Find the area of the region enclosed by

(x + 3y)dx – xdy = 0

2xdy/dx=1+y^2

$2xdy/dx=1+y^2$2xdy/dx=1+y^2

Find values of x, if any, at which f is not continuous.

f(x) = 5x^4 − 3x + 7

determine the following functions (v/g)x  

v= x² + 5x+4    g= x² + 2x-8

A bridge is built in the shape of a parabolic arc and is to have a span of 100 feet. The height of the arch at a distance of 40 feet from the center is to be 10 feet. Assume that the ground is the x-axis and the y-axis as the axis of the arc.

a. How high is the arch at its center approximated in two decimal places?

b. What is the horizontal length, approximated to two decimal places, of the arch from axis when it’s 15 feet high?

The function g is related to one of the parent functions g(x) = 2 √x a.) Identify the parent function f. b.) Use function notation to write g in terms of f.

4x x 3x -4 +19

 find the derivative of g(x)= x^2-4/x+0.5 by quotient rule 

How do you find a Least Common Factor in cases like

Example: 

2/5 x 5/7

The table that is needed has been provided in the images.
Given the differential ${\displaystyle =y{ }^{″}-8y^{\prime }+16y=-3e^{4x}}$.
a) Find the complementary solution ${\displaystyle =y_c}$ to the differential equation.
b) Write down the FORM of the particular solution ${\displaystyle =y_p}$ for a solution using undetermined coefficients. DO NOT solve for ${\displaystyle =y_p}$. Use Table. 4.4.1
$=y_p=$__________________________

Test x+y=9

The S&P portfolio pays a dividend yield of 1% annually. Its current value is 1,500. The T -bill rate is 4%. Suppose the S&P futures price for delivery in 1 year is 1,550. Construct an arbitrage strategy to exploit the mispricing and show that your profits 1 year hence equal the mispricing in the futures market (6 marks)

Suppose that the value of the S&P 500 stock index is 1,600

a.     If each futures contract costs $25 to trade with a discount broker, how much is the transaction cost per dollar of stock controlled by the futures contract? 

 b. If the average price of a share on the NYSE is about $40, how much is the transaction cost per “typical share” controlled by one futures contract? 

 c. For small investors, a typical transaction cost per share in stocks directly is about 10 cents per share. How many times the transactions costs in futures markets is this? 

Photographers often use multiple sources of light light

Control the shadows or to enlighten their subject in an artistic way. The light

from a light source is inversely proportional to the Carrá

The distance from the source and proportional has its intensity. Two Sone

of light, Li and L2, are placed 10 m away, and have respectively

An intensity of 4 units and 1 unit. Leclairage in any point P

corresponds to the sum of the lighting from the two sources.

An intense light source

Hint: when a quantity is inversely proportional to the square of a, where K is a constant.

a) Determine a function which represents the lighting in point P according to its

Distance from point L

b) How much distance is the lighting the most powerful in P?

) At what distance from the point is the lighting the weakest in P?

d) examine whether it would be more effective to increase the intensity of L2 or to place

L2 closer to Li so as to increase lighting at the point found in C).

Press your answer using mathematical evidence. Take into account the

time of lighting and rate of variation in lighting when the intensity

or the position of L2 varies.