Question

Find the area of one loop of the polar curve r=4*sin(3*theta + Pi/3)

Let f(x,y) = 3x^2 + cos(Pi*y). a) f has a saddle point at (0,k) whenever k is an odd integer b) f has a saddle point at (0,k) whenever k is an even integer) c) f has a local maximum at (0,k) whenever k is an even integer d) f has a local minimum at (0,k) whenever k is an odd integer

Answer #1

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Let f(x,y) = 3x^2 + cos(Pi*y). a) f has a saddle point at (0,k)
whenever k is an odd integer b) f has a saddle point at (0,k)
whenever k is an even integer) c) f has a local maximum at (0,k)
whenever k is an even integer d) f has a local minimum at (0,k)
whenever k is an odd integer.

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

Let f(x,y) = 3x^2y − 2y^2 − 3x^2 − 8y + 2.
(i) Find the stationary points of f.
(ii) For each stationary point P found in (i), determine whether
f has a local maximum, a local minimum, or a saddle point at P.
Answer:
(i) (0, −2), (2, 1), (−2, 1)
(ii) (0, −2) loc. max, (± 2, 1) saddle points

(3)If H(x, y) = x^2 y^4 + x^4 y^2 + 3x^2 y^2 + 1, show that H(x,
y) ≥ 0 for all (x, y). Hint: find the minimum value of H.
(4) Let f(x, y) = (y − x^2 ) (y − 2x^2 ). Show that the origin
is a critical point for f which is a saddle point, even though on
any line through the origin, f has a local minimum at (0, 0)

1. Find the area between the curve f(x)=sin^3(x)cos^2(x) and y=0
from 0 ≤ x ≤ π
2. Find the surface area of the function f(x)=x^3/6 + 1/2x from
1≤ x ≤ 2 when rotated about the x-axis.

find the equation of the tangent lines at the point where the
curve crosses itself: x=cos^3(theta) y=sin^3(theta) y=t^2

Find the Critical point(s) of the function f(x, y) = x^2 + y^2 +
xy - 3x - 5. Then determine whether each critical point is a local
maximum, local minimum, or saddle point. Then find the value of the
function at the extreme(s).

f (x, y) =(x ^ 4)-8(x ^ 2) + 3(y ^ 2) - 6y
Find the local maximum, local minimum and saddle points of the
function.
Calculate the values of the function at these points

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

Let f(x, y) = −x 3 + y 2 . Show that (0, 0) is a saddle point.
Note that you cannot use the second derivative test for this
function. Hint: Find the curve of intersection of the graph of f
with the xz-plane.

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