CPhill

Thanks, tertre   !!!

y  = 3(x)^(1/2) - x

y'  =  (3/2)x^(-1/2)  - 1

So..... the slope  at  x  =  4  =   (3/2) (4)^(-1/2)  - 1   =  (3/2) / √4  - 1  =  (3/4) - 1  =  -1/4

Correct  !!!

In the first one.....the degree of the polynomial in the numerator is one greater than the degree of the polynomial in the denominator...thus we will have a "slant"  asymptote.....so....no horizontal asymptote

Here's the graph of the second : 

https://www.desmos.com/calculator/ihiugvntuz

Well....we could use average rates of change, but here's something that's faster

Using the function ..... 24x - 10

Setting this to 0 and solving for x we get that x  = 10/24   = 5/12

This function be negative for all x values < 5/12.....so our given function will be concave down for all the same x values

For all x values > 5/12, this function will be positive so our given function will be concave up for the same x values

So....on the interval  [ -10, -6 ]....the given  function will be concave down

Here's a graph : https://www.desmos.com/calculator/5jagkhb5zl

The problem with using rates of change to calculate concavity is that we may "skip" over some interval  where the concavity changes

(3 √xy )^3       Write in exponential form, first

( 3^1  * x^(1/2)  * y^1)^3   =

3^3  * x^(3/2) * y^3  =

27 * x√x * y^3

11.  Notice NSS,  that taking the product of the number of volunteers with their associated time in minutes, we have that

2 * 220  =  440

5 * 88  =  440

8 * 55 +  440

11*40 + 440

So....it appears that

nt  =  440

10)    Joint variation might be a new one to you, NSS

We have this

A  = kds^2      

Where A is the amount of oil used, d is the distance  and s is the speed

As in other types of variation, we need to find the proportionality constant, k

So  we know that

30  =  k (85) (42)^2         divide both sides by  (85)(42)^2

30 / [ (85)(42)^2 ]  = k    1 / 4998

So we want to find A for this

A  =  (1 / 4998) (26) (54)^2   =    15.2  barrels

Your answer was better than mine, killersteve.....!!!!

This might explain it better than I can :

https://www.mathsisfun.com/numbers/imaginary-numbers.html

BTW....Isaac Asimov would argue with the term "Imaginary"....he claimed that they were just as "Real" as "Real" numbers !!!!!

y =  1/x       y   =   5 / (x + 6)

The second graph  shifts the first graph 6 units to the left  and  vertically "stretches" it by a factor of 5

See the graphs here, NSS  :

https://www.desmos.com/calculator/ckidk8ssrf