part 1)
Find the partial derivatives of the function
f(x,y)=xsin(7x^6y):
fx(x,y)=
fy(x,y)=
part 2)
Find the...
part 1)
Find the partial derivatives of the function
f(x,y)=xsin(7x^6y):
fx(x,y)=
fy(x,y)=
part 2)
Find the partial derivatives of the function
f(x,y)=x^6y^6/x^2+y^2
fx(x,y)=
fy(x,y)=
part 3)
Find all first- and second-order partial derivatives of the
function f(x,y)=2x^2y^2−2x^2+5y
fx(x,y)=
fy(x,y)=
fxx(x,y)=
fxy(x,y)=
fyy(x,y)=
part 4)
Find all first- and second-order partial derivatives of the
function f(x,y)=9ye^(3x)
fx(x,y)=
fy(x,y)=
fxx(x,y)=
fxy(x,y)=
fyy(x,y)=
part 5)
For the function given below, find the numbers (x,y) such that
fx(x,y)=0 and fy(x,y)=0
f(x,y)=6x^2+23y^2+23xy+4x−2
Answer: x= and...
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...
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) = ( x2 +
z2)ln(y)
a)Find the gradient of f.
b) Find...
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〉
6. (5 marks) Consider the function f defined by f (x, y) = ln(x
− y)....
6. Consider the function f defined by f (x, y) = ln(x
− y). (a) Determine the natural domain of f. (b) Sketch the level
curves of f for the values k = −2, 0, 2. (c) Find the gradient of f
at the point (2,1), that is ∇f(2,1). (d) In which unit vector
direction, at the point (2,1), is the directional derivative of f
the smallest and what is the directional derivative in that
direction?
4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y)...
4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...
Let X and Y have the joint probability density function f(x, y)
= ⎧⎪⎪ ⎨ ⎪⎪⎩...
Let X and Y have the joint probability density function f(x, y)
= ⎧⎪⎪ ⎨ ⎪⎪⎩ ke−y , if 0 ≤ x ≤ y < ∞, 0, otherwise. (a) (6pts)
Find k so that f(x, y) is a valid joint p.d.f. (b) (6pts) Find the
marginal p.d.f. fX(x) and fY (y). Are X and Y independent?
For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when...
For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when x > 0 and y
> 0 f(x,y) = 0 otherwise
a. Find the conditional density F xly (xly)
b. Find the marginal probability density function fX (x)
c. Find the marginal probability density function fY (y).
d. Explain if X and Y are independent
Let X and Y be continuous random variables with joint density
function f(x,y) and marginal density...
Let X and Y be continuous random variables with joint density
function f(x,y) and marginal density functions fX(x) and fY(y)
respectively. Further, the support for both of these marginal
density functions is the interval (0,1).
Which of the following statements is always true? (Note there
may be more than one)
E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)
E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx
E[Y^3]=∫0 TO 1 y^3 fX(x) dx
E[XY]=(∫0 TO 1 x fX(x)...
1). Consider the following function and point.
f(x) = x3 + x + 3; (−2,
−7)
(a)...
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 following vector field.
F(x, y, z) =
6yz ln x i + (3x −...
Consider the following vector field.
F(x, y, z) =
6yz ln x i + (3x −
7yz) j +
xy8z3 k
(a)
Find the curl of F evaluated at the point
(5, 1, 4).
(b)
Find the divergence of F evaluated at the
point (5, 1, 4).