) = 8xI + 9yJ + K f (x,y,z) = D. F(x,y) = (8siny)I + (4y + 8xcosy)J f (x,y) =

18 E. F(x,y,z) = 8x I + y J + 7z K f (x,y,z) = Note: Your answers should be either expressions of x, y and z (e.g. 3xy + yz ), or the letter N 8.( pt) Suppose C is any curve from (0,0,0) to (,,) and F(x,y,z) = (4z + y)i + (5z + x)j + (5y + 4x)K. Compute the line integral R C F dx. 9.( pt) Find the work done by the force field F(x,y,z) = xi+yj+7k on a particle that moves along the helix X(t) = 6cos(t)I + 6sin(t)J + 3tK,0 t π. 0.( pt) Calculate the divergence and curl of these vector fields: A. F (X) = ( x 3 3xy ) I + ( 3x y + y 3) J curl (F)= I+ J+ K div (F)= B. G(X) = x 3 yi + x zj + yz 3 K curl (F)= I+ J+ K div (G)= Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

19 Peter Alfeld Math 80-, Spring 006 WeBWorK Assignment 9 due 4//06 at :59 PM Independence of Path, Green s Theorem.( pt) Let C be the positively oriented circle x + y =. Use Green s Theorem to evaluate the line integral R C 4ydx + xdy..( pt) Let F = yi + xj. Use the tangential vector form of Green s Theorem to compute the circulation integral R C F dx where C is the positively oriented circle x + y = 5. 3.( pt) Let F = xi + yj and let n be the outward unit normal vector to the positively oriented circle x + y = 9. Compute the flux integral R C F nds. 4.( pt) Let F be the radial force field F = xi + yj. Find the work done by this force along the following two curves, both which go from (0, 0) to (0, 00). (Compare your answers!) A. ZIf C is the parabola: x = t, y = t, 0 t 0, then F dx = C B. If C is the straight Z line segment: x = 0t, y = 00t, 0 t, then F dx = C 5.( pt) Let F(x,y) = yi+xj and let C be the circle x +y X(t) = (cost)i + (sint)j, 0 t π. A. Compute R C F dx Note: Your answer should be a number B. Is F conservative? Type Y if yes, type N if no. 6.( pt) Let C be the positively oriented square with vertices (0,0), (,0), (,), (0,). Use Green s Theorem to evaluate the line integral R C y xdx+x ydy. 7.( pt) Find a parametrization of the curve x /3 + y /3 = and use it to compute the area of the interior. 8.( pt) Let F = 3x 3 I + 5y 3 J and let n be the outward unit normal vector to the positively oriented circle x + y = 4. Compute the flux integral R C F nds. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR

20 Peter Alfeld Math 80-, Spring 006 WeBWorK Assignment 0 due 4/7/06 at :59 PM Surface Integrals, Divergence and Stokes Theorems.( pt) Let F = 9xI + 7yJ + 9zK. Compute the divergence and the curl. A. div F = B. curl F = I+ J+ K.( pt) Let F = (8yz)I + (9xz)J + (5xy)K. Compute the following: A. div F = B. curl F = I+ J+ K C. div curl F = Note: Your answers should be expressions of x, y and/or z; e.g. 3xy or z or 5 3.( pt) A fluid has density 5 and velocity field v = yi + xj + zk. Find the rate of flow outward through the sphere x + y + z = 4 Z Z4.( pt) Use Stoke s theorem to evaluate curlf ds where F(x,y,z) = 7yzI + 7xzJ + S 3(x + y )zk and S is the part of the paraboloid z = x + y that lies inside the cylinder x + y =, oriented upward. Z 5.( pt) Use Stoke s Theorem to evaluate C F dr where F(x,y,z) = xi + yj + 5(x + y )K and C is the boundary of the part of the paraboloid where z = 64 x y which lies above the xy-plane and C is oriented counterclockwise when viewed from above. 6.( pt) Suppose F = F(x,y,z) is a gradient field with F = f, S is a level surface of f, and C is a curve on S. What is the value of the line integral R C F dr? 7.( pt) Evaluate Z Z S + x + y ds where S is the helicoid: r(u,v) = ucos(v)i + usin(v)j + vk, with 0 u 5,0 v 5π (Note, see Exercise 5 on page 77 of the text.) 8.( pt) Find the surface area of the part of the sphere x + y + z = 4 that lies above the cone z = x + y 9.( pt) Let S be the part of the plane 4x+3y+z = which lies in the first octant, oriented upward. Find the flux of the vector field F = 4I + 3J + 3K across the surface S. Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR