DOC PREVIEW
MIT 18 01 - Practice Midterm Questions - 18.02A

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

18.02a Practice Midterm Questions, Fall 2007Problems 1-5 cover material from the first unit. This will be on the midterm,but won’t be emphasized.Problems 1-5 take about 1-1.5 hours, problems 6-19 take 2-3 hours.The actual test will be shorter –designed to take 2 hours, with simpler arithmetic.Problem 1. Consider the point P = (20, 0, 0), the plane P : x + 2y + 3z = 6, and thepoint Q = (1, 1, 1) on P.a) Compute the distance from P to P.b) Give parametric equations for the line through P and perpendicular to P.c) Find the point of intersection between P and the line of part(b). For later reference,call this point R.d) Find the angle, ∠P QR.e) By computing |P R| directly, verify your answer to part (a).f) Find the area of the triangle with vertices P, Q and R.Problem 2. Suppose tape is unwound from a roll in such a way that it is always vertical.Assuming the roll is centered at the origin and has radius 2, and the end of the tape startsat the point (2, 0), give parametric equations for the path traced out by the end of the roll.For what values of your parameter does this make sense?Problem 3. The motion of a point P is given parametrically by−−→OP = r(t) = h4 sin t, 5 cos t, 3 sin ti.a) Find v,dsdt, T, κ.b) Show r is perpendicular to 3i − 4k. What information about the motion of the point Pdoes this give ?Problem 4. Let (sin t, cos 2t) be a parametrized path of a point P in the plane. Give thex-y equation of this path. Sketch and describe how P moves over time.Problem 5. Let Ac=1 0 13 2 11 1 c.a) Let B =143 1 −2−5 1 21 −1 2.Show B is the inverse of A2(the subscript indicates c = 2). Show your arithmetic carefully.b) Use part (a) to solve x + z = 1, 3x + 2y + z = 0, x + y + 2z = 4.c) For what c will the system of equations Acx = 0 have a non-zero solution?d) For the value of c found in part (c) find a non-zero solution to the system.e) Compute A−11.(continued)118.02a Practice Midterm Questions, Fall 2007 2Problem 6.a) Find the normal to the level surface x3+ y3z = 3 at the point (1, 1, 2).b) Use level surfaces to show the perp e ndicular to the graph of z = f(x, y) is given byh∂f∂x,∂f∂y, −1i. (You can use what you know about gradients and level surfaces.)Problem 7. Graph the surface and level curves of z = y2− x.Problem 8. Let w = f(x, y) and r(t) = x(t)i + y(t)j. Along the path given by r we havew = f(x(t), y(t)).a) Assuming everything is differentiable, showdwdt= ∇w ·drdt.b) Suppose the path is along a level curve of f. Show that ∇w is perp endicular to the levelcurve.Problem 9. Suppose w = f(xy) (i.e. w = f(u) with u = xy). This implies x∂w∂x−y∂w∂y= 0.a) Verify this for the function w = sin(xy).b) Show this in general.Problem 10. Supp ose w = x2+ y2+ z2and x, y, z are related by x = f(y, z).a) Suppose f(y, z) = yz., find∂w∂zx. Do this twice, once with the chain rule and once withdifferentials. Do not do it by solving for y in terms of z (except to check your answer).b) For arbitrary f, Find∂w∂zxin terms of the formal partials of f(y, z). (Again, do thistwice, once with the chain rule and once with differentials.)Problem 11.a) Starting from the basic chain rule:∂w∂u=∂w∂x∂x∂u+∂w∂y∂y∂u, derive the matrix equationrelating∂w∂u,∂w∂vand∂w∂x,∂w∂yb) Write down the matrix equation in the c ase where (x, y) is as usual and (u, v) = (r, θ)(polar coordinates).c) Same question for (u, v) = (x, θ ).Problem 12. Let w = x3y + x/y and P = (2, 1).a) Compute the gradient ∇w|P.b) Compute the directional derivative,dwdsat P in the direction of i + 3j.c) Find a direction at P in which w is not changing.d) Estimate the value of w at (2.1, 0.9).Problem 13. Give the equation for the tangent plane to the surface x2+ xy2+ yz2= 23at the point (1, 2, 3).(continued)18.02a Practice Midterm Questions, Fall 2007 3Problem 14. Suppose you have an open box of volume 4 with dimensions x, y, z. So weall use the same notation, assume the open end is one of the sides with dimensions y and z.a) By substituting for z write down the unconstrained equation for the surface area of thebox.b) Use part (a) to find the dimensions that minimize the area.c) Use the second derivative test to verify your answer to part (b) is a minimumd) In part (a) x and y can be anywhere in a region R. Describe R. What is its boundary?e) Why can’t the minimum area occur on the boundary?f) Redo the minimization using Lagrange multipliers.Problem 15. Find the critical points of x2−2xy2+2y2. Classify them as minima, maximaor saddle points.Problem 16. Evaluate by reversing the limits of integration:Z10Z1x13ey4dy dx.Problem 17. Let R be a circular region of radius a and uniform density. Set up (but donot evaluate) iterated integrals in polar coordinates for the following moments of inertia.You can make things easier by carefully choosing where to put R for each problem.a) Moment of inertia of R about its center.b) Moment of inertia of R about a point on its edge.c) Moment of inertia of R about a diameter.d) Moment of inertia of R about a line tangent to the circle.e) Le t S be the region in the first quadrant, bounded below by the x-axis, above by y = xand on the right by the circle of radius 1 with center at (1, 0) (NOTE CENTER). Assumeuniform density and write down (but don’t evaluate) an integral in polar coordinates forthe polar moment of inertia (i.e. moment of inertia about the origin) of S.Problem 18. Find the mass of the planar region inside the cardioid r = a(1 + cos θ) withdensity given by δ(r, θ) = 1/r.Problem 19. EvaluateR RR(2x − 3y)2(x + y)2dx dy, where R is the triangle bounded bythe positive x-axis, negative y-axis and the line y =23x −43, by making a change of variableu = x + y, v = 2x −


View Full Document

MIT 18 01 - Practice Midterm Questions - 18.02A

Documents in this Course
Graphing

Graphing

46 pages

Exam 2

Exam 2

3 pages

Load more
Download Practice Midterm Questions - 18.02A
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Practice Midterm Questions - 18.02A and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Practice Midterm Questions - 18.02A 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?