MATHEMATICS COLLEGE

Find each x-value at which f is discontinuous and for each x-value, determine whether f is continuous from the right, or from the left, or neither. f(x) = x + 4 if x < 0 ex if 0 ≤ x ≤ 1 8 − x if x > 1 x = (smaller value) continuous from the right continuous from the left neither

Answers

Answer 1

Answer:

Using continuity concepts, it is found that the function is left-continuous at x = 1.

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A function f(x) is said to be continuous at x = a if:

$\lim_(x \rightarrow a^(-)) f(x) = \lim_(x \rightarrow a^(+)) f(x) = f(a)$

If only $\lim_(x \rightarrow a^(-)) f(x) = f(a)$ , the function is left-continuous.

If only $\lim_(x \rightarrow a^(+)) f(x) = f(a)$ , the function is right-continuous.

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The piece-wise definition of the function $f(x)$ is:

$x + 4, x < 0$

$x, 0 \leq x \leq 1$

$8 - x, x > 1$

We have to check the continuity at the points in which the definitions change, that is, x = 0 and x = 1.

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At x = 0:

The definition at 0 is $f(0) = 0$
Approaching x = 0 from the left, we have values less than 0, thus:

$\lim_(x \rightarrow 0^(-)) f(x) = \lim_(x \rightarrow 0) x + 4 = 0 + 4 = 0$

Approaching x = 0 from the right, we have values greater than 0, thus:

$\lim_(x \rightarrow 0^(+)) f(x) = \lim_(x \rightarrow 0) x = 0$

Since the limits are equal, and also equal to the definition at the point, the function is continuous at x = 0.

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At x = 1:

The definition at 1 is $f(1) = 1$

Approaching x = 1 from the left, we have values less than 1, thus:

$\lim_(x \rightarrow 1^(-)) f(x) = \lim_(x \rightarrow 1) x = 1$

Approaching x = 1 from the right, we have values greater than 1, thus:

$\lim_(x \rightarrow 1^(+)) f(x) = \lim_(x \rightarrow 1) 8 - x = 8 - 1 = 7$

To the right, the limit is different, thus, the function is only left continuous at x = 1.

A similar problem is given at brainly.com/question/21447009

Answer 2

Answer:

Answer:

the function is continuous from the left at x=1 and continuous from the right at x=0

Step-by-step explanation:

a function is continuous from the right , when

when x→a⁺ lim f(x)=f(a)

and from the left when

when x→a⁻ lim f(x)=f(a)

then since the functions presented are continuous , we have to look for discontinuities only when the functions change

for x=0

when x→0⁺ lim f(x)=lim e^x = e^0 = 1

when x→0⁻ lim f(x)=lim (x+4) = (0+4) = 4

then since f(0) = e^0=1 , the function is continuous from the right at x=0

for x=1

when x→1⁺ lim f(x)=lim (8-x) = (8-0) = 8

when x→1⁻ lim f(x)=lim e^x = e^1 = e

then since f(1) = e^1=e , the function is continuous from the left at x=1

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Whitley park is a rectangular park with a perimeter of 70 yards. One side of Whitley park is 18 feet long. What is the area of Whitley park?

Answers

If Whitley park is a rectangular park with a perimeter of 70 yards. One side of Whitley park is 18 feet long then 174 yards is the area.

What is Area of Rectangle?

Area of rectangle is length times of breadth.

We know that 18 feet=6 yards.

It is given that One side of Whitley park is 18 feet long, so one side of length is 6 yards.

2(Lenght+breadth)=70

2(L+6)=70

2l+12=70

2l=70-12

2l=58

l=29 yards

Now

Area =Length×breadth

=29×6

= 174 square yards

Hence 174 square yards is the area of Whitley park.

To learn more on Area of Rectangle click:

brainly.com/question/20693059

#SPJ2

Answer:

174 yards squared

Step-by-step explanation:

Complete the sentence. 13 is 65% of _____

Answers

Step-by-step explanation:

is what i got i think its wrong tho

The answer : 20%
Hope it help u

Wisconsin Public Radio wants to duplicate a survey conducted in 2011 that found that 68% of adults living in Wisconsin felt that the country was going in the wrong direction. How many people would need to be surveyed for a 90% confidence interval to ensure the margin of error would be less than 3%? Be sure to show all your work and round appropriately

Answers

Answer:

655 people would need to be surveyed.

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of $\pi$ , and a confidence level of $1-\alpha$ , we have the following confidence interval of proportions.

$\pi \pm z\sqrt{(\pi(1-\pi))/(n)}$

In which

z is the zscore that has a pvalue of $1 - (\alpha)/(2)$ .

In this question, we have that:

$\pi = 0.68$

The margin of error is:

$M = z\sqrt{(\pi(1-\pi))/(n)}$

90% confidence level

So $\alpha = 0.1$ , z is the value of Z that has a pvalue of $1 - (0.1)/(2) = 0.95$ , so $Z = 1.645$ .

How many people would need to be surveyed for a 90% confidence interval to ensure the margin of error would be less than 3%?

We need to survey n adults.

n is found when M = 0.03. So

$M = z\sqrt{(\pi(1-\pi))/(n)}$

$0.03 = 1.645\sqrt{(0.68*0.32)/(n)}$

$0.03√(n) = 1.645√(0.68*0.32)$

$√(n) = (1.645√(0.68*0.32))/(0.03)$

$(√(n))^(2) = ((1.645√(0.68*0.32))/(0.03))^(2)$

$n = 654.3$

Rounding up

655 people would need to be surveyed.

Isaac works a job that pays him based oncommission. He earns 5% commission on all
sales after the first $5000. To determine the
amount of sales he earns commission on, he
uses the function f(x) = x - 5000. Then he uses
a different equation to determine how much
commission he actually earns, g(x) = 0.05x. He
wants to create a composite that includes
both. What is the composite function?
-
=

Answers

Answer: 1000000

Step-by-step explanation:

5x100

Use the Pythagorean Theorem and the square root property to solve the following problem. Express your answer in simplified radical form.

Then find a decimal approximation to the nearest tenth.

A rectangular park is 12 miles long and 4 miles wide. How long is a pedestrian route that runs diagonally across the park? In simplified radical form, the pedestrian route is 10 miles long.

Answers

Answer:

12.6 miles.

Step-by-step explanation:

Let L represent the length of the pedestrian.

We have been given that a rectangular park is 12 miles long and 4 miles wide. We are asked to find the length of a pedestrian route that runs diagonally across the park.

We will use Pythagoras theorem to find the length of the pedestrian (Hypotenuse).

$L^2=12^2+4^2$

$L^2=144+16$

$L^2=160$

Now, we will take positive square root of both sides:

$L=√(160)$

$L=√(16*10)$

$L=4√(10)$

$L=12.6491106$

Upon rounding to nearest tenth, we will get:

$L\approx 12.6$

Therefore, the length of the pedestrian is approximately 12.6 miles.

A popcorn company builds a machine to fill 1 kg bags of popcorn. They test the first hundred bags filled and find that the bags have an average weight of 1,040 grams with a standard deviation of 25 grams. 1.) Fill out the normal distribution curve for this situation. 2.) What percentage of people would receive a bag that had a weight greater than 1115 grams?

Answers

Answer:

Step-by-step explanation:

Given that a popcorn company builds a machine to fill 1 kg bags of popcorn. They test the first hundred bags filled and find that the bags have an average weight of 1,040 grams with a standard deviation of 25 grams.

i.e. Sample mean = 1040 and

Sample std dev s = 25 gm

Sample size n = 100

Hence by central limit theorem we have the sample mean follows a normal distribution with mean =1040 and std dev = s = 25 gm

$\bar X = N(1040,25)$

Normal curve would be with mean 1040 and std deviatin 25

b) P(X>1115)

= 1-0.9987

=0.0013

i.e. 0.13% would receive a bag that had a weight greater than 1115 grams

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