Math 223 Final Exam Review Packet Fall 2012: The Foll 565742

Math 223 Final Exam Review Packetfall 2012the Following Questions C

The following questions can be used as a review for Math 223. These questions are not actual samples of questions that will appear on the final exam, but they will provide additional practice for the material that will be covered on the final exam. When solving these problems keep the following in mind: Full credit for correct answers will only be awarded if all work is shown. Exact values must be given unless an approximation is required. Credit will not be given for an approximation when an exact value can be found by techniques covered in the course.

Paper For Above instruction

The review packet covers a broad range of topics relevant to Math 223, including applications of multivariable calculus, vector fields, surfaces, level sets, gradients, divergence theorem, curl, optimization, parametrization, and more. This comprehensive review is designed to reinforce understanding of key concepts necessary for success on the final exam.

Question 1: A sonic boom carpet is a region on the ground where the sonic boom is heard directly from the airplane and not as a reflection. The width of the carpet, W, can be expressed as a function of the ground temperature t, the vertical temperature gradient at the airplane’s altitude d, with W(t, d) = k·t·d for some positive constant k. Determine whether the width W increases or decreases with respect to t when d is fixed, and whether W increases or decreases with respect to d when t is fixed.

Question 2: Describe and sketch the following sets of points: (a) Points at a fixed distance of five units from the line L, where L is the intersection of the plane 3y = and the xy-plane. (b) Points at a fixed distance of three units from the yz-plane. (c) Points where the distance from the z-axis and the xy-plane are equal.

Question 3: Find the intersection of the plane with the surface 2 - 3x y z = in different scenarios: (a) setting one variable constant to obtain a parabola, (b) related to cosine curves representing waves, (c) line(s) in certain conditions.

Question 4: Given the function f(x, y) = y - x², (a) plot its level curves for specific constant z values, (b) analyze the surface height at a point, and (c) interpret how the height changes when moving parallel to the x- or y-axes.

Question 5: Describe the level surfaces of functions such as f(x, y, z) = x² + y² - z² and g(x, y, z) = e^{-(x² + y² + z²)}.

Question 6: Analyze temperature data modeled by level curves over time and depth setting, approximate times of sunrise, and graph the temperature variations at specific depths and times.

Question 7: Complete a table for a linear function using given points and derive its formula.

Question 8: For various planes, identify those parallel to axes, contain specific points, or are normal to given vectors. Determine tangent planes to surfaces where applicable.

Question 9: Find an equation for a linear function, the perpendicular plane, and the area of a specific triangular region based on a partially shown graph.

Question 10: Match functions to their appropriate graphical and contour representations based on formula characteristics.

Question 11: Work with vectors, find vectors parallel/perpendicular, compute angles, and components between vectors v and w.

Question 12: Analyze vectors u and v to determine perpendicularity and parallelism, find equations of planes, and parameterizations of lines.

Question 13: If v is a vector in the yz-plane with a specific angle to u, find v given the length of 12 units and the angle between u and v.

Question 14: Calculate partial derivatives and Laplacians for specific functions, relating them to physical interpretations and potential functions.

Question 15: Find tangent planes to given surface points, and model projectile motion to analyze physical parameters such as position, velocity, and tangent lines at specific times.

Question 16: Set up the differential ds and linear approximation of the travel distance s(v, α), interpreting the sign of changes and small variations in initial speed and angle.

Question 17: Use the given height function of the lake’s depth, analyze whether the boat moves into deeper or shallower water, and determine the direction along which the depth remains constant and the rate of change along a given path.

Question 18: Calculate gradient, curl, divergence, the greatest rate of change, and potential functions for specific vector fields in various coordinate systems.

Question 19: Maximize the flux of a radial vector field through a sphere, find the point of maximum flux, and its value.

Question 20: Set up integrals to find volumes bounded by spheres, cones, and other surfaces in Cartesian, cylindrical, and spherical coordinates.

Question 21: Sketch parametric equations of curves such as circles, lines, and intersections of surfaces, as well as describe the motion of a child sliding down a helical slide at specific times and conditions.

Question 22: Find parametric equations for various curves, including circles, lines, and intersections, based on given conditions and parameterizations.

Question 23: Analyze a scalar field for critical points, determine their type, and classify according to the second derivative test.

Question 24: Verify the nature of critical points of a quadratic form, find conditions on parameters for saddle points, minima, maxima, and compute the minimum distance from a surface to the origin.

Question 25: Find equations for surfaces in different coordinate systems and evaluate whether certain integrals are positive, negative, or zero over specified regions.

Question 26: Calculate multiple integrals over specified regions, including volumes between surfaces and in regions like tetrahedra and cylinders.

Question 27: Set up integrals for volume calculation, including specific surfaces and regions, in Cartesian, cylindrical, and spherical coordinates.

Question 28: Write integrals for the mass of a dirt pile with density proportional to height, and compute volume using given bounds and densities.

Question 29: Provide parametric equations for various curves including circles, lines, and intersections of surfaces, along with slopes, velocities, and tangent lines at specified points and times.

Question 30: Set up the flux of water through a paraboloid-shaped net submerged in the ocean, applying the Divergence Theorem and specific flow velocity functions.

Question 31: Analyze vector fields to determine positivity of divergence, curl, or other properties at points, and compare fluxes from spheres of different radii.

Question 32: In a region between circles, analyze a curl field for circulation around specified curves, given their orientations and the field’s properties.

Question 33: Find the circulation density and flux density of a given vector field at specified points, based on the field’s formula and properties.

Question 34: Determine whether the quantities are vectors, scalars, or undefined, based on typical differential operators, functions, and field behaviors, with assumptions about the given fields and regions.

Question 35: Using a contour diagram, estimate the gradient, critical points, and crossings of the function, and evaluate circulation integrals and surface integrals relating to the function’s behavior.

This comprehensive review covers key concepts essential for mastering multivariable calculus topics including vector calculus, surface integrals, optimization, parameterization, physical applications, coordinate transformations, and differential operators. Practice these problems thoroughly to prepare effectively for the Math 223 final exam.

References

• Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
• Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendental Functions. John Wiley & Sons.
• Thomas, G. B., & Finney, R. L. (2004). Calculus and Analytic Geometry. Pearson.
• Murphy, B. (2017). Vector Calculus. OpenStax.
• Lay, D. C. (2011). Linear Algebra and Its Applications. Addison Wesley.
• Marion, J. B. (2013). Classical Dynamics of Particles and Systems. Cengage Learning.
• Ross, K. A. (2004). Differential Equations: An Introduction. Springer.
• Shifrin, R. (2014). Multivariable Mathematics. Empresa Editorial, S.A.
• Fitzpatrick, R. (2018). Advanced Calculus. American Mathematical Society.
• Weinstock, R. (2011). Calculus of Several Variables. Dover Publications.