1 Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6 n. A) ln 6 B) C) e 6 D) does not exist 3. Find the limit of the sequence a n = n! 6 n +8 n. A) B) C) e 8 D) does not exist. Find a formula for the nth partial sum of the series it to find the series sum if the series converges. 6 9n +n+ n= and use A) 6n 5(3n+5) ; 5 B) 6n 5n+5 ; 5 C) 6n 5(n+) ; 6 5 D) 6n 5n+ ; Find the sum of the telescoping series e n e n 3, if it exists. n= A) e 5 B) e 3 C) e D) does not exist 6. Find the sum of the telescoping series ( ln n= n+ ) ln ( n+ A) ln B) ln 3 C) D) does not exist ), if it exists. 7. Find the sum of the geometric series n= A) 9 7 B) 7 7 C) 8 D) ( ) n 9 8 n. 8. Use the integral test to determine whether the series n e /n converges or diverges. A) converge B) diverge n= 9) Determine whether the series ln ( e / n) converges or diverges. n=
2 A) converge B) diverge. Use the ratio test to determine if the series n= 5(n!) (n)! converges or diverges. A) converge B) diverge. Use the comparison test to determine if the series ( n n 5n+) converges or diverges. A) converge B) diverge n=. Use the limit comparison test to determine whether the series 9 converges or diverges. n 3 ln n+ n= A) converge B) diverge 3. Determine whether the alternating series ( ( ) n ln diverges. n= A) converge B) diverge 6n+ 6n+3. Determine whether the alternating series ( ) n ( n n + ) converges absolutely, conditionally, or diverges. n= ) converges or A) converge absolutely B) diverges C) converges conditionally 5. Estimate the magnitude of the error involved in using the sum of the first four terms to estimate the sum of the entire series n= ( ) n+ (.) n+ n+. A) B) 3. C).5 8 D) Find the radius of convergence of the power series A) 8 B) C) 9 D) n= 7. Find the radius of convergence of the power series n= (x 5) n 9 n n. (x 5) n 9 n n!.
3 A) 8 B) C) D) 9 8. Find the interval of convergence of the power series n= ( ) n (x 5) n (n+)6 n. A) x B) < x C) x 6 D) x < 6 9. Find the first for tems of the McLaurin series for f (x) = ln ( + x). A) x x! + x3 3! x x! + B) x + + x3 3 + x C) x x + x3 3 x. Find the Taylor series sin x at x = π. + x + D) x +! + x3 3! + x! + A) ( ( ) ( ) x π + x π ( ) ) 6 x π 3 + B) ( + ( ) ( ) x π x π ( ) ) 3 x π 3 + C) ( ( ) ( ) + x π 6 x π ( ) ) x π 3 + D) ( + ( ) ( ) x π x π ( ) ) 6 x π 3 + E) None of the above. Find the angle between the two planes x + 3y 3z = and x + y 5z =. A).9 B).3 C).57 D).8. Which of the following vectors is parallel to the two planes x + 3y z = and x + y + z = 8? A) i + 3 j 5 k B) i 3 j 5 k C) i + 3 j 6 k D) i 3 j 6 k 3. Find parametric equations of the line of intersection of the two planes x + y + z = and x y + z =. A) x = t +, y = t, z = 3t B) x = t +, y = t, z = 3t C) x = t, y = 3 + t, z = 5 3t D) x = + t, y = t, z = 3 + 3t. Find the point of intersection of the line x x + y + z =. = y+ = z and the plane 3
4 A) (,, ) B) (,, ) C) ( 9,, ) D) (,, ) 5. Find symmetric equations of the line perpendicular to the plane 5x + y + z = and passing through the point (, 3, ). A) x+ = y + 3 = z+ 5 B) x+ 5 = y + 3 = z+ C) x 5 = y 3 = z D) x 5 = y 3 = z 6. Which of the vectors below is tangent to the curve r (t) = cos t i + sin t j + e t k at t =? A) j k B) j + k D) All of the above C) j k 7. Calculate the arc length of the curve r (t) = t i + t sin t j + t cos t k ; t. A) 5 B) C) D) 6 8. Find the acceleration at t = for r (t) = ( t t 3) i + 5t j + ( t 3 ln t ) k. A) i + k B) i + k C) 6 i + 6 k D) i + 5 j + k x 9. Find lim +y + (x,y)(,) x 6xy+7. A) B) 7 C) 7 D) No limit 3. Find lim x+y. (x,y)(,) x +y A) B) No limit C) 7 D) - 3. At what points is the function f (x, y) = xy x continuous? A) All (x, y) such that xy > and x B) All (x, y) such that x i y and x > C) All (x, y) such that x D) All (x, y) 3. Find all first order partial derivatives of the function f (x, y) = x ln (xy).
5 A) f f x = + ln (xy) ; y = x y C) f f x = x + ln (xy) ; y = x y B) f f x = ln (xy) ; y = x y D) f x = x + x y ; f y = y 33. Find all second order partial derivatives of the function f (x, y) = xye y. A) f x B) f x C) f x D) f x = ye y ; f y = xye y ( y 6 ) ; = ; f y = xye y ( y 3 ) ; = ye y ; f y = xye y ( y ) ; = ; f y = xye y ( y 3 ) ; 3. Find a chain rule formula for w z = k(r, s, t). f x y = f y x = ( ) y e y f x y = f y x = ( ) y e y f x y = f y x = ( ) y e y f x y = f y x = ( ) y e y if w = f(x, y, z), x = g(r, s), y = h(t), A) w C) w = w dx dt + w z = w dy y dt + w z z B) w D) w = w z z = dy dt + z 35. Compute the gradient of f(x, y, z) = ln ( x 5y + 7z ) at ( 5, 5, 5). A) 5 i + 3 j B) 5 i + 3 j 5 C) i + 3 j 5 k D) i + 3 j k k k 36. Find the derivative of f (x, y, z) = 3xy 3 z at the point ( 3, 7, 9) in the direction of v = i + j k. A) 53, B) 885, 735 C) 78, 588 D) 35, Find the direction in which the function f (x, y) = xe y ln x is decreasing most rapidly at the point (, ). A) 7 i + 7 j B) 7 i 7 j C) 7 i 7 j D) 7 i 38. Find the derivative of the function f (x, y) = arctan ( y x) at the point ( 8, 8) in the direction in which the function increases most rapidly. A) 3 B) 6 C) D) 3 6 5
6 39. A simple electrical circuit consists of a resistor connected between the terminals of a battery. The voltage V (in volts) is dropping at the rate of. volts per second as the battery wears out. At the same time the resistance R (in ohms) is increasing at the rate of ohms per second as the resistor heats up. The power P (in watts) disippated by the circuit is given by P = V R. How much is the power changing when R = 5 and V =? A).3 watts B).6 watts C).3 watts D).6 watts. Find an equation for the tangent plane to the surface ln z = 8x + 3y at the point (,, ). A) 6x + 6y z = 9 B) 6x + 6y + z = C) 6x 6y + z = 3 D) 6x 6y z =. Find all extreme values of the function f(x, y) = 5x y + 7xy and identify each as a local maximum, local minimum, or saddle point. A) f ( 7, ) 5 = 7 5, local minimum B) f (35, 35) = 5, 5, local max C) f ( 5, ) 7 = 35, local minimum D) f (, ) =, saddle point. Find all extreme values of the function f(x, y) = x 3 + y 3 3x 8y 8 and identify each as a local maximum, local minimum, or saddle point. A) f (, 6) =, local minimum; f (, 6) = 576, saddle point; f (, 6) = 56, saddle point; f (, 6) =, local maximum B) f (, 6) = 576, saddle point; f (, 6) = 56, saddle point C) f (, 6) =, local maximum D) f (, 6) =, local minimum; f (, 6) =, local maximum 3. Find the absolute maximum and minimum of the function f (x, y) = 7x + y on the trapezoidal region with vertices (, ) (, ), (, ), (, ). A) Absolute maximum: at (, ); absolute minimum: 7 at (, ) B) Absolute maximum: 8 at (, ); absolute minimum: at (, ) C) Absolute maximum: at (, ); absolute minimum: at (, ) D) Absolute maximum: at (, ); absolute minimum: 7 at (, ). Use Lagrange multipliers to find the maximum and minimum values of f (x, y, z) = x + y z subject to the constraint x + y + z = 9. A) Maximum: 8 at (,, ); minimum -8 at (,, ) B) Maximum: 9 at (,, ); minimum -9 at (,, ) C) Maximum: at (,, ); minimum - at (,, ) D) Maximum: at (,, 3); minimum - at (,, 3) 6
7 5. Find the point on the plane x + y z = that is nearest the origin. A) (,, ) B) (, 8, ) C) (,, ) D) (,, ) 6. Evaluate π 5π (sin x + cos y) dxdy A) π B) 9π C) 5π D) 8π 7. Write an integral equivalent to ln 8 ln integration reversed. A) 8 ln x ln 8 ln 7y dy dx B) 8 C) 8 ln x ln 7y dy dx D) 8 ln 8 e y 7y dx dy but with the order of ln x ln 8 7y dy dx ln x 7y dy dx ln 8 8. Write an integral equivalent to 6 integration reversed. 36 8y dx dy but with the order of y A) 36 x 8y dy dx B) 36 x 8y dy dx 6 C) 6 x 8y dy dx D) 6 x 6 8y dy dx 9. Express the area of the region bounded by x = y and the line y = x as a double integral. A) y+8 dx dy B) y C) 5. Evaluate ln y+8 dx dy D) e y dx dy. e y y+8 y dx dy y+8 dx dy A) 8 B) C) 8 D) 5. Integrate f (x, y) = ln x over the region bounded by the x axis, the line x = 3, and the curve y = ln x. A) B) C) 3 D) 5. Find the volume of the region bounded by the coordinate planes, the parabolic cylinder z = x, and the plane y = 5. 7
8 A) 8 3 B) 3 C) 8 D) Evaluate ln 8 ln 8 dx dy by reversing the order of integration. y/ ex A) 36 B) 35 C) D) 5 5. Change 6 dy dx to polar coordinates and evaluate. 36 x + x +y A) π(6+ln 7) π(6 ln 7) B) 55. Find the area of the region enclosed by r = 9 sin θ. π(6 ln 7) π(6+ln 7) C) D) A) 7 π B) 8 π C) 8 8 π D) 8 π 56. Write a triple integral in the order dz dy dx for the volume of the solid enclosed by the parabaloids z = x y and z = x + y. A) x x y dz dy dx B) x x x +y x y dz dy dx x x +y C) x x x y x +y dz dy dx D) 57. Write cylindrical coordinates. A) π C) π x x y x x +y dz dy dx as an equivalent integral in r r dz dr dθ B) π r 5 r r dz dr dθ D) π/ r x x y dz dy dx x x +y r r dz dr dθ r r r r dz dr dθ 58. Set up a triple integral for the volume of the solid in the first octant inside the cone ϕ = π 3 and between the spheres ρ = and ρ = 7. A) π/ π/3 C) π π/ 7 ρ sin ϕ dρ dϕ dθ B) π/ π/ π/3 7 ρ sin ϕ dρ dϕ dθ D) π/ π/3 7 π/ π/3 ρ sin ϕ dρ dϕ dθ 7 ρ sin ϕ dρ dϕ dθ 59. Use cylindrical coordinates to find the volume of the solid bounded below by the xy-plane, laterally by the cylinder r = sin θ, and above by the plane z = 7 x. A) 9π B) 9 π C) 7 π D) 7π 8
9 6. Find the mass of the solid in the first octant between the spheres x + y + z = 6 and x + y + z = if the density at any point is inversely proportional to the distance from the origin. A) 68kπ B) 8kπ C) k D) kπ Answer Key. C. B 3. D. A 5. B 6. D 7. C 8. A 9. B. A. A. B 3. A. C 5. D 6. C 7. D 8. B 9. C. D. A. B 3. C. D 5. D 6. D 7. C 8. C 9. B 3. B 3. A 3. A 33. D 3. C 35. B 36. A 37. C 9
10 38. B 39. A. D. D. A 3. C. B 5. B 6. D 7. C 8. A 9. B 5. C 5. D 5. A 53. B 5. A 55. D 56. C 57. A 58. B 59. D 6. C
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