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]
2 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]2 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
Need help ASAP please
Kay [80]

Answer:

because the length is 4 feet more than the width

=> the length is x + 4 (feet)

because the base has an area of 21 square feet

=> x(x + 4) = 21

<=> x² + 4x = 21

3 0
2 years ago
A cookie recipe calls for 1/4 cup of brown sugar for one batch of 24 cookies. How many cups of brown sugar would be needed to ba
mixer [17]

Answer:1 1/2

Step-by-step explanation:trust me bro

7 0
3 years ago
Read 2 more answers
I need to fill this out for notes .
docker41 [41]

Answer:

not an answer but how do you attach images because it would be more useful

Step-by-step explanation:

5 0
2 years ago
Heights of men have a bell-shaped distribution, with a mean of 176 cm and a standard deviation of 7 cm. Using the Empirical Rule
Vaselesa [24]

Answer:

a) 68% of the men fall between 169 cm and 183 cm of height.

b) 95% of the men will fall between 162 cm and 190 cm.

c) It is unusual for a man to be more than 197 cm tall.

Step-by-step explanation:

The 68-95-99.5 empirical rule can be used to solve this problem.

This values correspond to the percentage of data that falls within in a band around the mean with two, four and six standard deviations of width.

<em>a) What is the approximate percentage of men between 169 and 183 cm? </em>

To calculate this in an empirical way, we compare the values of this interval with the mean and the standard deviation and can be seen that this interval is one-standard deviation around the mean:

\mu-\sigma=176-7=169\\\mu+\sigma=176+7=183

Empirically, for bell-shaped distributions and approximately normal, it can be said that 68% of the men fall between 169 cm and 183 cm of height.

<em>b) Between which 2 heights would 95% of men fall?</em>

This corresponds to ±2 standard deviations off the mean.

\mu-2\sigma=176-2*7=162\\\\\mu+2\sigma=176+2*7=190

95% of the men will fall between 162 cm and 190 cm.

<em>c) Is it unusual for a man to be more than 197 cm tall?</em>

The number of standard deviations of distance from the mean is

n=(197-176)/7=3

The percentage that lies outside 3 sigmas is 0.5%, so only 0.25% is expected to be 197 cm.

It can be said that is unusual for a man to be more than 197 cm tall.

3 0
3 years ago
Area of a triangle with a base length of 4in and a height of 9in
Aleks04 [339]
18 inches sqa-wared. "Nailed it!"
4 \times 9 =  36 \\ 36 \div 2 = 18
4 0
3 years ago
Other questions:
  • Find the sum of the interior angle measures of a convex octagon
    13·1 answer
  • A green ribbon is 9 inches long. A red ribbon is 1/10 the length of the green
    13·1 answer
  • Evaluate. 5/8−(14)2= ________
    15·1 answer
  • A line passes through point (3,7) and has a slope of 3/4 what would the equation be
    14·1 answer
  • Amy usually swims 20 laps in 30 minutes. What is her rate in Laos per minute
    11·2 answers
  • What is the value of 9(n+2)−5n for n=14?<br> 3<br> 9<br> 10<br> 19
    10·1 answer
  • 14 3/5 - 3 ÷ 5 11/12​
    5·1 answer
  • Find the area of the trapezoid
    9·2 answers
  • Please help <br><br>worth 15 points
    14·2 answers
  • Can someone answer my earlier question in math its 20 points
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!