Consider the following planes.

x + y + z = 7,     x + 3y + 3z = 7

(a) Find parametric equations for the line of intersection of the planes. (Use the parameter t.)
(x(t), y(t), z(t)) =

(b) Find the angle between the planes. (Round your answer to one decimal place.)
°

Find the vertex and the x-intercepts (if any) of the parabola. (If an answer does not exist, enter DNE.)

f(x) = 3x² - 8x - 3

Suppose the first and second derivatives of f(x) are: f' (x) = 4x(x^2 − 9) f''(x) = 12(x^2 − 3).

(a) On what interval(s) is f(x) increasing and decreasing?

(b) On what interval(s) is f(x) concave up and concave down?

(c) Where does f(x) have relative maxima? Minima? Inflection points?

Solve differential equation by using undetermined coefficient method

y^''+2y^'+y= 4 sin2x

2-Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.)

f(θ) = 6 cos(θ) + 3 sin²(θ)

3- Consider the following.

f(x) = x⁵ − x³ + 9,    −1 ≤ x ≤ 1

(a) Use a graph to find the absolute maximum and minimum values of the function to two decimal places.

minimum

(b) Use calculus to find the exact maximum and minimum values.

maximum

minimum

A rectangular storage container with an open top is to have a volume of 8 m3. The length of this base is twice the width. Material for the base costs $6 per square meter. Material for the sides costs $10 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

Let S be the set of natural numbers which can be written as a non-empty string of ones followed by a non-empty string of zeroes. For example, 10, 111100 and 11100000 are all in S, but 11 and 1110011 are not in S. Prove that there exists a natural number n∈S, such that 2018 | n.

The business manager of a 90 unit apartment building is trying to determine the rent to be charged. From past experience with similar buildings, when rent is set at $400, all the units are full. For every $20 increase in rent, one additional unit remains vacant. What rent should be charged for maximum total revenue? What is that maximum total revenue?

To help solve the above scenario, perform an internet search for Profit Parabola or Applications of Quadratic Functions. List the URL of one of the applications that you find.

URL ___________________________________________________________________

Go to http://www.purplemath.com/modules/quadprob3.htm to see the process used for determining the quadratic function for revenues R(x) as a function of price hikes x on page 3 with the canoe-rental business problem. Use this process to determine the quadratic function that models the revenues R(x) as a function of price hikes x in the apartment building scenario above. SHOW ALL YOUR WORK!

3. What is the formula for revenues R after   x $20 price hikes in the apartment building?

Graph the function. Clearly label the graph (desmos.com is a great an on-line graphing resource).

Find the maximum revenue (or income) of the apartment building.

What is the rent that coincides with this maximum revenue?

What is the outcome of the rent hike of $20 results in 2 additional vacancies instead of 1 additional vacancy? Recalculate questions 3, 5, 6 for this new scenario.

Solve differential equation by using undetermined coefficient method

y^''+9y=x^2e^x+6 , y(0)=1 and y^'(0)=1

Four numbers have a sum of 9900. The second exceeds the first by 1/7 of the first. The third exceeds the sum of the first two by 300. The fourth exceeds the sum of the first three by 300. Find the four numbers.

The polynomial equation x^3−5x^2+11x−15=0 is known to have an integer solution. Complete the following table listing in the first column the candidate integer solutions (there are eight) of x^3−5x^2+11x−15=0 supplied by the Rational Root Test, and in the second column the values of P evaluated at the corresponding candidates. MAKE SURE THAT THE CANDIDATE ROOTS ARE IN INCREASING ORDER!

x. x^3−5x^2+11x−15

#1

#2

#3

#4

#5

#6

#7

#8

With this information, give all three roots of PP (with distinct roots separated by a comma):

Q1)

(a) Write implicit equations for two parallel planes, P1 containing the line <−3,−5,3>+t<3,−4,−5> and P2 containing the line <−1,−2,1>+t<5,4,1>

(b)  Let n = <16,-28,32>

The set of vectors w = <x,y,z> such that <3,−4,−5> x w = n forms a line. Write a parametric equation for that line, and make sure to use t as your parameter.

Please show the working clearly.

solve the following initial value problem y''+4y'=g(t),y(0)=0,y' (0)=1 if g(t) is the function which is 1 on [0,1) and zero elsewhere

Laplace Transform : y ' - y = e^-3t cos3t , y(0) =3

and, Show that, Differential Form ?

dU = Tds - Pdv , dH=Tds-Vdp , dF= -sdT-Pdv , dG= -sdT+VdP

Find the second-order Taylor polynomial for f(x,y)=8y^2e^(−x^2) at (1,1).

(Use symbolic notation and fractions where needed.)

p(x,y)p(x,y) = .