Without using a calculator, order the following numbers from least to greatest. 5 39 6

For what values of θ on the polar curve r=θ, with 0≤θ≤2π , are the tangent lines horizontal? Vertical?

Bond [772]

Given that $r=\theta$ , then $r'=1$

The slope of a tangent line in the polar coordinate is given by:

$m= \frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$

Thus, we have:

$m= \frac{\sin\theta+\theta\cos\theta}{\cos\theta-\theta\sin\theta}$

Part A:

For horizontal tangent lines, m = 0.

Thus, we have:

$\sin\theta+\theta\cos\theta=0 \\ \\ \theta\cos\theta=-\sin\theta \\ \\ \theta=- \frac{\sin\theta}{\cos\theta} =-\tan\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:

θ = 0

θ = 2.02875783811043

θ = 4.91318043943488

Part B:

For vertical tangent lines, $\frac{1}{m} =0$

Thus, we have:

$\cos\theta-\theta\sin\theta=0 \\ \\ \Rightarrow\theta\sin\theta=\cos\theta \\ \\ \Rightarrow\theta= \frac{\cos\theta}{\sin\theta} =\sec\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:

θ = 4.91718592528713

3 0

3 years ago

What is the sum of the complex numbers 6+5i and 8+3i2? (NOTE: −1‾‾‾√=i )

Ahat [919]

Answer:

11 + 5i

Step-by-step explanation:

6 + 5i + 8 + 3i² =

= 6 + 8 + 3(-1) + 5i

= 14 - 3 + 5i

= 11 + 5i

4 0

2 years ago

A sign company charges $28 per yard for each custom-made banner. Ms.Gill orders two banners that are each 178 yards long, and on

zhuklara [117]

Given,

A sign company charges $28 per yard for each custom-made banner.

Ms.Gill orders two banners that are each 178 yards long, and one banner that is 258 yards long.

To find,

Total money paid by Ms. Gill.

Solution,

Total length of 2 banners of 178 yards = 356 yards

Third banner is 258 yards long.

Total length of the banners = 356 + 258

= 614 yards

The cost of each banner = $28 per yards.

Total amount paid by Ms. Gill is :

= $28 × 614

= $17,192

Hence, she will pay $17,192 for all the three banners.

8 0

2 years ago

Which phrase best describes the transformation from the graph of y=(2x-15)^2+3 to the graph of y=2(x-11)^2+3?

weeeeeb [17]

Answer:

The graph's vertex would move 4 to the right and would become twice as wide.

Step-by-step explanation:

We can tell this because the x value of the vertex is the constant inside the parenthesis. In this case, the number changes from 11 to 15, which means it moves to the right by 4.

And with the 2 outside of the parenthesis instead of inside, it becomes more wide (since the 2 is not being squared).

7 0

2 years ago

What is the domain of the function in the graph above?

alexdok [17]

Answer: [-1,infinity)

Step-by-step explanation:

4 0

3 years ago