1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
dimulka [17.4K]
3 years ago
7

At what point does the curve have maximum curvature? Y = 4ex (x, y) = what happens to the curvature as x → ∞? Κ(x) approaches as

x → ∞.
Mathematics
1 answer:
MAXImum [283]3 years ago
6 0

<u>Answer-</u>

At x= \frac{1}{2304e^4-16e^2} the curve has maximum curvature.

<u>Solution-</u>

The formula for curvature =

K(x)=\frac{{y}''}{(1+({y}')^2)^{\frac{3}{2}}}

Here,

y=4e^{x}

Then,

{y}' = 4e^{x} \ and \ {y}''=4e^{x}

Putting the values,

K(x)=\frac{{4e^{x}}}{(1+(4e^{x})^2)^{\frac{3}{2}}} = \frac{{4e^{x}}}{(1+16e^{2x})^{\frac{3}{2}}}

Now, in order to get the max curvature value, we have to calculate the first derivative of this function and then to get where its value is max, we have to equate it to 0.

 {k}'(x) = \frac{(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})}{(1+16e^{2x} )^{2}}

Now, equating this to 0

(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x}) =0

\Rightarrow (1+16e^{2x})^{\frac{3}{2}}-(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})

\Rightarrow (1+16e^{2x})^{\frac{3}{2}}=(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})

\Rightarrow (1+16e^{2x})^{\frac{1}{2}}=48e^{2x}

\Rightarrow (1+16e^{2x})}=48^2e^{2x}=2304e^{2x}

\Rightarrow 2304e^{2x}-16e^{2x}-1=0

Solving this eq,

we get x= \frac{1}{2304e^4-16e^2}

∴ At  x= \frac{1}{2304e^4-16e^2} the curvature is maximum.




You might be interested in
What is the value of X? <br> X= <br><br> B=75<br> C=50<br> A=x <br><br> Thank u! Sorry i'm dumb:(
murzikaleks [220]
Angels on a triangle is 180°
so you add 75 and 50 which gives you 125 then u take away that by 180 to find x
and that your answer
I'm going to let you work it out since its easy question
8 0
3 years ago
2. (08.05 HC)
4vir4ik [10]

Answer:

option 3

Step-by-step explanation:

not sure if right

8 0
3 years ago
These angels are equal. What are they called?
Yanka [14]
Acute angles because they are less then 90 degrees which is a right angle.

7 0
4 years ago
PLEASE HELP MEEE
STALIN [3.7K]
If there are 16 ounces in a pound, and 8 ounces costs $8, The best answer would be D. $16 per pound
4 0
4 years ago
I don't understand this plz help explain how to do it
podryga [215]

Answer:

1. U. 190

2.U. 155

3. U. 240

Step-by-step explanation:

Explanation: To find an angle of a triangle we add the angles we know about

EXAMPLE: 50 + 120. We need to find the third angle.

We know that Angles of triangles should add up to 360

So to find the third angle we add the angle we know about.

EXAMPLE: 50 + 120 = 170 then we Subtract the sum of a triangle *360*

EXAMPLE: 50 + 120 = 170

EXAMPLE: 170 - 360

EXAMPLE: = 190

That's how we find a third angle!

If this doesn't clarify how do to it enough just let me know and i can try to explain more in detail!

Hope this helps you!!!

Have a nice day!

6 0
3 years ago
Other questions:
  • When three or more lines intersect at one point, they are ____________.
    11·1 answer
  • Please help me with domain and range of graphs Part 1​
    9·1 answer
  • Is 1/7 bigger than 1/4?
    15·1 answer
  • What is the perimeter of this rectangle?​
    10·1 answer
  • What is the unit rate for Bill's boffo bagel?​
    11·1 answer
  • Please awnser properly
    10·1 answer
  • Determine whether each statement about the system of the two linear equations is true or false:
    15·1 answer
  • Bags of sugar come in 3 sizes:
    12·1 answer
  • PLEASE HELP DUE TODAY
    5·2 answers
  • Find the value of each card and put them in order, from lowest to highest.
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!