CPhill

Correct

f'(x)  = 7 (x - 1)^6

Set to 0

7 ( x - 1)^6  =  0

(x - 1)^6  = 0

So....x  = 1 is a critical number

If  f'(x)  = 0  for all x, we just have a horizontal line....so....f(x)  will be the same for all values of x....so...

d) f(x1) = f(x2) for every x1 < x2 

If f'(x) > 0  for all x, then the function must be always increasing from left to right

Thus...if  x1 > x2

Then f(x1) > f(x2)

So...

a) f(x1) > f(x2) for every x1 > x2

7)The lateral surface area of a cylinder is 30% of its surface area. What is the ratio between the base radius and the height of the cylinder

Let S be the surface area....then....

S  =   2pi * r  ( r + h)     (1)

Lateral surface area  =   2pi * r * h  = .30S    ⇒  2pi * r  =  .30S / h    (2)

Sub (2)  into (1)

S  =  (.30S/h) (r + h)   ......  divide through by S

1 = (.30/h) (r + h)

h / .30   =  r + h

h / (3/10)  = r + h

(10/3) h = r + h

(7/3) h  = r

r / h  =  7/3

We will have a vertical asymptote

See an example here:  https://www.desmos.com/calculator/38rlm9mnnw

g (x)  = 2x^2 -1

f(x)  = x^3

g(0)  = 2(0)^2 -  1   =  -1

f(0) = (0)^3  =  0

Note...that at  x  = c  =  0......we have the vertical asymptote  x  =  0

[ 6 cos 45  - 3tan 45 ] 

__________________   =

         2 tan 60

6 (√2/2) -  3(1)

___________  =

     2√3

3√2 - 3

______        multiply  top/bottom by √3     

   2√3   

3√6 - 3√3

________   =    divide through by 3

    2 * 3  

√6  - √3

______

    2

f(x)  =  (x^2  - 16)  / ( x - 4)    =  (x + 4) (x - 4) / (x - 4)  =  x + 4

If x  could be 4,  we  would have   ( 4) + 4  =  8

So  k  =  8     will make the function continuous

Population density  =  population / land area  =

8550405  / 1213.4    ≈ 7047 people / km^2 

Mass  = Density * Volume

Mass / Volume  = Density

The volume of the board is  0.3 * 2.1 * 0.1  = .063 miles^3

So....the density is

0.07 kg  / .063 miles^3   =

1.1 kg / miles^3

x  = 2y + 5    (1)

x^2 + y^2  = 5  (2)

Sub (1) into (2) to find the y intersection of these functions

(2y + 5)^2  + y^2  =  5      simplify

4y^2 + 20 y + 25 + y^2  = 5

5y^2 + 20y +20  = 0      divide through by 5

y^2 + 4y + 4  =  0      factor

(y + 2)^2  = 0        take the square root of both sides

y + 2   = 0

y  = -2

And x  =    2(-2) + 5   =  1

So....(1, -2)  is the tangent point  because it  is the only point that makes both equations true

1 = 2(-2) + 5   is true      and

(1)^2  + (-2)^2  = 5     is also true

Here's a graph :  https://www.desmos.com/calculator/ty1aww3bmp