Math 21a (F99) Review Problems• The problems given below sample the material from the course which you will be responsiblefor on the final examination.• The level of difficulty of these problems should roughly correspond to the average level ofdifficulty of those which will appear on the exam. Of course, there may be problems on thefinal exam which are somewhat longer or more involved.• Since electronic aids will not be allowed in the exam room, use these aids in your review only asa last resort.• The answers to the problems below appear at the end.• Problems 1-35 are relevant for all sections of Math 21a.• Problems 36-50 are not relevant for students in the BioChem sections. These problems areonly for those in the Regular and Physics sections.• Problems 50-62 are meant for the students in the BioChem sections and are not relevant forthose in the Regular or Physics sections.• Students can obtain additional answered review problems by working the relevant oddnumbered problems from Chapters 9.4-9.9, 10.1-10.5, 10.7, 11.1, 11.3, 12.1-12.10, 13.1-13.6,14.1-14.8 and at the ends of Chapters 9-14 in Part II of the 9’th edition of Thomas andFinney’s book Calculus. The latter is on reserve in Cabot Library. Moreover, almost any bookon multi-variable calculus will cover essentially the same subjects as we did here. Thus, evenmore problems for review can be had by working answered problems in other multivariablecalculus books. (Please don’t take such books out of Cabot; zerox some problems instead sothat other students can have access to the same resource.) BioChem section students who wishto work additional problems in Probability and Statistics should work more of the answeredproblems from Rosner’s book.PROBLEMS:1. Give an equation of the form f(x, y) = 0 for the following parametrized curves in R2: a) x = (t2 + 1)1/4, y = 1 - t. b) x = 2 tan(t), y = 1/cos(t) for -π/2 < t < π/2. c) x = 4 cos(t), y = 3 sin(t). 2. In each case, give an equation for the line in R2 which is tangent to the given curve at theindicated point: a) The curve is parametrized as x = (t2 + 1)1/4, y = t and the point is where t = 0. b) The curve is where x3 + y2 = - 23 and the point is (-3, 2). c) The curve is where x + y3 = 1 and the point is (2, -1).3. Find the length of the curve parametrized by x = e2t - 2t, y = 4 et for 0 ≤ t ≤ 1. 4. a) Find a parametrization of the form t → (x(t), y(t)) for the curve in R2 which is parameterized in polar coordinates by r = t, θ = t3 with t ≥ 0. b) Write this curve in the form f(x, y) = 0. 5. In each of the cases below, write the vector B as a sum of a vector which is parallel to the vectorA and which is perpendicular to A. a) A = (1, 2, 2) and B = (1, 2, -1). b) A = (3, -4, 0) and B = (5, 1, 1). c) A = (2, -1, -2) and B = (3, 3, 3). 6. Suppose that v and w are vectors in R3 with |v| = 2 and |w| = 3. In each case below |2v - w| isgiven. Decide whether v and w are perpendicular or not, or whether there is not enoughinformation to decide. a) |2v - w| = 12. b) |2v - w| = 7. c) |2v - w| = 5. 7. Find the distance from the point (1, 2, 1) to the following: a) The plane where 2 x + y - 2 z = 0. b) The line parameterized by t → (6 t, 3 t + 2, 2 t + 1). 8. In each case, find an equation of the form f(x, y, z) = 0 for the indicated plane: a) The plane through the point (1, 0, 0) which is normal to A = (2, 1, -1). b) The plane containing the points (1, 1, -1), (2, 1, 0) and (3, 3, 3). c) The plane through the point (-1, 0, 0) for which A = (-1, -1, 1) and B = (1, 1, 3) are tangent. 9. Find the absolute value of the cosine of the angle between the planes x = 5 and 6 x + 3 y + 2 z= 2. 10. Write a parametric equation for the line through the origin which is normal to the plane throughthe three points (0, 1, 0), (1, -1, 1) and (1, 1, -1). 11. In each of the cases below, the given vector function of the parameter t is meant to be thevelocity vector of a parametrized curve in R3. Decide whether the given curve lies entirely in asingle plane.a) v(t) = (5 cos(t), 3 sin(t), cos(t)). b) v(t) = (8 t2, 3 t, cos(t)). c) v(t) = (8 t2, cos(t), -7 cos(t)). 12. In each case, find the linear approximations to the given function at the indicated points: a) f(x, y, z) = 10 x2 + y z - z2 + 1; and the points are (1, 1, 1) and (1, 1, -1). b) f(x, y, z) = sin(xyz2) + z; and the points are (1, 2, 0) and (3, 0, 1). 13. In each case, find the equation for the tangent plane to the given surface at the indicated points: a) The surface is where exyz - 2 + z = 0 and the points are (0, 0, 1) and (0, 1, 1). b) Thus surface is where x2 + y2 - xyz = 1 and the points are (1, 0, 1) and (1, 1, 1). 14. Write down the linear approximation at (1, 1, 1) for any function f(x, y, z) on R3 with followingproperties: a) f(1, 1, 1) = -2 b) Both A = (1, 1, 3) and B = (3, 1, -1) are tangent to the level set f = -2 at (1, 1, 1). c) The directional derivative of f at (1, 1, 1) in the direction (1, 0, 0) is 2. 15. Let f(x, y, z) = x2 - yz + 3. In each case below, the given point lies on a parametrized curve andthe given vector v is the tangent vector to that curve at the given point. Give the instantaneousrate of change of f along the curve at the given point. a) The point is (1, 1, 1) and v = (1, 0, 0). b) The point is (1, -1, 1) and v = (0, -1, 0). c) The point is (0, 2, 2) and v = (1, 1, 0). 16. Suppose that f(x, y) is a function on R2 whose gradient at the origin in (1, -3). Also, supposethat (x(u, v), y(u, v)) is a function from R2 to R2 which sends (0, 0) to (0, 0). Also, suppose thepartial derivatives of x(u, v) and y(u, v) at (0, 0) are: xu = 1, yu = 1, xv = -1, yv = 1. Give thegradient vector at (0, 0) of the function g(u, v) = f(x(u, v), y(u, v)). 17. In each …
View Full Document
Unlocking...