Calculus Answers

Find the critical points of the following function, and classify it as degener-
ate or non-degenerate. If the latter, further classify it as local maxima, local
minima, or as saddle points.
f(x, y) = x4 + y4 - 4xy + 1.

Find the critical points and determine maxima or minima subject to the following constraint.
f(x,y,z) = 8x - 4z.
x^2+ 10y^2 + z^2 = 5.

Let D={(x,y,z)∈ℝ3:x2+y2≤a2 and 0≤z≤b}
where a
and b
are positive constants. The answer to ∭D2zdV
is

The vector field F(x,y,z)=⟨2ze^(xy),xy^2z,sin(x+y)cos(z)⟩

is continuous on
Select one:
a. only the point (0,0)
.
b. all of R^2
.

c. all of R^3
.
d. all of R
.

As an engineer, you have been tasked to design a cooling tower int he shape of a hyperboloid of one sheet.the horizontal cross sections of the cooling tower are circular with 10m. the cooling tower is 40m tall with maximum cross-sectional radius of 15m. A) Construct a mathematical equation for this cooling tower. B) If x=a cos(u)cosh(v), y=b sin(u)cosh(v) and z=c sin h(v), show that (x,y,z) lies on your equation in Q1(A). C) A colleague wants to construct the cooling tower using a hyperbolic cylinder, give reasons for your result in Q1(A) as the best model for the design of cooling tower.

2. Suppose that a fluid has density p (function of space and time) and velocity v with no sources or sinks. (A) Show that the rate of change of the mass m of the fluid contained in a region Q is dm d! du. -SITE (B) Suppose further that, if the fluid crosses the boundary, show that dt -- S6w) - neds. (©) Using your results in Q2(A) and 02(B), show that де V. (e) + at = 0. (D) Why the contimity equation for water is given by V. (pv) = 0.

An equilateral triangle with side length 8cm has its base on the x-axis and top vertex on the y-axis. A rectangle is inscribed in the triangle. Determine the largest area of the rectangle.

Two numbers have a product of 20. Determine the 2 numbers (exactly) such that the sum of one number and 3 times the other number is a minimum.

A spring of natural length 10 inches stretches 1.5 inches under a
weight of 8 pounds. Find the work done in stretching the spring.
(a) from its natural length to a length of 14 inches
(b) from a length of 11 inches to a length of 13 inches.

Let f be a continuous function on the interval [a,b], and let c = a+b 2 . Suppose that F is any antiderivative of f with F(a) = 3 and F(b) = 7.
Find: (a)Rb a f(x)dx (2 marks) (b) −Ra b f(x)dx (2 marks) (c)Rc a f(x)dx +Rb c f(x)dx (2 marks) (d) Dx[−(Rc a f(x)dx +Rb c f(x)dx)] (2 marks) 3. Find the area bounded by the curve y = |x−2|, the x-axis and the lines x = −7 and x = 11. (5 marks)