1. (a) Throughout Question 1 part (a), let f be the function given by f(x, y) = 6+x^3+y^3−3xy.

(i) At the point (0, 1), in what direction does the function f have the largest directional derivative?

(ii) Find the directional derivative of the function f at the point (0, 1) in the direction of the vector [3, 4] .

(iii) The function f has critical points at (0, 0) and at (1, 1). Classify the natures of these critical points by using the Hessian. Justify your answer.

(iv) Suppose that x = t^2 and y = 1 − t^3 . Use the chain rule to calculate df/dt. You should write your function as a function of t but there is no need to simplify your answer.

(b) Consider optimisation of the function f(x, y) = 4x − 2y subject to the constraint x^2 + y^2 = 125. Use the method of Lagrange multipliers to find the critical points of this constrained optimisation problem. You do not need to determine the nature of the critical points.

1. Determine if the point (-1, -1, 0) lies in the plane with equation 2x + 3y -4z + 5 = 0.
2. Find the scalar equation of the plane through the points M(1,2,3) and N(3,2,-1) that is perpendicular to the plane with equation 3x + 2y + 6z +1 = 0.

Solve the problem.

Let C(x) be the cost function and R(x) the revenue function. Compute the marginal cost, marginal revenue, and the marginal profit functions.
C(x) = 0.0004x³ - 0.036x² + 200x + 30,000
R(x) = 350x

Select one:

A. C'(x) = 0.0012x² - 0.072x + 200
R'(x) = 350
P'(x) = 0.0012x² - 0.072x - 150

B. C'(x) = 0.0012x² + 0.072x + 200
R'(x) = 350
P'(x) = 0.0012x² + 0.072x + 150

C. C'(x) = 0.0012x² - 0.072x + 200
R'(x) = 350
P'(x) = -0.0012x² + 0.072x + 150

Question No.1: Solve the following system of two linear equations with two variables x and y by “Equating the equations” method. ? = ?? − ?? ??? ? = −? + 5

Question No.2: Is this matrix ? = [ ? ? ? ? ] singular or non-singular?

Question No. 3: Solve the following operations with the help of “PEMDAS”. ? ? − (?? ÷ ?) × ? ÷ ? − ? × ? + ?? ÷ ?3

Question No.4: A car was purchased for 5400 RO and is sold for 4300 RO. What is the percentage loss?

Question No.5 The total revenue function is given as ?? = ?? ? + ?? + ? 1. Find the average revenue 2. Find the marginal revenue 3. Find the marginal revenue when x = 3

Question No.6 : Calculate the rate of interest required for an investment ????? R.O to earn ???? R.O interest over ? years.

Question No.7: A salesman discounts a watch marked at 125 RO by 15%. 1. How much is the discount? 2. How much will a customer pay for the watch?

Consider the following in Euclidean geometry: Suppose that you want to translate a figure in the coordinate plane along the vector ( 0 2020 ). Find, with a brief explanation, the equations of two lines in the coordinate plane (call them ℓ and m) such that ρ m ∘ ρ ℓ is a translation along the vector ( 0 2020 ).

Use Lagrange multipliers to find the distance from the point

(2, 0, −1)

to the plane

4x − 3y + 8z + 1 = 0.

Use the formula for the general term​ (the nth​ term) of an arithmetic sequence to find the sixth term of the sequence with the given first term and common difference.

a₁ =15; d=7

a₆ =

4. [22 pts] For the function ?(?) = 2?5 − 9?4 + 12?3 − 12?2 + 10? − 3 answer the following:

Find the area of the region that lies inside the curve r=1+cos(theta) but outside the curve r=3cos(theta)

Answer the following questions thoroughly. Use correct grammar and punctuation.

1) A student observes the following spinner and claims that the color red has the highest probability of appearing, since there are two red areas on the spinner. What is your reply? Your answers should include facts, rules, any definitions necessary to explain why the student is correct or incorrect.

2) In class, an experiment of flipping a coin is performed.

Pretend that you are introducing the concept of probability and introducing the terms 1) experiment, 2) outcomes, 3)sample space, 4) theoretical probability, and 5) empirical probability. Explain these terms, as if you were speaking to a 5th grade class, as if they knew nothing about probability. You will need to tell the class what each term , 1-5, is represented by in this particular example. Use correct grammar and punctuation.

∫−9ln(x2−1)dx

Solve:

y' = (y²-6y-16)x²

y(4) = 3

The two basic facts about the quantifiers you need to understand, and from which all of the logical properties of the quantifiers follow are:

Basic Fact 1: A universal quantifier (x) Fx is equivalent to an infinite conjunction: Fa & Fb & Fc & Fd & ........

where a, b, c, d, are the names of objects in the universe picked out by the 'x' in the universal quantifier '(x)'.

Basic Fact 2: An existential quantifier is equivalent to an infinite disjunction

Fa v Fb v Fc v Fd v ......

Expand in a two-element universe

(a) ~(x) ((Fx v Gy) v Ka)

Water is flowing at the rate of 5m3/min into a tank in the form of a cone of altitude 20 m and a base radius of 10 m, with its vertex in the downward direction.
a) How fast is the water level rising when the water is 8m deep?
b) If the tank has a leak at the bottom and the water level is rising at 0.084 m/sec when the water is 8 m deep, how fast is the water leaking?

The quantity demanded of a certain electronic device is 1000 units when the price is $665. At a unit price of $640, demand increases to 1200 units. The manufacturer will not market any of the device at a price of $90 or less. However for each $50 increase in price above $100, the manufacturer will market an additional 1000 units. Assume that both the supply equation and the demand equation are linear.

(a) Find the supply equation.

(b) Find the demand equation

(c) Find the equilibrium price.

(d) Find the equilibrium quantity