Question

Consider the function:

Z = 5x2 + 3x2y + 2y3

Suppose that x = 3, y = 2, Δx = 0.2, and Δy = 0.2

so that The change in the portfolio value by using the Taylor series expansion is?

Answer #1

Find directional derivative of the function f(x, y, z) =
5x2 + 2xy – 3y2z at
P(1, 0, 1) in the direction v = i +
j – k .

1. Compare the values of dy and Δy for the
function.
Function
x-Value
Differential of x
f(x) = 5x + 1
x = 1
Δx = dx = 0.01
dy
=
Δy
=
2. Use differentials to approximate the change in profit
corresponding to an increase in sales (or production) of one unit.
Then compare this with the actual change in profit.
Function
x-Value
P = −0.2x3 + 800x − 80
x = 40...

For this problem, consider the function f(x) = ln(1 + x).
(a) Write the Taylor series expansion for f(x) based at b = 0. Give
your
final answer in Σ notation using one sigma sign. (You may use 4
basic Taylor
series in TN4 to find the Taylor series for f(x).)
(b) Find f(2020) (0).
Please answer both questions, cause it will be hard to post them
separately.

Write the Taylor series for the function f(x) = x 3− 10x 2 +6,
using x = 3 as the point of expansion; that is, write a formula for
f(3 + h). Verify your result by bringing x = 3 + h directly into f
(x).

Consider the function w = x^(2) + y^(2) + z^(2) with x =
tsin(s), y = tcos(s), and z = st^(2)
(a) Find ∂w/∂s and ∂w/∂t by using the appropriate Chain
Rule.
(b) Find ∂w/∂s and ∂w/∂t by first substituting and writing w as
a function of s and t
before differentiating.

Consider the function given by f(x,y) = 3x2 −6xy + 2y3 +
23.
(a) Find all critical points of f(x,y) and determine their
nature.
(b) What are the minimum and maximum values of f(x,y) on the
straight line segment given by 0 ≤ x ≤ 3, y = 2?

The hyperbolic cosine function, cosh x = (1/2) (e^x + e^-x).
Find the Taylor series representation for cosh x centered at x=0 by
using the well known Taylor series expansion of e^x. What is the
radius of convergence of the Taylor Expansion?

4. Consider the function z = f(x, y) = x^(2) + 4y^(2)
(a) Describe the contour corresponding to z = 1.
(b) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane x = 1.
(c) Write down the equation of the curve obtained as the
intersection of the graph of z and the plane y = 1.
(d) Write down the point of intersection of the curves in (b)
and...

Suppose X, Y, and Z are random
variables with the joint density function
f(x, y, z) = Ce−(0.5x + 0.2y + 0.1z) if x ≥ 0, y ≥ 0,
z ≥ 0, and f(x, y, z) = 0 otherwise.
(a) Find the value of the constant C.
(b) Find P(X ≤ 0.75 , Y ≤ 0.5).
(Round answer to five decimal places).
(c) Find P(X ≤ 0.75 , Y ≤ 0.5 ,
Z ≤ 1). (Round answer to six decimal...

For the function g(x)=5x2+x
A. Find the instantaneous rate of change at x=-3
B. Find the equation of the tangent line at x=-3
Show Work

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