Let f(x,y) = 3x3 + 3x2 y − y3 − 15x.
a) Find and classify the critical points of f. Use any method taught during the course (the second-derivative test or completing the square).
b) One of the critical points is (a,b) = (1,1). Write down the second-degree Taylor approximation of f about this point and motivate, both with computations and with words, how one can see from this approximation what kind of critical point (1,1) is. Use completing the square.
In: Math
1. Consider the spiral S in the xy-plane given in polar form by r = e ^θ . (a) (5 points) Find an arc length parametrization for S for which the reference point corresponds to θ = 0.
(b) (5 points) Compute the curvature of S at the point where θ = π.
In: Math
The consumer demand equation for tissues is given by
q = (100 − p)2,
where p is the price per case of tissues and q is the demand in weekly sales.
(a) Determine the price elasticity of demand E when the
price is set at $34. (Round your answer to three decimal
places.)
E =
Interpret your answer.
The demand is going ? up down by % per 1% increase in price at that price level.
(b) At what price should tissues be sold to maximize the revenue?
(Round your answer to the nearest cent.)
$
(c) Approximately how many cases of tissues would be demanded at
that price? (Round your answer to the nearest whole number.)
cases per week
In: Math
Find the coordinates of all local extrema and inflections of y = x − 4√x. Give the intervals where the function is increasing, decreasing, concave up, and concave down.
In: Math
Find two positive integers such that the sum of the first number and four times the second number is 100 and the product of the numbers is as large as possible.
please double check answer
In: Math
Consider the equation below. (If an answer does not exist, enter DNE.)
f(x) = e^(3x) + e^(−x)
(a)
Find the interval on which f is increasing. (Enter your answer using interval notation.)
Find the interval on which f is decreasing. (Enter your answer using interval notation.)
(b)
Find the local minimum and maximum values of f.
local minimum value
local maximum value
(c)
Find the inflection point. (x, y) =
Find the interval on which f is concave up. (Enter your answer using interval notation.)
Find the interval on which f is concave down. (Enter your answer using interval notation.)
In: Math
use laplace transform to solve the initial value problem x'' +3x'-10x=5e^-4t; x(0)=3, x'(0)=-10
In: Math
consider the following planes.
-3x+y+z=3
18x-6y+3z=9
a.) find the angle between the two planes. (round your answer to
two decimal places.)
b.) find a set of parametric equation for the line of intersection
of the planes. (use t for the parameter. enter you answers as a
comma-separated list of equations)
In: Math
|
1) Solve the given quadratic equation by using Completing the Square procedure and by Quadratic formula ( you must do it both ways). Show all steps for each method and put your answer in simplest radical form possible. 2) Which part of the Quadratic Formula can help you to find the Nature of the roots for the Quadratic Equation. Explain how you can find the nature of the roots and provide one Example for each possible case with solution. |
In: Math
Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. (Let n represent an arbitrary positive number.)
y''+λy=
0，
y（0）= 0，
y'（π）= 0
In: Math
In: Math
Find the Taylor series for f(x)= ln(2+x) centered at x=0 and compute its interval of convergence. Show all work and use proper notation.
In: Math
Abdul is making a map of his neighborhood. He knows the following information:
What theorem can Abdul use to determine the two triangles are similar? (6 points)
|
Pieces of Right Triangles Similarity Theorem |
|
|
Side-Side-Side Similarity Theorem |
|
|
Midsegment Theorem |
|
|
Side-Angle-Side Similarity Theorem |
In: Math
Use the simplex method to solve the linear programming problem.
Maximize objective function: Z= 6x1 + 2x2
Subject to constraints:
3x1 + 2x2 <=9
x1 + 3x2 <= 5
when x1, x2 >=0
In: Math