Calculus

MATH 273 - Exam 2 Review

1. For the following problems find a ∈ R such that f(x,y) is continuous or state that no such a exists. Justify your answer.

i.

f(x,y) =

{ a x = y = 0 (x+y)2

x2+y2 otherwise

ii.

f(x,y) =

{ a x = y = 0 (x+y)5+(x2+y2)2

x4+2x2y2+y4 otherwise

2. Let f(x,y,z) = x2 + y2 + z2, g(x,y) = x2/2 + y2/2, and ~r(t) =< t2, t, 1 >.

i. Find a point where the surfaces f(x,y,z) = 1 and z = g(x,y) intersect.

ii. Find two planes passing through this point. One tangent to f(x,y,z) = 1 and one tangent to z = g(x,y).

iii. Find a line passing through this point which is tangent to both surfaces.

iv. Use the chain rule to find ∂ ∂t f(~r(t)) in terms of t. (Note that there is a slight abuse

of notation here. If ~r(t0) =< a,b,c > at some t0 then take f(~r(t0)) = f(a,b,c).)

3. Let f(x,y,z) = ln(1 + x + ey + e2z). Find the linear approximation to f at the point (0, 0, 0).

4. Find the maximum and minimum of f(x,y) = (x−1)2+y2 over S = {(x,y) : x2+y2 ≤ 4}.

5. Sketch the following sets in R2 and set up double integrals to find their areas.

i. D = {(x,y) : 1 ≤ x2 + y2 ≤ 4, 0 ≤ y ≤ x} ii. D = {(x,y) : 1 ≤ x2 + y2 ≤ 4, 0 ≤ y ≤ 1}

iii. D = {(x,y) : x2 + y2 ≤ 4, (x− 2)2 + y2 ≤ 4} iv. D = {(r,θ) : 0 ≤ θ ≤ π, r ≤ θ} v. D = {(r,θ) : 0 ≤ θ ≤ π

2 , r(sin θ + cos θ) ≤ 1}

6. Set up triple integrals to find the volumes of the following sets in R3. Do this twice for each set in two different coordinate systems. In any case the outermost integral should have constant upper and lower bounds of integration. Sketch the intersection of your domain with the surface given by setting your outermost variable to be a constant. (For example, if we are in cylindrical coordinates and our outermost integral has bounds 0 ≤ θ ≤ π, we intersect D with the half-plane θ = θ0 for some general fixed value of θ0 ∈ (0,π)).

i. D = {(x,y,z) : x2 + y2 + z2 ≤ 9}

ii. D = {(x,y,z) : 0 ≤ z ≤ x2 + y2 ≤ 9, x ≥ 0}

7. Let D be the region in R3 bounded by three circular cylinders with radius R > 0 which have mutually orthogonal axes. For example, we can take

D = {(x,y,z) : x2 + y2 ≤ R2, y2 + z2 ≤ R2, z2 + x2 ≤ R2}.

Show that the volume of D is |D| = 8R3(2− √

2). Hint: use cylindrical coordinates and symmetry.