Question

f(x, y) = 4 + x^3 + y^3 − 3xy

(a,b)=(0.5,0.5)

u = ( √ 1 /2 , − √ 1 /2 )

a) Calculate the rate of change of f along the curve r(t) = (t, t2 ), at t = −1

b)Classify the critical points of f using the second derivative test.

Answer #1

Consider
f(x, y) = (x ^2)y + 3xy − x(y^2)
and point P (1, 0).
Find the directional derivative of f at P in the direction of ⃗v
= 〈1, 1〉. Starting from P , in what direction does f have the
maximal rate of change? Calculate the maximal rate of change

Consider the equation: x^ 4 + 3xy^3 − y ^4 = 9x^2 y
a) find the points on the curve where x = 3
b) for each of the points found in pt. a find the slope of
tangent line at that point
c) for each of the points found in pt. a , write an equation for
the tangent line to the curve at that point.

The function
f ( x , y ) = x 3 + 27 x y 2 − 27 x
has partial derivatives given by
f x = 3 x 2 + 27 y 2 − 27,
f y = 54 x y,
f x x = 6 x,
f y y = 54 x,
f x y = 54 y, and
f y x = 54 y,
AND has as a critical point. (You need NOT check
this.)
Use the second...

Consider the following.
f(x, y) = x/y, P(4,
1), u =
3
5
i +
4
5
j
(a) Find the gradient of f.
(b) Evaluate the gradient at the point P.
(c) Find the rate of change of f at P in the
direction of the vector u.

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

find the total differential
(a) f(x,y)=x^2+3xy+2y
(b) f(x,y)=x-y/x+1

Given f(x,y) = x2−3y2−8x+9y+3xy for each and any
point that is critical, use the second-partial-derivative test to
determine whether the point is a relative maximum, relative
minimum, or a saddle point.

Can you find a functionv (x,y) so that u+iv is entire with
u(x,y) =x^3+ 3xy^2 ? (cauchy- riemann equations---hint)

consider the 2 variable function f(x,y) = 4 - x^2 - y^2 - 2x -
2y + xy
a.) find the x,y location of all critical points of f(x,y)
b.) classify each of the critical points found in part a.)

10. (6pts.) Show that the derivative of f(x) = 1 + 8x^ 2
is f ‘(x) = 16x by using the definition of the derivative as the
limit of a difference quotient.
11. (5pts.) If the area A = s^ 2 of an expanding square
is increasing at the constant rate of 4 square inches per second,
how fast is the length s of the sides increasing when the area is
16 square inches?
12. (5pts.) Find the intervals where...

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