Bi. lkent Calculus II Exams

Transcription

1 Bi. lkent Calculus II Exams Spring 206 Midterm I Spring 206 Midterm II Spring 206 Final Spring 205 Midterm I Spring 205 Midterm II Spring 205 Final Spring 204 Midterm I Spring 204 Midterm II Spring 204 Final Spring 203 Midterm I Spring 203 Midterm II Spring 203 Final Spring 202 Midterm I Spring 202 Midterm II Spring 202 Final Spring 20 Midterm I Spring 20 Midterm II Spring 20 Final Spring 200 Midterm I Spring 200 Midterm II Spring 200 Final Spring 2009 Midterm I Spring 2009 Midterm II Spring 2009 Final Spring 2008 Midterm I Spring 2008 Midterm II Spring 2008 Final Spring 2007 Midterm I Spring 2007 Midterm II Spring 2007 Final Spring 2006 Midterm I Spring 2006 Midterm II Spring 2006 Final Spring 2005 Midterm I Spring 2005 Midterm II Spring 2005 Final Spring 2004 Midterm I Spring 2004 Midterm II Spring 2004 Final Spring 2003 Midterm I Spring 2003 Midterm II Spring 2003 Final Spring 2002 Midterm I Spring 2002 Midterm II Spring 2002 Final Spring 200 Midterm I Spring 200 Midterm II Spring 200 Final Spring 2000 Midterm I Spring 2000 Midterm II Spring 2000 Final Spring 999 Midterm I Spring 999 Midterm II Spring 999 Final Spring 998 Midterm I Spring 998 Midterm II Spring 998 Final Spring 997 Midterm I Spring 997 Midterm II Spring 997 Final Spring 996 Midterm I Spring 996 Midterm II Spring 996 Final Spring 995 Midterm I Spring 995 Final Spring 994 Midterm I Spring 994 Midterm II Spring 994 Final Spring 993 Midterm I Spring 993 Midterm II Spring 993 Final Spring 992 Midterm II Spring 992 Final Spring 99 Midterm I Spring 99 Midterm II Spring 99 Final Spring 990 Midterm II Spring 989 Midterm II Spring 988 Midterm II Spring 206 F version

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3 Bilkent Calculus II Exams Spring 206 Midterm I. Consider the sequence {a n } satisfying the conditions a = A and a n+ = 3a n a n for n where A is a real number such that a n 0 for all n. a. Assume that the sequence converges and let L = lim n a n. Show that, depending on A, there are at most two possible values for L. Give an example of A for which the sequence converges. Explain your reasoning. c. Give an example of A for which the sequence diverges and a n < for all n. Explain your reasoning. d. Show that the sequence is increasing if A =. e. Determine whether the sequence converges or diverges if A =.

4 2 Bilkent Calculus II Exams a. In each of ➊-➍, indicate all possible completions of the sentence that will make it into a true statement by ing the corresponding s. No explanation is required. ➊ n = n + is a convergent sequence a divergent sequence a convergent series a divergent series none of these ➋ { } = {, 2 n, 3, 4,, n }, a convergent sequence is a divergent sequence a convergent series a divergent series none of these ➌ { } { =, 2 n 2, 4, } 8,, 2, n a convergent sequence is a divergent sequence a convergent series a divergent series none of these ➍ 2 = + n is 2n a convergent sequence a divergent sequence a convergent series a divergent series none of these 2 In each of ➎-➏, if there exists a sequence {a n } satisfying the given conditions, write its general term inside the box; and if no such sequence exists, write Does Not Exist inside the box. No explanation is required. ➎ The sequence {a n } diverges and the sequence {( ) n a n } converges. a n = ➏ The series a n converges and the series ( ) n a n diverges. a n =

5 Bilkent Calculus II Exams Determine whether each of the following series converges or diverges. a. c. n=2 (ln n) 206 ( ) n n/n ( 206 /n ) 4. Consider the power series f(x) = n=2 x n (n 2 ) n!. a. Find the radius of convergence R of the power series. Show that f() < 2 e 7 6. c. Show that f( ) < Spring 206 Midterm II. Consider the plane P : 3x y + 2z = 7. a. Give an example of a nonzero vector n normal to the plane P. No explanation is required. n = i + j + k Give an example of a point P 0 in the plane P. No explanation is required. P 0 (,, ) c. Give an example of a nonzero vector v parallel to the plane P. No explanation is required. v = i + j + k d. Write inside the box parametric equations of one of the lines lying in the plane P. No explanation is required.

6 4 Bilkent Calculus II Exams L : e. Find an equation of the plane that passes through the point (,, ) and contains the line L in Part d. Show all your work. 2. Suppose that f(x, y, z) is a dierentiable function with the gradient f = (3x 2 y 2 z)i 2xyzj + (2z xy 2 )k and consider the point P 0 (,, 2). Find a unit vector u for which the directional derivative D u f(p 0 ) has its largest possible value. c. Find a unit vector u for which D u f(p 0 ) = 0. d. Find a unit vector u for which D u f(p 0 ) = 5. e. Give an example of a function f whose gradient is the one given in this question. No explanation is required. f(x, y, z) = 3. Each of the following functions has a critical point at (0, 0). [You do not need to verify this.] Determine whether this critical point is a local maximum, a local minimum, a saddle point or something else. a. f(x, y) = ( x 2 )( y 2 ) f(x, y) = x 2 y 4 c. f(x, y) = x 2 xy + y 4 4. Find the absolute maximum and minimum values of the restriction of the function f(x, y, z) = xy + xz to the unit sphere x 2 + y 2 + z 2 =.

7 Bilkent Calculus II Exams Spring 206 Final. Determine the smallest of the real numbers A, B, C, D, E where : A = ( ) n 2 n B = n2 n C = ( ) n+ n! D = n 3 n E = ( ) n 3 n (2n + ) 2. The Dym equation u t = u 3 u xxx is a nonlinear evolution equation which arises in the study of the motion of the interface between a viscous and a nonviscous uid with surface tension. Find all nonzero constants a, b, c such that the function u(x, t) = (ax + bt) c satises the Dym equation for all (x, t) with ax + bt > Suppose that g(x, y) = f(x 2 y 2, 2xy) where f(x, y) is a dierentiable function. Find an equation of the tangent plane to the graph of z = f(x, y) at the point (3, 4, 7) if an equation of the tangent plane to the graph of z = g(x, y) at the point ( 2,, 7) is z = 5x 6y Consider the double integral I = (x 2 + y 2 ) da 2 R where R is the region in the rst quadrant lying outside the circle x 2 + y 2 = 2, and bounded by the line x = 2 on the right and the line y = 2 at the top. y 2 2 R a. Express I in terms of iterated double integrals in Cartesian coordinates. 2 2 x Express I in terms of iterated double integrals in polar coordinates. c. Evaluate I. 5. Let V be the volume of the solid cone whose base is the unit disk in the xy-plane and whose tip is at the point (0, 0, 2) in the xyz-space.

8 6 Bilkent Calculus II Exams z 2 x y a. Only two of ➊-➌ will be graded. Mark the ones you want to be graded by putting a in the corresponding s. ➊ Express V in terms of iterated integrals in Cartesian coordinates by lling in the rectangles. ˆ ˆ ˆ V = dz dy dx ➋ Express V in terms of iterated integrals in cylindrical coordinates by lling in the rectangles. ˆ ˆ ˆ V = r dz dr dθ ➌ Express V in terms of iterated integrals in spherical coordinates by lling in the rectangles. ˆ ˆ ˆ V = ρ 2 sin ϕ dρ dϕ dθ

9 Bilkent Calculus II Exams Compute V using its expression in terms of iterated integrals in one of the coordinate systems in Part a. Spring 205 Midterm I. /2 /2 /3 /3 /4... /4 We have a /n /n square for each positive integer n. For each of (a-c), indicate by ing the to the left of Yes or No whether it is possible or not to place these squares in the xy-plane in such a way that they completely cover the given set. If Yes, describe how this can be done (you might also want to draw a picture) and then fully justify your claim. If No, explain why this cannot be done. a. The entire xy-plane : Yes No The line dened by the equation y = x : Yes No c. The region between the graph of y = e x and the x-axis for x 0 : Yes No 2. In each of the following, if the reasoning in the given sentence is correct, then the corresponding ; otherwise, leave it blank. No explanation is required. For each of the parts (a-e), to get full points you must check exactly the squares corresponding to the correct reasonings. Note that in each sentence all the statements to the right of because" are true. You must decide whether they lead to the statement to the left of because", possibly using a test you have seen in this course. a. converges because n lim n n = 0. converges because 0 < n n + < for all n. n n diverges because > ln(n + ) for all n. n n diverges because n > + n 2 ( ) n converges because ( ) n for all n 0. for all n 2. ( ) n diverges because the sequence,,,,,... diverges. ( ) n diverges because the sequence, 0,, 0,,... diverges. ( ) n diverges because ( ) n for all n 0.

10 8 Bilkent Calculus II Exams c. d. converges. converges because lim n+/n n n = 0. +/n n converges because 0 < (/(n + +/n )+/(n+) )/(/n +/n ) < for all n. converges because 0 < n+/n n < for all n 2 and +/n n ( diverges because lim n+/n (/n +/n )/(/n) ) = and n ( ) n 2 n ( ) n 2 n ( ) n 2 n n=2 n=2 n=2 n=2 ( ) n 2 n converges because (( n2) lim /n = e n n) <. n diverges. n diverges. ( converges because lim ) n 2 = 0. n n ( converges because 0 < ) n 2 < n 2 for all n 2 and n 2 n diverges because ( 3 < ) n 2 for all n 6 and n n 3 n converges. e. ( ) /n converges because lim = 2n n 2 n 2 <. ( converges because lim 2n (/2 n+ )/(/2 n ) ) = n 2 <. 2 n converges because the sequence, 3 2, 7 4, 5 8, 3 6,... converges. 2 diverges because 0 for innitely many n 0. n 2n 3. Determine whether each of the following series converges or diverges. Explain your reasoning in full by clearly stating the name and the conditions of the test you are using, and explicitly checking the validity of these conditions for the given series. a. c. ( 5 n 3 n) sin 5( π/ 3 n ) n=2 cos 5( π/ 3 n )

11 Bilkent Calculus II Exams Consider f(x) = x n 5 n (n 2 + ). a. Find the radius of convergence R of the power series. Show that 4 3 < f(3) < 3 2. c. Show that 3 4 < f( 3) < 4 5. Spring 205 Midterm II. In each of the following if the series converges, then write the exact value of its sum in a form as simplied as possible in the box; otherwise, write Div. No explanation is required. a. 3 n n! = ( ) n 3 n 2n + = c. ( ) n π 2n (2n + )! = d. ( ) n n 3 = n e. 3 n n =

12 0 Bilkent Calculus II Exams In the xyz-space where a yscreen lies along the plane with the equation 2x + y 2z =, the trajectory of a bee as a function of time t is given by for < t <. r = t i + t 2 j + t 3 k a. Find the velocity v of the bee as a function of time. Give an example of a nonzero vector n perpendicular to the screen. c. Find all times t when the bee is ying parallel to the screen. d. Find all times t when the bee is ying perpendicular to the screen. e. There are holes in the screen through which the bee passes. Find the coordinates of all of these holes. 3a. Evaluate the following limits. [ Show all your work. ] ➊ ➋ lim (x,y) (0,0) lim (x,y) (0,0) x 3 y 2 x 6 + y 6 x 3 y 2 x 6 x 2 y 2 + y 6 Your choice: Fill in the with +" or ", then evaluate! 3 Evaluate the following statements. [ Just check and fill, no further explanation is required. ] ➌ If f(x, y) approaches 0 as (x, y) approaches (0, 0) along any line through (0, 0), then lim f(x, y) = 0. (x,y) (0,0) True and can be proven using the False because does not hold for f(x, y) = ➍ If, for (x, y) (0, 0), the line y = x and every circle tangent to it at (0, 0) are level curves of f(x, y) belonging to dierent values, then lim f(x, y) does not exist. (x,y) (0,0) True and can be proven using the False because does not hold for f(x, y) =

13 Bilkent Calculus II Exams In Genetics, Fisher's Equation, p t = p ( p) + 2 p x 2 describes the spread of an advantageous allele in a population with uniform density along a -dimensional habitat, like a shoreline, as a result of both reproduction and dispersion of the ospring. Here p(x, t) is the frequency of the allele as a function of the position x and the time t. Find all possible values of the pair of constants (a, b) for which the function satises the Fisher's Equation. p(x, t) = ( + e ax+bt ) 2 Spring 205 Final. Consider the following conditions for a dierentiable function f(x, y) : ➊ f(2, ) = 8 ➋ An equation for the tangent line to the level curve f(x, y) = 8 in the xy-plane at the point (2, ) is 3x 5y = Let P be the tangent plane to the graph of z = f(x, y) at the point (2,, 8). In each of the parts (a-e) below a ➌ rd condition is given. If there is no function satisfying the conditions ➊-➌, then the next to None. If there are functions satisfying the conditions ➊-➌, but they do not all have the same tangent plane P, then the next to Many. If there are functions satisfying the conditions ➊-➌ and all of these functions have the same tangent plane P, then the next to Unique and write an equation of P inside the box. a. ➌ f(3, 2) = None Many Unique P : ➌ f x (2, ) = None Many Unique P : c. ➌ d dt f(t2 +, t 3 ) = 6 t=

14 2 Bilkent Calculus II Exams None Many Unique P : d. ➌ The line with parametric equations x = 4t+2, y = 2t+, z = t+8, ( < t < ), lies in P None Many Unique P : e. ➌ The line with parametric equations x = t+2, y = 2t+, z = t+8, ( < t < ), is perpendicular to P None Many Unique P : 2. In the gure below some of the level curves and the corresponding values of a function f are shown. a. Indicate the signs of the following derivatives at (0, 0) by ing the corresponding. No explanation is required. ➊ f x (0, 0) is positive negative ➋ f y (0, 0) is positive negative ➌ f xx (0, 0) is positive negative ➍ f yy (0, 0) is positive negative ➎ f xy (0, 0) is positive negative Draw the gradient vector f(0, 0) at the origin as best you can on the gure. No explanation is required.

15 Bilkent Calculus II Exams Find the absolute maximum and minimum values of the function f(x, y) = 2(x 2 + y 2 ) 2 + x 2 y 2 on the unit disk D = {(x, y) : x 2 + y 2 }. 4. Evaluate the following integrals. a. ( + x y) da where R = {(x, y) : x y 2/3 and 0 x and 0 y } R ˆ ˆ x 2 0 sin ( π(x 2 + y 2 ) ) dy dx 5. Let V be the volume of the ball B = {(x, y, z) : x 2 + y 2 + z 2 2z}. a. Only two of ➊-➌ will be graded. Mark the ones you want to be graded by putting a in the corresponding s.

16 4 Bilkent Calculus II Exams ➊ Express V in terms of iterated integrals in Cartesian coordinates by lling in the rectangles. ˆ ˆ ˆ V = dz dy dx ➋ Express V in terms of iterated integrals in cylindrical coordinates by lling in the rectangles. ˆ ˆ ˆ V = r dr dz dθ ➌ Express V in terms of iterated integrals in spherical coordinates by lling in the rectangles. ˆ ˆ ˆ V = ρ 2 sin ϕ dρ dϕ dθ Compute V using its expression in terms of iterated integrals in one of the coordinate systems in part (a). Spring 204 Midterm I a. In each of ➊-➑, indicate the kind of geometric object dened in the xyz-space by the given set of equations by marking the corresponding with a. No explanation is required. ➊ x + 3z = 7y + 5 A point A line A plane Something else ➋ x = 3t 2, y = 4, z = 5t + ; < t < A point A line A plane Something else ➌ x 4 = (y 3)/5 = (z )/8 A point A line A plane Something else ➍ 2x + y 3z = 0, x + 6y + z = 8 A point A line A plane Something else

17 Bilkent Calculus II Exams ➎ y = 3 A point A line A plane Something else ➏ x = t 3 + 2, y = 4t 3 + 5, z = t 3 ; < t < A point A line A plane Something else ➐ x =, z = 3 A point A line A plane Something else ➑ x 2 + y 2 + z 2 = 0 A point A line A plane Something else Write inside the box an equation for the plane that passes through the point (3, 2, ) and contains the line of intersection of the planes with equations x + y + z = 3 and x + 2y + 3z = 6. No explanation is required. c. The equation of the tangent plane to the graph of the function z = f(x, y) at the point with (x, y) = (3, 5) is 4z 7x + y = 2. Write inside the box an equation of the tangent line to the level curve of the function f in the xy-plane that passes through the point (3, 5). No explanation is required. 2. Mark two s in each of the sentences in parts (a) and (b) with s to make them true statements. Then prove your claim by using the method you chose. x a. 3 y 2 x 2 y 3 The limit lim (x,y) (0,0) x 4 + y 6 is 0 does not exist, and this can be shown by using the Sertöz Theorem the 2-Path Test the Squeeze Theorem. x a. 4 y 2 The limit lim (x,y) (0,0) x 4 + y 6 x 2 y 2 is 0 does not exist, and this can be shown by using the Sertöz Theorem the 2-Path Test the Squeeze Theorem.

18 6 Bilkent Calculus II Exams Sertöz Theorem: Let a and b be nonnegative integers, let c and d be positive even integers, and let Then: If a c + b d >, then If a c + b d, then f(x, y) = xa y b x c + y d. lim f(x, y) = 0. (x,y) (0,0) lim f(x, y) does not exist. (x,y) (0,0) 3a. Find the largest possible value of the directional derivative D u f(2,, ) for the function: f(x, y, z) = yze xy+2z2 3 A y walks with a speed of cm/s in any direction on this page which is identied with the xy-plane. If the y walks from the origin in the direction of the vector A = i 7j, the temperature it measures increases at a rate of 3 C/s. If the y walks from the origin in the direction of the vector B = i+j, the temperature it measures decreases at a rate of 2 C/s. Find how fast the temperature it measures changes if the y walks from the origin in the positive x-direction. 4a. Find and classify the critical points of f(x, y) = 2x 2 + y 2 + 2x 2 y. 4 Next week in Section 4 of Math 02 the following quiz will be given: Q. Find the absolute minimum value of f(x, y) = 2x 2 + y 2 + 2x 2 y on the square S = {(x, y) : x + y 4}. (The gure is on the next page.)

19 Bilkent Calculus II Exams A friend of yours in Section 4 tells you their solution to this quiz: f has three critical points in the interior of S where its values are 0, and, respectively. Moreover f(4, 0) = f( 4, 0) = 32 and f(0, 4) = f(0, 4) = 6. So the absolute minimum value of f on S must be 0." You respond by saying that: Your answer cannot be correct, because." Fill in the box with less than 3 characters to make your response mathematically valid. a. Evaluate the iterated integral Evaluate the double integral Spring 204 Midterm II ˆ ˆ 0 R dy dx. x (y 2 + ) 2 da where R is the region shown in the gure. (x 2 + y 2 ) 2 (The gure is on the next page.)

20 8 Bilkent Calculus II Exams The region R bounded by the curve z 2 = y 2 y 4 in the right half of the yz-plane is rotated about the z-axis to obtain a solid D in the xyz-space. Let V be the volume of the solid D. a. Only two of ➊-➌ will be graded. Mark the ones you want to be graded by putting a in the corresponding s. ➊ Express V in terms of iterated integrals in Cartesian coordinates by lling in the rectangles. ˆ ˆ ˆ V = dz dy dx ➋ Express V in terms of iterated integrals in cylindrical coordinates by lling in the rectangles.

21 Bilkent Calculus II Exams ˆ ˆ ˆ V = r dz dr dθ ➌ Express V in terms of iterated integrals in spherical coordinates by lling in the rectangles. ˆ ˆ ˆ V = ρ 2 sin ϕ dρ dϕ dθ Compute V. 3a. Fill in the rectangles to make each of the following sentences a true statement. is a convergent sequence. is a convergent series. is a divergent sequence. is a divergent series. 3 Consider the question: Q. Find the sum of the series n=2 ( ) n ln. n + Your friend from Section 4 shows you their solution: ( ) ( ) ( ) n 2 3 ln = ln + ln +ln n n=2 ( 4 5 ) +... = (ln 2 ln 3) + ( ln 3 ln 4) + ( ln 4 ln 5) +... = ln 2

22 20 Bilkent Calculus II Exams Express your opinion of your friend's solution by ing the corresponding. The solution is correct. The solution is incorrect, but the answer is correct. Both the solution and the answer are incorrect. If you ed the rst, you are done with Question 3 If you ed the second or third, write the correct solution below. 4. In each of the parts (a-d), mark the appropriate s with s and then give a brief explanation consisting of complete and meaningful mathematical sentences. a. The series is convergent divergent. 2n n A positive integer n such that exists does not exists, 2i i= because: Explain! The series 5 n is convergent divergent. n A positive integer n such that i 205 exists does not exist, because: Explain! ( ) n 2999 c. The series is convergent divergent n ( ) i 2999 A positive integer n such that exists does not exist, 3000 i= because: Explain! d. The series is convergent divergent. n n A positive integer n such that exists does not exist, i i= because: i= Explain! Spring 204 Final. Suppose that f(x, y) is a dierentiable function that satises f(, 2) = 7, f x (, 2) = 3, f y (, 2) =, and ( ) x f x 2 + y, y = f(x, y) 2 x 2 + y 2 for all (x, y) (0, 0). a. Find an equation of the tangent plane to the graph of z = f(x, y) at the point with (x, y) = (, 2). Find parametric equations of the normal line to the surface z = f(x, y) at the point (x, y, z) = (, 2, 7).

23 Bilkent Calculus II Exams c. Find an equation of the tangent line to the level curve f(x, y) = 7 at the point (x, y) = (, 2). d. Compute f x (/5, 2/5). 2. Consider the transformation T : x = a. Compute the Jacobian (x, y) (u, v) of T. u u + v +, y = v u + v +. Show that there is a constant C such that the inequality for all regions G contained in the rst quadrant of the uv-plane. G (x, y) (u, v) du dv C holds 3. In each of the following, if there exists a sequence {a n } with nonzero terms satisfying the given conditions, write its general term inside the box; and if no such sequence exists, write Does Not Exist inside the box. No explanation is required. No partial credit will be given. a. a n converges and a n converges. a n = a n converges and a n diverges. a n = c. a n diverges and a n diverges. a n = d. {a n } converges and { } converges. a n a n =

24 22 Bilkent Calculus II Exams { } e. {a n } converges and diverges. a n a n = f. {a n } diverges and { } diverges. a n a n = 4. Determine whether each of the following series converges or diverges. a. (ln 2) n c. d. e. n ln 2 ( ) n n ln 2 ( n ln 2) ( n ln 2) n 5. Consider the power series n=2 x n n 2. a. Determine the interval of convergence of the power series. Find the exact sum of the power series at x =. Spring 203 Midterm I. Consider the sequence dened by: a = Write the last digit of your Bilkent ID number in the box! a n+ = 90 + a n for n

25 Bilkent Calculus II Exams In Parts a-c, mark the in the appropriate box or ll in the statements and then prove them. to make these into true a. The sequence {a n } is bounded below by. The sequence {a n } is bounded above by. c. The sequence {a n } is increasing decreasing. d. Show that the sequence {a n } converges. e. Let L = lim n a n. Find L. 2. Determine whether each of the following series is convergent or divergent. Explain your reasoning in full. ( ) n 2 n a. n + c. (2 /n ) n=2 (ln n) ln n 3. In each of the following, If the given statement is true for all sequences {a n }, then mark the to the left of True with a and ll in the blank with the name of a test; Otherwise, mark the to the left of False and give a counterexample. a. If a n converges, then {a n } converges. True, as can be shown using. False, because it does not hold for a n = for n. If {a n } converges, then a n converges. True, as can be shown using.

26 24 Bilkent Calculus II Exams False, because it does not hold for a n = for n. c. If a n converges conditionally, then na n diverges. True, as can be shown using. False, because it does not hold for a n = for n. d. If a n converges conditionally, then na n converges. True, as can be shown using. False, because it does not hold for a n = for n. 4. Consider the function f dened by the following power series: f(x) = n n (n!) 2 xn a. Write the rst ve nonzero terms of the power series with their coecients in a form as simplied as possible. Find the radius of convergence of the power series. c. Show that f() 5 4. You may use the fact that n n (n!) 64 for n n 5. For each of the following series, in the Write the exact value of the sum in a form as simplied as possible if the series converges; and Write Div if the series diverges. No explanation is required. No partial credit will be given.

27 Bilkent Calculus II Exams a. ( ) n π 2n 4 n (2n)! = ( ) n 3 n (2n + ) = c. 2 3n 3 2n = d n = Spring 203 Midterm II. Use power series to determine the value of the constant a for which the limit cos x e ax2 lim x 0 sin(x 4 ) exists and evaluate the limit for this value of a. (Do not use L'Hôpital's Rule!) 2. Consider the following lines: L : x = 2t, y = t + 2, z = 3t + L 2 : x = s + 5, y = 2s + 3, z = s a. Find a nonzero vector v perpendicular to both L and L 2. Find a parametric equation of the line L that intersects both L and L 2 perpendicularly. 3. Consider the function x a y b if (x, y) (0, 0) x f(x, y) = 4 + y 6 0 if (x, y) = (0, 0)

28 26 Bilkent Calculus II Exams where a and b are nonnegative integers. In each of the following, mark the corresponding with a and ll in the s where necessary to form a true statement. No explanation is required. a. f(x, y) is continuous at (0, 0) for a = and b =. for no values of a and f(x, y) goes to as (x, y) approaches (0, 0) along the line y = x, and f(x, y) goes to as (x, y) approaches (0, 0) along the line y = x for a = and b =. for no values of a and c. f(x, y) goes to 0 as (x, y) approaches (0, 0) along any line through the origin, and the limit f(x, y) does not exist lim (x,y) (0,0) for a = and b =. for no values of a and d. f(x, y) goes to 0 as (x, y) approaches (0, 0) along any line through the origin except the y-axis, and f(x, y) goes to as (x, y) approaches (0, 0) along the y-axis for a = and b =. for no values of a and e. f x (0, 0) and f y (0, 0) exist, and f(x, y) is not dierentiable at (0, 0) for a = and b =. for no values of a and 4. Let z = f(x, y) be a dierentiable function such that f(3, 3) =, f x (3, 3) = 2, f y (3, 3) =, f(2, 5) =, f x (2, 5) = 7, f y (2, 5) = 3. a. Find an equation of the tangent plane to the graph of z = f(x, y) at the point (2, 5, ).

29 Bilkent Calculus II Exams Suppose w is a dierentiable function of u and v satisfying the equation f(w, w) = f(uv, u 2 + v 2 ) for all (u, v). Find w u at (u, v, w) = (, 2, 3). 5. Find the values of the constants c and k for which the function satises the equation for all (x, y). u(x, y) = + e x cy u u x + u y = k 2 u x 2 Spring 203 Final. Consider the functions f(x, y) = x 2 y and g(x, y) = x 3 + 5y 2. a. Compute f and g. Find all unit vectors u along which the directional derivatives of f and g at the point P 0 (2, ) are equal. c. Find all points P (x, y) in the plane at which the directional derivatives of f and g in the direction of v = i + j are zero. 2. Find and classify the critical points of f(x, y) = xy 2 x 2 + 2x 3y Three hemispheres with radiuses, x and y, where x y 0, are stacked on top of each other as shown in the gure. Find the largest possible value of the total height h. y x h 4. Evaluate the following integrals.

30 28 Bilkent Calculus II Exams a. ˆ ˆ 2x 0 D x e y2 dy dx (x 2 + y 2 ) 2 da where D = {(x, y) : x2 + y 2 and 0 x and 0 y } 5a. Express the given triple integral as an iterated integral by lling in the boxes, where D is the region shown in the gure. ˆ ˆ ˆ f(x, y, z) dv = D f(x, y, z) dy dx dz 5 Evaluate D + (x 2 + y 2 + z 2 ) 3/2 dv where D = {(x, y, z) : x2 + y 2 + z 2 }. Spring 202 Midterm I a. Write in the box an equation for one of the planes that contain the line x = 2t + 3, y = 5t +, z = 4t, < t <. No explanation is required. Write in the box parametric equations for one of the lines that are contained in the plane 7x y 2z =. No explanation is required.

31 Bilkent Calculus II Exams c. Find an equation of the plane that passes through the point P ( 3, 2, ) and is perpendicular to both of the planes with equations x + 3y 8z = 2 and 2x y + 6z =. 2a. Find the length of the parametric curve x = (+cos t) cos t, y = (+cos t) sin t, 0 t 2π. 2 Find parametric equations of the tangent line to the parametric curve r = t i + t 2 j + t 3 k, < t <, at the point with t = The level curves of the following ve functions are shown in the gures below. Match these with their functions by lling in the boxes with the corresponding letters. A. f(x, y) = sin x + 2 sin y B. f(x, y) = (4x 2 + y 2 )e x2 y 2 C. f(x, y) = x 2 y 2 D. f(x, y) = xye y2 E. f(x, y) = 3x 2y (The gures are on the next page.)

32 30 Bilkent Calculus II Exams

33 Bilkent Calculus II Exams Let f(x, y) = x 3 y 4 x 4 + x 3 y 2 + y 0. a. Show that the limit of f(x, y) as (x, y) approaches (0, 0) along any line through the origin is the same. Show that lim f(x, y) does not exist. (x,y) (0,0) 5. Let u = x + y + z, v = xy + yz + zx, w = xyz, and suppose that f(u, v, w) is a dierentiable function satisfying f(u, v, w) = x 4 + y 4 + z 4 for all (x, y, z). Find f u (2,, 2). Spring 202 Midterm II a. In (i-iii), if there is a dierentiable function f(x, y) whose derivatives at (0, 0) in the directions of the vectors A, B, C are all positive, give an example of such a function; if there is no such function, write Does not exist in the box. No explanation is required. i. A = i + 2j, B = i j, C = i f(x, y) = ii. A = i + 2j, B = i j, C = i j f(x, y) = iii. A = 3i + j, B = i j, C = i j f(x, y) = A bug is standing on the ground at a point P. If it moves towards north from P, the temperature decreases at a rate of 4 C /m. If it moves towards southeast from P, the temperature increases at a rate of 3 2 C /m. In which direction should the bug move to go to cooler points as fast as possible? (Choose a coordinate system on the ground with the positive x-axis pointing east and the positive y-axis pointing north, and express your answer as a unit vector.) 2. Find the absolute maximum value of f(x, y) = x 3 xy y 2 + 2y on the square R = {(x, y) : 0 x and 0 y }. 3. Find and classify the critical points of f(x, y) = x 3 2x 2 + xy Evaluate the following integrals.

34 32 Bilkent Calculus II Exams a. cos(x + y) da where R = {(x, y) : 0 x π/2 and 0 y π/2}. R ˆ ˆ y y dx dy (x 2 + y 2 (x 2 + y 2 ) 2 ) /2 5. Let D be the region in the rst octant bounded by the coordinate planes, the plane x+y = 4, and the cylinder y 2 + 4z 2 = 6. Choose two of the following rectangular boxes by putting a in the small square in front of them, and then choose one of the orders of integration in each of the selected boxes by putting a in the small square in front of them. dx dy dz dy dx dz dz dx dy dx dz dy dy dz dx dz dy dx Express the volume V of the region D as iterated integrals in both of your selected orders of integration (a) and (b). (Do not evaluate the integrals! ) Spring 202 Final a. Find all points on the surface z = x 2 + y 2 where the tangent plane is parallel to the plane x + 2y + 3z = 6. A bug is standing on the ground at a point P. If it moves towards north from P, the temperature decreases at a rate of 4 C /m. If it moves towards southeast from P, the temperature increases at a rate of 3 2 C /m. In which direction should the bug move to go to cooler points as fast as possible? (Choose a coordinate system on the ground with the positive x-axis pointing east and the positive y-axis pointing north, and express your answer as a unit vector.) 2a. Suppose that T is a one-to-one transformation from the xy-plane to the uv-plane whose Jacobian satises the condition 0 < (u, v) (x, y) (x 2 + y 2 + ) 2 for all (x, y). Show that if G is a region in the xy-plane then the area of its image T (G) in the uv-plane is not more than π. 2 Let f(x) = e x + e 2x e 3x +. Find the domain and the range of f. 3. Let D be the region in space bounded on the top by the sphere x 2 + y 2 + z 2 = 2 and on the bottom by the paraboloid z = x 2 + y 2. Fill in the boxes in parts (a-c) so that the right sides of

35 Bilkent Calculus II Exams the equalities become iterated integrals expressing the volume V of D in the given coordinates and orders of integration. No explanation is required. ˆ ˆ ˆ a. V = dz dy dx ˆ ˆ ˆ V = dz dr dθ ˆ ˆ ˆ c. V = dρ dϕ dθ ˆ ˆ ˆ + dρ dϕ dθ 4. In each of the following indicate whether the given series converges or diverges, and also indicate the best way of determining this by choosing exactly one of the tests and lling in the corresponding blank if any. (You must choose a test to get any points.) a. n 2 + n nth Term Test Integral Test converges diverges Direct Comparison Test with Limit Comparison Test with Ratio Test nth Root Test Alternating Series Test 3 n sin(π n ) converges diverges

36 34 Bilkent Calculus II Exams nth Term Test Direct Comparison Test with Integral Test Limit Comparison Test with Ratio Test nth Root Test Alternating Series Test c. 2 n2 n! nth Term Test Integral Test converges diverges Direct Comparison Test with Limit Comparison Test with Ratio Test nth Root Test Alternating Series Test d. ( ) n ln n converges diverges n=2 nth Term Test Direct Comparison Test with Integral Test Limit Comparison Test with Ratio Test nth Root Test Alternating Series Test e. n(ln n) 2 n=2 nth Term Test Integral Test converges diverges Direct Comparison Test with Limit Comparison Test with Ratio Test nth Root Test Alternating Series Test 5. Let f(x) = c n x n for all x in the interval of convergence of the power series where c 0 = ( and c n+ = c n + ) (n+) for n 0. n + a. Find c, c 2 and c 3. Find the radius of convergence of the power series c. Find f (0). c n x n.

37 Bilkent Calculus II Exams d. Express the sum of the series n 2 c n in terms of A = f(), B = f () and C = f (). Spring 20 Midterm I. Consider the power series x n2 2 n n. a. Write the rst three nonzero terms of the power series. Find the radius of convergence of the power series. c. Find the exact value of the sum of the power series at x =. 2a. Find the coecient of x 20 in the Maclaurin series of f(x) = xe x2. 2 Exactly one of the following statements is true. Choose the true statement and mark the box in front of it with a. If a 2 n converges, then ( ) n+ a n converges. If a 2 n converges, then ( ) n+ a 3 n converges. Now either Prove the true statement, or Give an example that shows the other statement is not true. Indicate the task you choose with a. 3a. Find the equation of the plane containing the line L : x = 2t + 3, y = 4t, z = t + 2, < t <, and parallel to the line L 2 : x = 2s + 3, y = s + 2, z = 2s 2, < s <. 3 When a wheel of unit radius rolls along the x-axis the path traced by a point P on its circumference is given by r = (t sin t) i + ( cos t) j, < t <. Find the distance traveled by P during one full turn of the wheel. x 4a. 3 y 2 Show that lim (x,y) (0,0) x 6 + y = 0. 2 x 4 6 y 4 Show that lim does not exist. (x,y) (0,0) (x 6 + y 2 ) 3 5. Find all possible values of the constants a and b such that the function f(x, y) = y a e bx2 /y

38 36 Bilkent Calculus II Exams satises the equation for all (x, y) with x > 0 and y > 0. 2 f x + 2 f 2 x x = f y Spring 20 Midterm II. Let P 0 (2, 2, ) and suppose that f(x, y, z) and g(x, y, z) are dierentiable functions satisfying the following conditions: i. f(p 0 ) = and g(p 0 ) = 6. g ii. x =. P0 iii. At P 0, f increases fastest in the direction of the vector A = 4i j 8k and its derivative in this direction is 7. iv. The tangent plane of the surface dened by the equation f(x, y, z) + 2g(x, y, z) = 3 at the point P 0 has the equation 5x + y z =. Find g z. P0 2. Each of the following functions has a critical point at (0, 0). Indicate the type of this critical point by marking the corresponding box with a. No explanation is required. No partial credit will be given. a. f(x, y) = x 3 y 3 has a local maximum local minimum saddle point none of the above at (0, 0). f(x, y) = x 2 y 2 has a local maximum local minimum saddle point none of the above at (0, 0).

39 Bilkent Calculus II Exams c. f(x, y) = y 2 yx 2 has a local maximum local minimum saddle point none of the above at (0, 0). d. f(x, y) = x 2 x 2 y + y 2 has a local maximum local minimum saddle point none of the above at (0, 0). 3. A at circular plate has the shape of the region x 2 + y 2. The plate, including the boundary where x 2 + y 2 =, is heated so that the temperature at a point (x, y) is T (x, y) = x 2 + 2y 2 x. Find the temperatures at the hottest and coldest points on the plate. 4. Evaluate the following integrals. a. x2 + y 2 da where R is the region shown in the gure. R y y = 3 x R x 2 + y 2 = 4 x ˆ ˆ /x 2 0 xy 2 e y2 dy dx 5. Let D be the region in space lying inside the sphere x 2 + y 2 + z 2 = 4, outside the cone z 2 = 3(x 2 + y 2 ), and above the xy-plane. Fill in the boxes in parts (a-c) so that the right sides of the equalities become iterated integrals expressing the volume V of D in the given coordinates and orders of integration. No explanation is required.

40 38 Bilkent Calculus II Exams ˆ ˆ ˆ a. V = dρ dϕ dθ ˆ ˆ ˆ V = dz dr dθ ˆ ˆ ˆ + dz dr dθ ˆ ˆ ˆ c. V = dr dz dθ z y x Spring 20 Final a. Write the rst three nonzero terms of the Maclaurin series of x where a is a constant. + ax2

41 Bilkent Calculus II Exams Write the rst three nonzero terms of the Maclaurin series of sin(bx) where b is a constant. c. Find the constants a, b, c, d if interval containing x = 0. x + ax 2 sin(bx) = x3 + cx 4 + dx 5 + on some open 2. Let u = x 2 y 3, v = sin(πx), and z = f(u, v) where f is a function with continuous second order partial derivatives satisfying: a. Find z x. (x,y)=( 2,) Find 2 z y x. (x,y)=( 2,) f(4, 0) = 0 f u (4, 0) = 5 f v (4, 0) = 7 f uu (4, 0) = 2 f uv (4, 0) = f vv (4, 0) = 3 3. Evaluate e (y x)/(y+x) da where R is the region shown in the gure. R y R x 4a. Evaluate 4 Evaluate the line integral D dv x2 + y 2 + (z 2) 2 where D is the unit ball x 2 + y 2 + z 2. boundary of the region R shown in the gure. C (6xy + sin(x 2 )) dx + (5x 2 + sin(y 2 )) dy where C is the (The gure is on the next page.)

42 40 Bilkent Calculus II Exams y C R x 2 + y 2 = 4 x 2 + y 2 = x 5. Consider the parametrized surface S : r = u 2 i + 2 uvj + v 2 k, < u <, 0 v <. Find the area of the portion of the surface S that lies inside the unit ball x 2 + y 2 + z 2. Spring 200 Midterm I xy a. 2 Show that lim (x,y) (0,0) x 6 + y = 0. 2 xy Show that lim does not exist. (x,y) (0,0) x 6 + y2 x y c. a Consider lim where a is a constant. (x,y) (0,0) x 6 + y2 There is a real number A such that this limit is 0 if a > A, and this limit does not exist if a < A. What is A? No explanation is required and no partial points will be given in this part. Write your answer here A = 2. Assume that f(x, y, z) is a dierentiable function and at the point P 0 (,, 2), f increases fastest in the direction of the vector A = 2i + j 2k. Exactly one of the following statements can be true about this function. Mark this statement with an and nd ( f) P0 assuming the statement to be true. Mark the other statement with an and explain why it cannot be true. The directional derivative of f at P 0 in the direction of the vector B = 2i + 6j + 3k is 5. The directional derivative of f at P 0 in the direction of the vector B = 3i + 2j + 6k is Find the points on the surface xy + yz + zx x z 2 = 0 where the tangent plane is parallel to the xy-plane.

43 Bilkent Calculus II Exams Find all possible values of the constants C and k such that the function f(x, y) = C(x 2 +y 2 ) k satises the equation f xx + f yy = f 3 for all (x, y) (0, 0). 5. Find the absolute maximum and minimum values of the function f(x, y) = 2x 3 +2xy 2 x y 2 on the unit disk D = {(x, y) : x 2 + y 2 }. Spring 200 Midterm II. Evaluate the integral ˆ π ˆ 0 x/π y 4 sin(xy 2 ) dy dx. 2. Let D be the region in space bounded by the plane y + z = on the top, the parabolic cylinder y = x 2 on the sides, and the xy-plane at the bottom. Choose two of the following rectangular boxes by putting a in the small square in front of them, and then choose one of the orders of integration in each of the selected boxes by putting a in the small square in front of them. dx dy dz dy dx dz dz dx dy dx dz dy dy dz dx dz dy dx Express the volume V of the region D as iterated integrals in both of your selected orders of integration (a) and (b). (Do not evaluate the integrals! ) 3. V = ˆ 3 ˆ 3 x ˆ 2 x 2 2 y x 2 x 2 +y 2 dz dy dx expresses the volume V of a region D in space as an iterated integral in Cartesian coordinates. Fill in the boxes in (a) and (b) so that the right sides of the equalities become iterated integrals expressing the volume of D in cylindrical and spherical coordinates, respectively. No explanation is necessary in this question. (Do not evaluate the integrals! ) ˆ ˆ ˆ a. V = dz dr dθ ˆ ˆ ˆ V = dρ dϕ dθ

44 42 Bilkent Calculus II Exams ˆ ˆ ˆ + dρ dϕ dθ 4. Find the value of the line integral (3x 2 y 2 + y) dx + 2x 3 y dy where C is the cardioid r = + cos θ parameterized counterclockwise. C 5. Let D be the closed ball x 2 + y 2 + z 2 6. a. Show that for any vector eld F that has continuous partial derivatives and satises the condition F 5 on D, F dv 320π. D Give an example of a vector eld F that has continuous partial derivatives and satises the condition F 5 on D such that F dv = 320π, and verify that the equality holds. D Spring 200 Final. Let T be a transformation from the uv-plane to the xy-plane given by x = f(u, v) and y = g(u, v) where f and g are functions with continuous partial derivatives. Assume that T satises the condition that (Area of T (G)) = (u 2 + v 2 ) du dv for every closed subset of the uv-plane on which T is one-to-one. a. Let G = {(u, v) : u 2 + v 2 4, u 0 and v 0}. Find the area of the image of G under T assuming that T is one-to-one on G. If f(u, v) = uv, nd a g(u, v) that satises the conditions of the question. (You do not have to explain how you found g(u, v), but you must verify that the conditions are satised.) G 2. Determine whether each of the following series is convergent or divergent. a. (ln 2) n

45 Bilkent Calculus II Exams c. (ln n) 3 n=2 n 2 (5n)! 30 n (2n)!(3n)! 3. Determine whether each of the following series is convergent or divergent. ( ) π a. ( ) n+ cos n c. n + n sin n 5 n 2 n 7 n 6 n 4. Consider the power series x n 9n 2. a. Find the radius of convergence of the power series. Determine whether the power series converges or diverges at the right endpoint of its interval of convergence. If it converges, determine the type of convergence. c. Determine whether the power series converges or diverges at the left endpoint of its interval of convergence. If it converges, determine the type of convergence. 5. In parts (a-b) of this question, if the series converges, write the exact sum of the series inside the box; and if the series diverges, write Diverges inside the box. No explanation is required. No partial points will be given. a. 4 n (2n + ) = n=2 n 2 = In parts (c-d) of this question, if there exists a sequence {a n } satisfying the given conditions, write its general term inside the box; and if no such sequence exists, write Does Not Exist inside the box. No explanation is required. No partial points will be given. c. a n < a n+ for all n and lim n a n.

46 44 Bilkent Calculus II Exams a n = d. lim n na n = 0 and a n diverges. n=2 a n = Spring 2009 Midterm I. Determine whether each of the following series is convergent or divergent. n+ 2n a. ( ) c. 3 n + (ln n) 2 cos(/n) 2. Determine whether each of the following series is convergent or divergent. n!(n + )! a. (2n + )! c. 0 n2 /(n+) (e /n2 ) 3. Consider the power series x n (n 2 + )(2 n + ). a. Find the radius of convergence of the power series. Determine whether the power series is absolutely convergent, conditionally convergent or divergent at the right endpoint of its interval of convergence.

47 Bilkent Calculus II Exams c. Determine whether the power series is absolutely convergent, conditionally convergent or divergent at the left endpoint of its interval of convergence. 4. Find the coecient of the rst nonzero term in the Maclaurin series generated by x f(x) = sin x + x 2 /6. 5a. Determine whether the sum of the series 5 Find the sum of the series n=2 2 n (n 2 ). ( 4) n n!(n + )! is positive or negative. x a. 4 y 4 Show that lim (x,y) (0,0) x 2 + y = 0. 4 Spring 2009 Midterm II x 4 y 4 Show that lim does not exist. (x,y) (0,0) (x 2 + y 4 ) 3 2. A dierentiable function f(x, y, z) increases fastest at the point P 0 (2, 5, ) in the direction of i + 2j 3k and at a rate of 7. a. Find f z. P0 Find the equation of the tangent plane to the level surface of f passing through the point P 0. 3a. Show that if f(z) is a dierentiable function and u(x, y) = f(x 2 y 2 ), then yu x + xu y = 0 for all (x, y). 3 Show that the parametric curve r(t) = e t cos t i + e t sin t j, < t <, cuts every circle with center at the origin at the same angle, and nd this angle. 4. Find and classify the critical points of f(x, y) = x 3 6x + x 2 y Evaluate the following integrals ˆ ˆ π/6 π/3 a. 0 R x 2 x 3 sin(y 3 ) dy dx x 2 sin(x 2 + y 2 ) da where R = {(x, y) : x 2 + y 2 π}.

48 46 Bilkent Calculus II Exams Spring 2009 Final. In each of the following indicate whether the given series converges or diverges, and also indicate the best way of determining this by marking the corresponding boxes and lling in the corresponding blanks. a. n +/n converges diverges nth Term Test Ratio Test Direct Comparison Test with Limit Comparison Test with nth Root Test n=2 n n5 nth Term Test Ratio Test nth Root Test Alternating Series Test converges diverges Direct Comparison Test with Limit Comparison Test with Alternating Series Test c. n 2 2 n nth Term Test Ratio Test nth Root Test converges diverges Direct Comparison Test with Limit Comparison Test with Alternating Series Test d. ( ) n n ln n n=2 nth Term Test Ratio Test nth Root Test converges diverges Direct Comparison Test with Limit Comparison Test with Alternating Series Test e. e n 2 n π n 3 n converges diverges

49 Bilkent Calculus II Exams nth Term Test Direct Comparison Test with Ratio Test Limit Comparison Test with nth Root Test Alternating Series Test 2. Find the closest and farthest points on the sphere x 2 + y 2 + z 2 = 4 to the point P (3,, ). 3a. Evaluate the integral D z 2 (x 2 + y 2 + z 2 ) 3 dv where D = {(x, y, z) : x2 + y 2 }. 3 Express the iterated integral in cylindrical coordinates iterated integrals in spherical coordinates. Do not evaluate. ˆ π ˆ ˆ 4 r dz dr dθ in terms of 4a. Find a function f(x, y) such that, for any region G in the rst quadrant of the xy-plane, the double integral f(x, y) dx dy gives the area of T (G) where T is the transformation u = x 2 /y, v = x/y 2. 4 Evaluate G ˆ F dr where F = 3x 2 y 2 z i + 2(x 3 z + z 2 )y j + (x 3 + 2z)y 2 k and C C : r = (t 3 2t) i + (t 4 4t 2 ) j + cos(πt) k, 0 t. 5. Let C be the unit circle in the xy-plane, and S be the boundary of the square K = {(x, y) : 0 x and 0 y }. If a is a constant such that (y 3 + xy 2 + xy) dx + (3xy 2 + x 2 y + ax) dy =, evaluate C S (y 3 + xy 2 + xy) dx + (3xy 2 + x 2 y + ax) dy. Spring 2008 Midterm I (n!). a. 2 (3n) n Evaluate lim. n n n (2n)! ( ) sin n Does the series + converge or diverge? Justify! n 2 n(ln n) 2 n=2 2. a. Find the Taylor series generated by f(x) = ( + x 2 ) /3 at x = 0. Use the series in part (a) to estimate ˆ /2 0 ( + x 2 ) /3 with error less than 0.0.

50 48 Bilkent Calculus II Exams Find the radius and the interval of convergence of the series n (x + 2)n ( ). n(n + ) 4. a. Find parametrization for the line in which the planes 5x 2y = and 4y 5z = 7 intersect. Write u = j + k as the sum of a vector parallel to v = i + j and a vector orthogonal to v. 5. a. Let r (t) = cos t i + sin t j + t k and r 2 (t) = sin s i + cos s k be two curves. Find the intersection points of r and r 2. Moreover, nd the angle between the tangent vectors at each intersection point. Find the point of intersection of the lines x = 2t +, y = 3t + 2, z = 4t + 3 and x = s + 2, y = 2s + 4, z = 4s. Then nd the plane determined by these lines. Spring 2008 Midterm II. Let D be the solid bounded by the surfaces x 2 + y 2 + z 2 = 9 and x 2 + y 2 + z 2 =, and lying above the surface z 2 = 3(x 2 + y 2 ) with z 0. Write down three integrals in rectangular, cylindrical, and spherical coordinates that give the volume of the solid. Do not evaluate these integrals. 2. a. Evaluate Evaluate ˆ 9 ˆ 3 0 y sin(x 3 ) dx dy. ˆ 2 ˆ (x ) x + y dy dx. x 2 + y2 3. Find the shortest distance from the origin to the surface xyz 2 = 2. (Explain why it is the shortest and not the longest.) 4. a. Let f(x, y) = x 2 y 2xy + y 2 5y. Find and classify the critical points of f. Find the direction of most rapid increase for f at the point (, ) and the rate of change of f in this direction. 5. a. Let z = f(x, y) be a dierentiable function of two independent variables x and y such that f(2, ) = 3, f x (2, ) = 2, f y (2, ) =. Dene another function z = g(x, y) of two independent variables x and y as follows: ( 2 g(x, y) = f x 2 + y, y ) 2 x exy Find the equation of the tangent plane to the surface z = g(x, y) at the point (x, y) = (, 0). Using the tangent plane in part (a) approximate g(., 0.2).