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3 years ago

The inverse function of f(x) = ex has a asymptote at

Mathematics

2 answers:

gogolik [260]3 years ago

6 0

Answer:

x=0

Step-by-step explanation:

You are given the function $f(x)=e^x.$ To find it inverse function, express x in terms of y:

$y=e^x\\ \\\ln y=x$

Now change x into y and y into x:

$y=\ln x\\ \\f^{-1}(x)=\ln x$

The graph of the function $f^{-1}(x)$ has vertical asymptote x=0.

kherson [118]3 years ago

4 0

Vertical asymptote at x = 0

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Will a large-sample confidence interval be valid if the population from which the sample is taken is not normally distributed? e

Dovator [93]

A normal distribution is a type of continuous probability distribution for a real-valued random variable in statistics.

Yes, the large-sample confidence interval will be valid.

<h3>What is meant by normal distribution?</h3>

A normal distribution is a type of continuous probability distribution for a real-valued random variable in statistics.

The normal distribution, also known as the Gaussian distribution, is a symmetric probability distribution about the mean, indicating that data near the mean occur more frequently than data far from the mean.

The confidence interval will be valid regardless of the shape of the population distribution as long as the sample is large enough to satisfy the central limit theorem.

<h3>What does a large sample confidence interval for a population mean?</h3>

A sample is considered large when n ≥ 30.

By 'valid', it means that the confidence interval procedure has a 95% chance of producing an interval that contains the population parameter.

To learn more about normal distribution, refer to:

brainly.com/question/23418254

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8 0

1 year ago

Find the values of the sine, cosine, and tangent for ZA C A 36ft B <br> 24ft

Reptile [31]

<h2>Question:</h2>

Find the values of the sine, cosine, and tangent for ∠A

a. sin A = $\frac{\sqrt{13} }{2}$ , cos A = $\frac{\sqrt{13} }{3}$ , tan A = $\frac{2 }{3}$

b. sin A = $3\frac{\sqrt{13} }{13}$ , cos A = $2\frac{\sqrt{13} }{13}$ , tan A = $\frac{3}{2}$

c. sin A = $\frac{\sqrt{13} }{3}$ , cos A = $\frac{\sqrt{13} }{2}$ , tan A = $\frac{3}{2}$

d. sin A = $2\frac{\sqrt{13} }{13}$ , cos A = $3\frac{\sqrt{13} }{13}$ , tan A = $\frac{2 }{3}$

<h2>Answer:</h2>

d. sin A = $2\frac{\sqrt{13} }{13}$ , cos A = $3\frac{\sqrt{13} }{13}$ , tan A = $\frac{2 }{3}$

<h2>Step-by-step explanation:</h2>

The triangle for the question has been attached to this response.

As shown in the triangle;

AC = 36ft

BC = 24ft

ACB = 90°

To calculate the values of the sine, cosine, and tangent of ∠A;

<em>i. First calculate the value of the missing side AB.</em>

<em>Using Pythagoras' theorem;</em>

⇒ (AB)² = (AC)² + (BC)²

<em>Substitute the values of AC and BC</em>

⇒ (AB)² = (36)² + (24)²

<em>Solve for AB</em>

⇒ (AB)² = 1296 + 576

⇒ (AB)² = 1872

⇒ AB = $\sqrt{1872}$

⇒ AB = $12\sqrt{13}$ ft

From the values of the sides, it can be noted that the side AB is the hypotenuse of the triangle since that is the longest side with a value of $12\sqrt{13}$ ft (43.27ft).

<em>ii. Calculate the sine of ∠A (i.e sin A)</em>

The sine of an angle (Ф) in a triangle is given by the ratio of the opposite side to that angle to the hypotenuse side of the triangle. i.e

sin Ф = $\frac{opposite}{hypotenuse}$ -------------(i)

Ф = A

opposite = 24ft (This is the opposite side to angle A)

hypotenuse = $12\sqrt{13}$ ft (This is the longest side of the triangle)

<em>Substitute these values into equation (i) as follows;</em>

sin A = $\frac{24}{12\sqrt{13} }$

sin A = $\frac{2}{\sqrt{13}}$

<em>Rationalize the result by multiplying both the numerator and denominator by </em> $\sqrt{13}$ <em />

sin A = $\frac{2}{\sqrt{13}} * \frac{\sqrt{13} }{\sqrt{13} }$

sin A = $\frac{2\sqrt{13} }{13}$

<em>iii. Calculate the cosine of ∠A (i.e cos A)</em>

The cosine of an angle (Ф) in a triangle is given by the ratio of the adjacent side to that angle to the hypotenuse side of the triangle. i.e

cos Ф = $\frac{adjacent}{hypotenuse}$ -------------(ii)

Ф = A

adjacent = 36ft (This is the adjecent side to angle A)

hypotenuse = $12\sqrt{13}$ ft (This is the longest side of the triangle)

<em>Substitute these values into equation (ii) as follows;</em>

cos A = $\frac{36}{12\sqrt{13} }$

cos A = $\frac{3}{\sqrt{13}}$

<em>Rationalize the result by multiplying both the numerator and denominator by </em> $\sqrt{13}$ <em />

cos A = $\frac{3}{\sqrt{13}} * \frac{\sqrt{13} }{\sqrt{13} }$

cos A = $\frac{3\sqrt{13} }{13}$

<em>iii. Calculate the tangent of ∠A (i.e tan A)</em>

The cosine of an angle (Ф) in a triangle is given by the ratio of the opposite side to that angle to the adjacent side of the triangle. i.e

tan Ф = $\frac{opposite}{adjacent}$ -------------(iii)

Ф = A

opposite = 24 ft (This is the opposite side to angle A)

adjacent = 36 ft (This is the adjacent side to angle A)

<em>Substitute these values into equation (iii) as follows;</em>

tan A = $\frac{24}{36}$

tan A = $\frac{2}{3}$

6 0

3 years ago

On a snow day, Mason created two snowmen in his backyard. Snowman A was built to a height of 51 inches and Snowman B was built t

mr_godi [17]

Answer:

A ( t ) = -4t + 51

B ( t ) = -2t + 29

t < 11 hours ... [ 0 , 11 ]

Step-by-step explanation:

Given:-

- The height of snowman A, Ao = 51 in

- The height of snowman B, Bo = 29 in

Solution:-

- The day Mason made two snowmans ( A and B ) with their respective heights ( A(t) and B(t) ) will be considered as the initial value of the following ordinary differential equation.

- To construct two first order Linear ODEs we will consider the rate of change in heights of each snowman from the following day.

- The rate of change of snowman A's height ( A ) is:

$\frac{d h_a}{dt} = -4$

- The rate of change of snowman B's height ( B ) is:

$\frac{d h_b}{dt} = -2$

Where,

t: The time in hours from the start of melting process.

- We will separate the variables and integrate both of the ODEs as follows:

$\int {} \, dA= -4 * \int {} \, dt + c\\\\A ( t ) = -4t + c$

$\int {} \, dB= -2 * \int {} \, dt + c\\\\B ( t ) = -2t + c$

- Evaluate the constant of integration ( c ) for each solution to ODE using the initial values given: A ( 0 ) = Ao = 51 in and B ( 0 ) = Bo = 29 in:

$A ( 0 ) = -4(0) + c = 51\\\\c = 51$

$B ( 0 ) = -2(0) + c = 29\\\\c = 29$

- The solution to the differential equations are as follows:

A ( t ) = -4t + 51

B ( t ) = -2t + 29

- To determine the time domain over which the snowman A height A ( t ) is greater than snowman B height B ( t ). We will set up an inequality as follows:

A ( t ) > B ( t )

-4t + 51 > -2t + 29

2t < 22

t < 11 hours

- The time domain over which snowman A' height is greater than snowman B' height is given by the following notation:

Answer: [ 0 , 11 ]

8 0

3 years ago