Quiz problem bank. Quiz 1 problems. 1. Find all solutions (x, y) to the following:

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1 Quiz problem bank Quiz problems. Find all solutions x, y) to the following: xy x + y = x + 5x + 4y = ) x. Let gx) = ln. Find g x). sin x 3. Find the tangent line to fx) = xe x at x =. 4. Let hx) = x 3 9x + x π. Find the critical points of h and classify them as local max, local min, or neither. 5. Find the second order Taylor polynomial of the function fx) = arctan x centered around x =. x + 3x 6. Find dx. x 7. Find d x ) sint ) dt. dx x 8. Find e x + dx. 9. Identify and sketch the curve r = sin θ. 8. A 4 mph wind is blowing due south 7 ), you are on a plane that has bearing of N 6 E aka 3 ), but the plane is traveling due east ). What is the airspeed of the plane? How fast is the plane moving relative to the ground? 9. Find the projection of u = i + 5j + 3k onto the vector v = i + j 3k. Quiz 3 problems. Find the equation of the plane corresponding to the set of points x, y, z) equidistant from the points,, 4) and 6, 3, ).. Find cos θ where θ is the angle between the planes x + y z = 7 and 4x + 3z = Which of the following do not make sense. Explain your answer.) u v) w; u v w); u v) w; u v) w 4. Find the plane which goes through the points,, 3),,, 3) and,, ). 5. Find the area of the triangle with,, ),,, ) and,, ) as vertices. 6. Find the plane passing through the point,, 3) and perpendicular to both the planes x 3y + z = 7 and x y z = e. 7. While holding an egg you travel along Quiz problems. Find the volume for the set of points x, y, z) satisfying y + z x + y + z y Find the equation of the sphere which has the line segment joining, 3, 7) and 4,, 5) as a diameter. 3. Rewrite φ = π/6, θ = π/3, ρ = 6 as a point in artesian coordinates. 4. Describe, by picture and short explanation, the set of points satisfying φ π/4 and ρ. 5. Rewrite ρ = secφ) in artesian coordinates and cylindrical coordinates. 6. Rewrite ρ = in ylindrical coordinates and sketch the cos φ + sin φ surface. 7. Write φ = 4π in cylindrical and artesian coordinates. rt) = t 3 + t)i + 6t 7)j + 7 t 3 )k. At time t = you let go of the egg. Determine where the egg will hit the xy-plane. 8. Find the line in parametric form containing the point,, ) and which perpendicularly intersects the line x = + t, y = t, z = t. 9. Let rt) = cosπt), sinπt), t 3. Find the plane perpendicular to this curve at time t =. Quiz 4 problems. A particle moving through three dimensional space has vt) = sec t, sec t tan t, tan t as its velocity function. If at t = the particle is at,, find the position of the particle at t = π 4. t. If Ft) = arctan t, e t, siny ) dy, find F t) and F t).

2 3. Find e t i + cosπt)j t ) t + k dt. 4. Find the distance a particle travels along the curve rt) = sint), 3 t3/, cost) from t = to t = Find the distance a particle travels along the curve e t, e t, t ) from t = to t =. 6. Find curvature κt) for rt) = sin t, cos t, t. 7. Find the curvature at time t = for the curve rt)= 7+lnt+), tan t+t + π, e t cos t eπ 6 Quiz 5 problems. Find a T and a N for rt) = ti+t j+ 3 t3 k. Simplify your answers.. Find a N for rt) = e t + 7, e t π, 37 t.. 3. Find all times t that the parametric curve given by t 3 + e t +, t + cos t +, t + e t + cos t ) intersects the plane x + y z = Sketch and label the level curves for the function fx, y) = y/x + ) for the values,,,,. 4. Identify the common name for the level surfaces corresponding to fx, y, z) = x + y z for the values,,. Draw with label) one of these three surfaces. 5. Find f x, ) and f y, ) for the function fx, y) = sin x + y cos x + y 4 arctanxy )) + lne sin x ) secxy) tany ). 6. Find all the second partial derivatives for the function gx, y) = xe xy. 7. Show hx, y) = x 3 y xy 3 satisfies h xx +h yy =. x sin x 8. Determine lim x,y),) x + y 4. x y sin x 9. Determine lim x,y),) x + y 4. x 4 y 4. Determine lim x,y),) x + y. 4. Find si s j + 3k) e t i + sint)k). 5. Give a formula for, and classify, the quadric surface containing the parametric curve e t sin t + cos t), e t, e t sint) ). 6. Find the point on the plane x+y = 5+3z closest to the point 4, 4, 7). Hint: the line between these two points is perpendicular to the plane.) 7. The following two lines are intersecting: x = t y = + t z = 3 t and x = t y = 3 t z = 3 + t Find the line which goes through the intersection point and is perpendicular to both given lines. Quiz 6 problems. Sketch the domain of the following function: fx, y) = lnx y ) 4 x y. Describe the domain of the following function: x y gx, y, z) = z ln z Quiz 7 problems. Given fx, y, z) = x yz + y z 3 find f.. Find all points x, y) so that fx, y) = where fx, y) = x 6x + y y + xy On the surface z = x 3 3xy + y a marble is placed over the point, ). When released the marble initially move in the direction of steepest decrease. Find a vector pointing a, b pointing in the direction the marble will initially move. 4. Find the directional derivative for the function fx, y, z) = xz 3xy + xyz 3x + 5y 7 from the point, 6, 3) in the direction of the origin. 5. Let fx, y, z) = x y 3 z + 5z. Find a unit vector u in the direction in which f increases most rapidly at the point p =,, ), and find the rate of change of f in this direction. 6. Find an equation of the tangent plane to the surface x + xy + y + z = 6 at the point,, 3). 7. On the amazing can machine, which has the ability to alter the dimensions of cylindrical cans, the gauges currently read as follows: h = 5, dv dr dt = 6π, dt = dh, and dt =, where V, h, r are volume, height and radius. But the gauge for r is broken. Find the possible values) for r.

3 8. Given the implicitly relationship z + sin z = xy find z/ x y only in terms of z. 9. Suppose that fx, y) is differentiable and satisfies ft 3 t+, t ) = t 4 4t 3 +4t+6. Find f x, ) and f y, ).. A machine is being set up to make cones to be used for storage. The volume of a cone is V = 3 πr h where r is a radius and h is a height. The machine ideally makes the cones with r = 3 and h = 5 for a total volume of 5π units however there tend to be slight imperfections. Given that the volume of a cone must be within π units i.e., V ±π) and that the height of the cone as the machine is set up has h ±., then estimate the tolerance that we need to have for the radius i.e., find r). Quiz 8 problems. Find the second order Taylor polynomial for fx, y) = e x+y + x sin y at the point, ).. Find and classify the critical points for fx, y) = x 3 8xy + y 3x + 4y Find and classify all of the critical points for the function fx, y) = e y y x ). 4. Find the critical points for gx, y) = x 3 x y + 6xy + y x Let hx, y) = y sin x x. Verify that, ) is a critical point and classify the point as a maximum, minimum or saddle. 6. The Nook corporation maker of the finest Yops) has recently merged with the Jibboo conglomerate maker of the finest Zans). urrently Yops sell for three dollars each and Zans sell for nine dollars each. By combining their production the new company enjoys economy of scope and is now able to produce y Yops and z Zans at a cost of + y + 3 z3 yz dollars. Determine how many Yops and Zans respectively should be made in order to maximize profit. Also, verify that your answer is a maximum by using the second derivative test. 7. Use the technique of Lagrange multipliers to find the maximum value of fx, y) = xy+y given that 9x + y 4 = You have recently been hired to paint three large non-overlapping dots a red dot, a blue dot, and a green dot) on the side of a building. They have left the design of the dots to you, the only instruction they gave is that the total area must be π m. The contract allows you to charge 3πr thousand dollars for painting a red circle of radius r, while for a blue circle you charge 4πr thousand and for a green circle you charge 5πr thousand. What radii should you choose for the various circles to maximize how much you charge? Quiz 9 problems. Approximate the volume of the solid above the region R in the xy-plane consisting of the points x 5 and y 5, and below the surface fx, y) = x, by splitting R into four equally + y sized pieces and using the center point of each to approximate the height of each piece.. Let [a,b] [c,d] fx, y) da denote the integral of fx, y) over the region with a x b and c y d. Find [,3] [,] fx, y) da given the following: [,] [,] fx, y) da = 4, [,] [,] fx, y) da =, [,3] [,] fx, y) da = 3, fx, y) da = 7. [,3] [,] 3. Let R be the region with x and y 3, let R be the region with x 5 and y 3, and let R be the union of the two regions. Given that R gx, y) da = 8 and R gx, y) da = 3 find ) R gx, y) + da. 4. Find 5. Find 37 y dy dx. + x xe y 3 e y3 ) dy dx. 6. Find da where R is the triangular region with vertices at, ),, ) and, R + x ). 7. Let R be the bounded region between y = x and y = x. Write R fx, y) da as an iterated integral in both ways, i.e., dx dy and dy dx. 8. By changing the order of integration write the following as a single integral 3+3x fx, y) dy dx x fx, y) dy dx fx, y) dy dx.

4 9. Find. Find 8 4 y/ x x sinx ) dx dy. dy dx. x + y Quiz problems. Find the volume of the set of points z x + y and z x + y ).. Find the volume of the solid consisting of points above the xy-plane, below z = x + 3y and satisfying x + y Find the surface area of the cone z = x + y above the region R given by x 37 and y. 4. Set up but do not evaluate) an integral for the surface area for fx, y) = x y y over the region x 4 and y Evaluate S xy da where S is the region in the first quadrant inside x + y = 9 and outside x + y = Find the volume of the solid region consisting of points x z 4 y. 7. Let S be the solid region consisting of points x z 4 y. Find S sinx 37 )e z y 3 arctan37z) ) dv. 8. Find x x 4 x y e z dz dy dx. 9. onsider the following integral: 6 4 /3)x /3)x /)y fx, y, z) dz dy dx hange the order of integration to dx dy dz. Quiz problems. The Archimedean spiral is given by r = θ. Let R be the region between the origin and the Archimedean spiral for θ π. Given that the density is thrice the distance to the origin, find the mass of the region.. Let R be the region x and x y. Given that δx, y) = xy find x, y). 3. Find the mass and center of mass of the region R consisting of the triangle with vertices, ),, ) and, ) with δx, y) = y. 4. onvert the integral π 3r r sin θ dz dr dθ from cylindrical coordinates to cartesian coordinates. 5. onvert the integral π 3r r sin θ dz dr dθ from cylindrical coordinates to spherical coordinates. 6. Set up, but do not evaluate, an integral to calculate the volume of the region inside the surfaces x +y +z = 9, below the surface z = x + y and above the plane z =. 7. The Archimedean Spiral is the curve r = θ. For the homogenous volume i.e., density is constant) bounded below by the xy-plane, bounded above by the surface z = x + y and over the region from the origin to the Archimedean Spiral for θ π find z. 8. Let R be the region in the first quadrant bounded by the curves x = y, x = y 4, x = 9 y and x = 6 y. Making the substitutions u = x y and v = x + y describe this region in terms of u and v and determine Ju, v). 9. Evaluate the integral 4 y y ye x+y dx dy by making the change of variables u = x + y and v = y. Find x x x 6x cos x + y) 3) dy dx by making the substitutions u = x +y and v = x. Quiz problems. Find I z for the region between the origin and the Archimedian spiral, r = θ, where θ π.. Find I z where δx, y) = x + y for the region x and y. 3. Let R be the set of points with x and y and δx, y) = x. Find the inertia when R is rotated around the line x = y. The distance between a, b) and the line x = y is a b / ). 4. Find the mass of the cube x, y, z given that δx, y, z) = xy + xz + yz. 5. Evaluate the integral x y) sinx + y) dx dy over the tilted square region with corners at, ), π/, π/), π, ) and π/, π/) by doing a change of variables u = x + y and v = x y.

5 6. Given a wire which is bent in the shape of a helix sin3t), cos3t), 4t) for t π and where the density of the wire is δx, y, z) = z, find the mass of the wire. 7. Evaluate x3 +y) ds where is the curve given by x = 3t and y = t 3 for t. 8. Let = sin t, cos t for t π. Find x dx + x y dy. Quiz 3 problems. Given F = y sin x, x + z, xy + 37, find div F.. Given F = y sin x, x + z, xy + 37, find curl F. 3. Find the work done by moving a particle along the curve given that F = x 3 y 3, xy and the curve is x = t and y = t 3 for t. 4. Determine if the following vector valued function is conservative. If so find an f so that F = f. F = 3x y + e x, x 3 y + sin y 5. Show the following vector valued function is conservative and find an f so that F = f. F = xyz + 3x z + e x, x z + z, x y + x z 6. Evaluate the line integral yz dx + xz dy + xy + z) dz where is cos t, sin t, t) for t π. 7. Evaluate the line integral y + x) dx + x y) dy where is the curve y = x 37 with x. 8. Let be the curve that travels on the unit circle counterclockwise from, ) to, ). alculate xe y cosx ) ) dx + e y sinx ) + x ) dy. 9. Evaluate the line integral xy dx + x dy where is the curve on the cardiod r = + cos θ traveled counterclockwise.. Let R be the annulus consisting x + y 4 and let be the boundaries of this region oriented so that the region is always on the left while traveled along. Find x dy. Quiz 4 problems. Let G be the surface corresponding to the portion of the sphere x + y + z ) = 4 below the plane z =. Find xy y, xz + x y, x + y T ds G where the orientation on the boundary is clockwise when viewed from above.. alculate curlf) n dsa) G where F = yzi + 4xzj + xzk and G is the part of the sphere x + y + z = 9 below the plane z =, and n is the outward pointing normal. 3. Let G be the surface z = x y intersecting the solid) cylinder x y. Find x, x + y 3, x + y z T ds G where G is oriented counter-clockwise when viewed from above. 4. Let S be the solid cube with one corner at,, ) and the opposite corner at,, ). Given that find F = x 3 y, sinz ) yz, z + 37x 6 F n dsa). 5. Let S be the solid cone consisting of the points below the plane z = 4 and above z = x + y. Find F n dsa), where F = xy + e y+cos y, x y + sin z, z + cos x 6. Let S be the solid consisting of the set of points satisfying x + y 4 and z 3. Find e y xz, z 4 y sine x ), z +3z ln+x ) 7. [Mystery problem] n dsa)