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Practice Problems
Math 2500 April 15, 2008
These problems are not in any particular order. The exam will be shorter. The last 8 problems form the final exam from the last time I taught this course.
- For each of the following pairs of vectors, compute ~u · ~v, ~u × ~v and the orthogonal projection of ~u onto ~v. (a) ~u = (1, 0 , 1), ~v = (1, 1 , 0) (b) ~u = (0, 1 , 0), ~v = (0, 1 , −1)
- Let Π 1 be the plane through (1, 1 , 1) with normal vector n~ 1 = (1, − 1 , 0), and let Π 2 be the plane defined by x + y + z = 1. (a) Write down a linear equation for Π 1. (b) Find the cosine angle between Π 1 and Π 2. (c) Parameterize the line l of intersection between Π 1 and Π 2. (d) Find the distance between Π 2 and (1, 1 , 1).
- Consider the curve c(t) = (cos(2t), sin(t)) for 0 ≤ t ≤ 2 π. (a) Sketch this curve. (b) Write down the tangent line to c at the point (− 1 , 1). (This point corresponds to the parameter value t = π/2). (c) Is the tangent line to this curve ever parallel to the line y = −x? Be sure to explain your answer. (d) Set up, but do not evaluate, the integral to compute the arclenth of this curve.
- Consider the function f (x, y) =
∫ (^) ex y
1 + t^2 dt.
(a) Compute the partial derivatives of f. (b) Does f have any critical points? Be sure to explain your answer.
- Consider the function f (x, y) = x^2 y − xy^3. (a) Does f have an upper or lower bound? Explain your answer. (b) Compute the partial derivatives of f. (c) Compute the directional derivative of f in the (1/
- direction, at the point (1, 1). (d) Find the direction of steepest ascent for f , starting at (1, 1). Make sure to write down a unit vector. (e) Notice that f (1, 1) = 0. Write down the equation of the tangent line to the f = 0 level set, at the point (1, 1).
- Consider the function f (x, y) = exy (^3) −x 2 . (a) Find and classify all the critical points of f. (b) Find the absolute minimum of f restricted to the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
- Recall that two tangent directions to the graph of f are given by the vectors
(1, 0 , ∂f∂x ), (0, 1 , ∂f∂y ).
(a) Compute a normal vector for the graph. (b) Can the tangent plane of the graph ever be parallel to the x − z plane? Explain your answer.
- Find the absolute maximum and minimum of the function f = xy on the ellipse 4x^2 + y^2 = 4.
- Evaluate
D
4 − x^2 − y^2 dA where D = {(x, y) | 1 ≤ x^2 + y^2 ≤ 4 , y > 0.
- Set up, but do not evaluate
D xex
(^2) y (^) dA, where D is the region bounded by the curves y = x (^3) and y = x. (Be careful of signs.)
- Evaluate
D
1 − x^2 dA where D is the triangle with vertices (0, 0), (1, 0), and (0, 1).
- Consider the domain D := {(x, y) | |x + y| ≤ 1 }.
(a) Sketch D. (b) If we change variables by u = √^12 (x + y), v = √^12 (−x + y), what is D in the u, v coordinate system? (c) Set up, but do not evaluate, the integral
D cos(πx^ −^ πy)dA^ in the (u, v) coordiante system.
- Consider the vector field F~ = (−y, x).
(a) Sketch F~. (b) Compute
γ F~^ ·^ d~s^ where^ γ(t) = (cos^ t,^ sin^ t) for 0^ ≤^ t^ ≤^2 π.
- Consider the vector field F~ = (2xex^2 +y^2 , 2 yex^2 +y^2 + cos y).
(a) Show that F~ = ∇f for some f. (b) Find a function f such that F~ = ∇f. (c) Compute
γ F~^ ·^ d~s, where^ γ(t) = (cos^ t,^ sin^ t) for 0^ ≤^ t^ ≤^ π/2.^ (Hint: you don’t need to actually do an integral.)
- Consider F~ = (−y + x^2 − y cos(xy), x − y^3 − x cos(xy)).
(a) Is F~ = ∇f for some f? Be sure to explain your answer. (b) Compute
γ F~^ ·^ d~s. (Hint: don’t actually compute the line integral.)
- Consider the vector field F~ = (z + x^3 − yzexyz^ , y − xzexyz^ , −x + z^2 − xyexyyz^ ).
(a) Compute ∇ · F~ and ∇ × F~. (b) Is F~ = ∇f for some f? Be sure to explain your answer. (c) Compute
Σ ∇ ×^ F~^ ·^ ~ndA, where Σ is the upper unit hemisphere, centered at (0,^0 ,^ 0), with the outward unit normal.
- Consider the surface ~r(u, v) = (u cos(v), u sin(v), v).
(a) Compute the tangent vectors ∂~r/∂u and ∂~r/∂v. (b) Verify that this is a good parameterization, by checking that ∂~r/∂u and ∂~r/∂v are never parallel. (c) Find the equation of the tangent plane to ~r for the parameter values u = 1, v = π. (d) Is the tangent plane ever parallel to the x − y plane? Be sure to explain your answer. (e) What is this surface? Can you draw a sketch of it? (Hint: fix a value of u, for instance u = 1 or u = 0, and draw the resulting curve.)
- Compute
Σ ∇ ×^ F~^ ·^ ~ndA, where^ F~^ = (−y, x,^ 0) and Σ is the upper unit hemisphere, centered at the origin, with the outward unit normal.
- Compute
Σ F~^ ·~ndA, where^ F~^ = (x^ +^ yz^ −^ cos^ y, y^ −^ exz^ +^ z^2 , z^ −^ x^ cos(x^2 y)) and Σ is the unit sphere (centered at the origin) with the outward unit normal.
- Compute
Σ F~^ ·~ndA, where^ F~^ = (x+zey
(^2) +z (^) , y −z cos(x+z (^2) ), z) and Σ is the upper unit hemisphere, centered at the origin, with the outward unit normal. (Hint: what is F~ restricted to the plane z = 0?)
(c) (3 points) Find the length of the section of c corresponding to 0 ≤ t ≤ 2 π.
- Consider the function f (x, y) = x^3 − x^2 y + 3xy. (a) (5 points) Verify that the only critical points of this function are (0, 0), (3, 9). (b) (5 points) Classify these critical points as local minima, local maxima, saddle points, or none of the above. (c) (5 points) Observe that f (1, 1) = 3. Write down an equation for the tangent line to the f = 3 level set of at (1, 1).
- Consider the composition g(t) = f (x(t), y(t)), where x(1) = 1, y(1) = 2, x′(1) = −1, y′(1) = 1 and ∇f (1, 2) = (3, 2). (a) (4 points) What is g′(1)? (b) (3 points) Suppose x′(2) = 0 and y′(2) = 0. Is it necessarily true that g′(2) = 0? Be sure to explain your answer. (c) (3 points) Suppose g′(0) = 0. Is is necessarily true that x′(0) = 0 and y′(0) = 0? Be sure to explain your answer.
- Below is a sketch of some of the level curves of a function f = f (x, y).
(!1,2)
x
y
f=
f=
f=! 1 f=! 2 f=! 3
(!1,0)
f=
f=3 f=
f=
f=
(a) (5 points) In which direction does ∇f (− 1 , 0) point? Be sure to explain your answer. (b) (5 points) If (− 1 , 2) is a critical point of f , what kind of critical point do you expect it to be? Be sure to explain your answer.
- (5 points) Set up, but do not evaluate, the integral
D x^ ln(1 +^ x^2 +^ y^2 )dA^ as an iterated integral, where^ D is the domain bounded by x = y^2 − 2 y and y = x − 1. (It might help to draw a picture.)
- (10 points) Evaluate
D
x^2 + y^2 − 1 dA, where D = {(x, y) | 1 ≤ x^2 + y^2 ≤ 4 }.
- Consider the vector field F~ = (yexy^ + 2x, xexy^ ) in the plane. (a) (5 points) Verify that F~ = ∇f for some function f = f (x, y). (b) (5 points) Find a function f such that F~ = ∇f. (c) (5 points) Evaluate
γ F~^ ·^ d~s, where^ γ(t) = (cos^ t,^ sin^ t) for 0^ ≤^ t^ ≤^ π/2. (Think before you compute.)
- Consider the vector field F~ = (z ln(1 + y^2 ) − x, 2 y − xexz^ , −1 + xy sin(1 + x^2 + y^2 )) in three–space. (a) (5 points) Compute the divergence of F~.
(b) (5 points) Evaluate
Σ F~^ ·^ ~ndA, where Σ is the unit sphere centered at the origin, oriented with the inward normal. (Think before you compute.)
- Consider the vector field F~ = (yz cos(xyz), xz cos(xyz), y + xy cos(xyz)) in three–space.
(a) (5 points) Compute the curl of F~. (b) (5 points) Is F~ the gradient of a function f (x, y, z)? Be sure to explain your answer. (c) (5 points) Evaluate
Σ ∇ ×^ F~^ ·^ ~ndA, where Σ is the upper unit hemisphere centered at the origin parammeterized by {(
1 − u^2 cos v,
1 − u^2 sin v, u) | 0 ≤ u ≤ 1 , 0 ≤ v ≤ 2 π}, oriented with the upward normal. (Hint: you don’t have to integrate over Σ, you can replace it with another surface which has the same boundary.)
- Consider the curve γ(t) = (t^3 , t^2 , t).
(a) (4 points) Find the equation of the tangent line to γ at t = 1. (b) (3 points) The the tangent line to γ ever parallel to the plane z = 1? Be sure to explain your answer. (c) (3 points) Set up, but do not evaluate the integral to find the length of the section of γ for 0 ≤ t ≤ 2.
- Consider the function
f (x, y) =
∫ (^) ey cos x
(1 + t^2 )dt.
(a) (5 points) Compute the partial derivative ∂f /∂x and ∂f /∂y. (b) (5 points) Does f have any critical points? Be sure to explain your answer.
- (10 points) Evaluate (^) ∫ ∫
D
ln(1 + x^2 + y^2 )dA, where D = { 1 ≤ x^2 + y^2 ≤ 4 , y ≤ 0 }. (Hint:
ln tdt = t ln t − t + const.)
- Consider the function f (x, y) = xy − x + ey−^1. (a) (5 points) Show that (− 1 , 1) is the only critical point of f. (b) (5 points) Classify the critical point as a local minimum, local maximum, or saddle point. (c) (5 points) Find the directional derivative of f in the ~u = (1/ 2 ,
3 /2) direction at the point (x 0 , y 0 ) = (1, 2). (d) (5 points) Observe that f (1, 2) = 1 + e. Find the equation of the tangent line of the level set {f = 1 + e} at the point (1, 2).
- Suppose f = f (x, y), while both x and y are functions of t. Let g be the composition g(t) = f (x(t), y(t)).
(a) (4 points) If x(0) = 1, x′(0) = −1, y(0) = 2, y′(0) = 4, and ∇f (1, 2) = (− 2 , 2), find g′(0). (b) (3 points) If x′(1) = 0y′(0), must it be true that g′(1) = 0? Be sure to explain your answer. (c) (3 points) If g′(−1) = 0, must it be true that x′(−1) = 0 and y′(−1) = 0? Be sure to explain your answer.
- Consider the vector field F~ = (F 1 , F 2 ) = (y + yexy^ , xexy^ ) on the plane.
(a) (5 points) Is F~ the gradient of a function? Be sure to explain your answer. (b) (5 points) Compute the path integral
γ F~^ ·^ d~s, where^ γ(t) = (2 cos^ t,^ sin^ t) for 0^ ≤^ t^ ≤^2 π. (Hint: the area of an ellipse with semi-major axis a and semi-minor axis b is πab.)
- Consider the vector field F~ = (− 2 x + y^2 ln(1 + z^2 ), zex^2 +2xz^ , 2 z − xy cos(x^2 − y^2 )).
(a) (5 points) Compute the divergence ∇ · F~. (b) (5 points) Evaluate the flux
Σ F~^ ·~ndA, where Σ is the boundary of the solid cylinder^ {x^2 +^ y^2 ≤^1 ,^ −^1 ≤ z ≤ 1 }, oriented with the outward normal ~n. (c) (5 points) Evaluate the flux
S F~^ ·^ ~ndA, where^ S^ is the hemisphere^ {x^2 +^ y^2 +^ z^2 = 1, z^ ≥^0 }, oriented with the upward normal ~n. (Hint: you may want to use a theorem to avoid doing the flux integral, but you would need to “supplement” S with another surface. Choose the supplemental surface to fit the problem. It may also help to draw a picture.)
