Question

Consider a function F(x,y)=(1/y,x^2) find a subset that is regular (meaning it is 1-1, and continiously differentiable and JF(X,Y) is not equal to 0

Answer #1

Consider the function f(x,y)=y+sin(x/y)
a) Find the equation of the tangent plane to the graph offat the
point(1,3)
b) Find the linearization of the function f at the point(1;3)and
use it to approximate f(0:9;3:1).
c) Explain why f is differentiable at the point(1;3)
d)Find the differential of f
e) If (x,y) changes from (1,3) to (0.9,3.1), compare the values
of ‘change in f’ and df

Consider the following function.
f (x, y) = [(y +
2) ln x] − xe7y −
x(y − 5)7
(a)
Find fx(1, 0) .
(b)
Find fy(1, 0) .

Consider a function f(x; y) =
2x2y
x4 + y2 .
(a) Find lim
(x;y)!(1;1)
f(x; y).
(b) Find an equation of the level curve to f(x; y) that passes
through the point (1; 1).
(c) Show that f(x; y) has no limits as (x; y) approaches (0;
0).

Consider the function f(x,y) = xe^((x^2)-(y^2))
(a) Find f(1,−1), fx(1,−1), fy(1,−1). Use these values to find a
linear approximation for f (1.1, −0.9).
(b) Find fxx(1, −1), fxy(1, −1), fyy(1, −1). Use these values to
find a quadratic approximation for f(1.1,−0.9).

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

1). Consider the following function and point.
f(x) = x3 + x + 3; (−2,
−7)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =
2) Consider the following function and point. See Example
10.
f(x) = (5x + 1)2; (0, 1)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =

Consider the function below. y=f(x)= x/x^2+x+1
Find all critical numbers of (f), if any.
Find interval(s) on which f is decreasing
Final all local maximum/minimum points of f.

Consider the function f(x,y) = ( x2 +
z2)ln(y)
a)Find the gradient of f.
b) Find the rate of change of f at the point (2, 1, 1) in the
direction of ?⃗ = 〈−2, 4, −4〉

1) Consider the function.
f(x) = x5 − 5
(a) Find the inverse function of f.
f −1(x) =
2)
Consider the function
f(x) = (1 + x)3/x.
Estimate the limit
lim x → 0 (1 + x)3/x
by evaluating f at x-values near 0. (Round
your answer to five significant figures.)
=

Consider function f(x, y) = x 2 + y 2 − 2xy and the 3D graph z =
x 2 + y 2 − 2xy. (a) Sketch the level sets f(x, y) = c for c = 0,
1, 2, 3 on the same axes. (b) Sketch the section of this graph for
y = 0 (i.e., the slice in the xz-plane). (c) Sketch the 3D
graph.

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