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a) Graph π(π₯) = π₯ π ππ(1βπ₯) to estimate lim π(π₯), zooming in on the origin as necessary π₯β0
(b) Confirm your estimate in part (a) with a proof.
Graph y = tan x and y = cot x together for β7 β€ x β€ 7. Comment on behavior of cot x in relations to the signs and values of tan x
Suppose that the speed v (in ft/s) of a skydiver t seconds after leaping from a plane is given by the equation π£ = 190(1 β π β 0.168π‘).
(a) Graph π£ versus π‘.
(b) By evaluating an appropriate limit, show that the graph of π£ versus π‘ has a horizontal asymptote π£ = π for an appropriate constant π.
(c) What is the physical significance of the constant π in part (b)?
Let
Ζ(x) = π₯1/(1βx) .
Make tables of values of Ζ at values of x that approach x = 1 from above and below. Does Ζ(x)
appear to have a limit as x approaches 1? If so, what is it? If not, why not?
curve that bounds the surface. (b) Consider the surface
S =
(x, y, z) | z = x
2 + y
2
; z β€ 9
.
Sketch the surface S in R
3 and show the curve that bounds S on your sketch. Then use Stokesβ Theorem
to evaluate the flux integral
Z Z
S
(curl F)
Z
C
(xy + z) ds
where C is the line segment in R
3 with initial point (1, 0 β 1) and endpoint (2, 1, 1). [8]
9. Let D be the region in R
3
that lies inside the sphere x
2 + y
2 + z
2 = 2 and above the plane z = 1.
(a) Sketch the region D in R
3
. (3)
(b) Express the volume of D in terms of a triple integral, using spherical coordinates. DO NOT
EVALUATE THE INTEGRAL. (7)
[10]
10. Let S be that part of the surface z =
p
x
2 + y
2 that lies between the plane z = 1 and the plane z = 3.
(a) Sketch the surface S, together with its XY-projection. (3)
(b) Use a surface integral to determine the area of S
2 ΓΒ’ Β Β R function f defined by
f (x, y) = ΓΒ’ Β Β
xy
and the R ΓΒ’ Β Β R
2
function r defined by
r (t) = ΓΒ― ΒΏ ΒΎ
e
2t
, cost
.
Use the General Chain Rule to determine the value of (f ΓΒ’ Β Β¦ r)
0
(0). [7]
6. Consider the R
2 ΓΒ’ Β Β R function f defined by
f (x, y) = 2x + 3y
2 ΓΒ’ Β Β x
2 + y
3
.
Determine how many saddle points the function f has. (Make use of the Second Order Partial Derivatives
Test for Local Extrema.) [9]
7. Use the Method of Lagrange to determine the largest possible volume that a rectangular block with a square
base can have if the height of the block plus the perimeter of its base equals 30cm
3
. A parametric equation for L1 is
(x, y, z) = (1, 0, 2) + t(2, 4, β1); t β R.
L2 is parallel to L1 and contains the point (3, 4, 3). V is the plane that contains both the lines L1 and L2. Find
an equation for V . [7]
2. Prove from first principles that
lim
(x,y)β(2,2)
(x β y) = 0.
[8]
3. Prove that lim
(x,y)β(0,0)
x
2 + y
x β y
does not exist. [8]
4. Consider the R
2 β R function f defined by
f (x, y) = x
2 β 2xy + y
2
.
Let V be the plane that is tangent to the graph of f at the point (x, y, z) = (1, β1, 4).
(a) Find a vector that is perpendicular to the plane V . (4)
(b) Find an equation for the plane V .
A closed rectangular box whose length is double its width has a total surface area of 600 cmΒ². Find the dimensions of the box with maximum volume.
A bee flies on a trajectory such that its polar coordinate at time t are given by "r=bt\/T"2(2T-t) "\\theta=t\/T" (0<t<2T) where b and T are positive constants. Find the velocity vector of the bee at time t. Show that the least speed achieved by the bee is b/T. Find the acceleration of the bee at this instant.
