Answers
1) Problem # 15.
(i) [tex] \lim_{x \to \ 60^{-} } f(x) [/tex]
You have to approach the value of x by the left. That is the horizontal segment at y = 56 (with the open circle to the left and the solid circle to the right).
So, the value of the limit es 56.
(ii) [tex] \lim_{x \to \ 60^{+} } f(x)[/tex]
You have to approach the x value by the right. That is the horizontal segment at y = 68.
So the value of the limit is 68.
(iii) Since, the existence of the limit requires that both limits from the left and the right be equal, the conclusion is that the limit does not exist.
That is shown in the graph because the way the function is defined is different if the value of x is greater or less than 60.
(iv) What causes the graph to jump vertically by the same amount at discontinuites is that the rates are defined in intervals and not in a continuous way.
2) Problem 16. Mathematical induction.
Mathematical induction requires 3 steps: 1) Initial hypothesis, 2) assume the equation is valid for n = k, and 3) prove the equation is valid for n = k+1.
This is the solution step by step.
1) Initial hypothesis: first term ⇒ n = 1
left side: 8
rigth side: 4n(n+1) = 4(1+1) = 4(2) = 8
8 = 8 ⇒ check
2) Assume n = k
letf side: 8 + 16 + 24 + ... + 8k
right side: 4k(k + 1)
3) Proove for n = k + 1
left side: 8 + 16 + 24 + ... + 8k + 8(k+1)
right side: 4(k+1) (k + 1 + 1) = 4(k+1)(k+2)
Since 8 + 16 + 24 + ... 8k is assumed to be equal to 4k(k+1), then the left side ends as:
left side: 4k(k+1) + 8(k+1)
take common factor k + 1 → (k+1)(4k + 8)
take common factor 4 for the second parenthesis → (k+1) 4(k + 2) = 4(k+1)(k+2)
So, we have shown that the left side is equal fo the right side, which completes the proof by induction.
Related Questions
What is the number six million, thirty thousand, expressed in scientific notation? 6.03 × 10^6
6.3 × 10^5
6.3 × 10^6
6.03 × 10^4
Answers
6.3 * 10^6 = 6,300,000
Hope this helps!
So, your answer is: B) 6.3 x 10^6
Wo trains leave stations miles apart at the same time and travel toward each other. one train travels at miles per hour while the other travels at miles per hour. how long will it take for the two trains to meet?
Answers
time = distance * speed
time = (distance of x+y) (speed x+y)
Thomas bought a new backpack that was 30% off the original price. The sale resulted in a $15 discount. What was the original price of the backpack?
a.) $45
b.)$50
c.)$65
d.)$70
Answers
1% = $15 ÷ 30 = $0.50
100% = $0.50 x 100 = $50
Answer: $50 (Answer B)
Final answer:
To calculate the original price of Thomas's backpack, we set up an equation with the information that a 30% discount resulted in a $15 savings, which, when solved, indicates the original price was $50. The correct option is b.)$50
Explanation:
Thomas bought a new backpack at a 30% discount, which saved him $15. To find the original price, we can set up an equation to solve for the original price. Let's denote the original price as 'P'. The equation would therefore be 0.30 * P = $15, meaning 30% of the original price is equal to the $15 discount he received.
First, divide both sides of the equation by 0.30 to isolate 'P':
P = $15 / 0.30
P = $50
Hence, Thomas's backpack originally cost $50 before the discount was applied. Therefore, the correct answer is b.) $50.
A jar contains 5 pink and 8 navy marbles. A marble is drawn at random. What is the theoretical probability of NOT drawing pink?
Answers
8 (navy marbles) over the total amount of marbles = 0.53
0=impossible
between 0 and 0.25 =likely impossible
between 0.25 and 0.50 = less likely
0.50 = equal , even, or same
between 0.50 ans 0.75 = likely
between 0.75 and 1 = most likely
1 = certain
An equation of the line passing through (6,−3) having slope −35 is
Answers
Now we plug in the point given and solve for b.
- 3 = - 35(6) + b
- 3 = - 210 + b
207 = b
So your equation is:
y = - 35x + 207
Which of the following is the best linear approximation for f(x) = cos(x) near x= π/2
Answers
f(x) ≈ f(a) + f'(a)(x-a)
Evaluating f at π/2
f(π/2) = cos(π/2) = 0
Since f(x) = cos(x), differentiating gets us
f'(x) = -sin(x)
f'(π/2) = -sin(π/2) = -1
So the local liner approximation is
f(x) ≈ 0+ -1(x-π/2)
f(x) ≈ -x+π/2
The answer to this question is -x+π/2
The best linear approximation for f(x) = cos(x) near x = π/2 is L(x) = π/2 - x. This conclusion is reached by finding the slope of the tangent line to the function at x = π/2 (which is the cosine derivative at this point), and then applying the formula for linear approximation.
Explanation:The linear approximation of a function at a certain point is the line tangent to the function at that point. In this case, we want the linear approximation for f(x) = cos(x) near x = π/2. The slope of the tangent line at a point is equal to the derivative of the function at that point, and the cosine derivative is -sin(x). So at x = π/2, the slope is -sin(π/2) = -1.
Now, we use the formula for the linear approximation, which is L(x) = f(a) + f'(a)(x-a). In this case, a is π/2, so we get L(x) = cos(π/2) - 1(x - π/2). Since cos(π/2)=0, the best linear approximation is L(x) = 0 - 1(x - π/2) = π/2 - x.
Note:We use the
cosine derivative
to find the slope of the tangent line and the
linear approximation formula
to find the equation of the line. We also assume a
small-angle approximation
since we are close to x = π/2.
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Ten vendors are selling at an event, and an attendance of 42,000 is projected. It is anticipated that 65 percent of the attendees will make purchases. Assuming that each vendor has an equal chance of having an attendee prefer their products, what is the average number of customers each will serve?
Answers
We have then:
(42000) * (0.65) = 27300
We are now looking for the number of attendees that each vendor will attend.
We have then:
N = (27300) / (10)
N = 2730
Answer:
the average number of customers each will serve is:
N = 2730
Answer:
2,730
Step-by-step explanation:
The first term of an arithmetic sequence is -5 , and the twelfth term is 1.7 .find the common difference
Answers
a12 = 1.7
a12 = a+(n-1)d
1.7 = -5+(12-1)d
1.7+5 = 11d
6.7/11 = d
0.60 = d
You are selling cookies for your organization. One box of cookies is sold for $3. Thirty percent of the sale price goes to the bakery to pay for the cookies. The rest of the sale price is split evenly between the regional headquarters for your organization, and your local chapter. How much does your local chapter earn for each box of cookies that are sold? How many boxes of cookies do you have to sell to earn $100 for your local chapter?
Answers
(1/2)×($3 - 0.30×$3) = (1/2)×$3×0.70 = $1.05 for each box of cookies
B) If the number you must sell to earn $100 is represented by n, then you have
$1.05×n = $100
n = 100/1.05 ≈ 95.24
To earn $100 for your local chapter, you must sell 96 boxes of cookies.
Use the guess-and-check method to solve the equation: 6(x + 3) + 1 = 31
Answers
After the data was recorded, Mary gives her latest book a rating of 1 star. How will this new rating affect the mean rating of her books? The mean book rating will not change.
Answers
A fence on a hill uses vertical posts L and M to hold parallel rails N and P. If m∠1 = 6x and m∠12 = 4x, then what is the measure of ∠5?
Answers
Answer;
D: 108 degrees
Answer:
Measure of ∠5 is 108°
Step-by-step explanation:
Given m∠1 = 6x and m∠12 = 4x, we have to find the measure of ∠5.
As ∠1 and ∠5 are alternate exterior angle hence equal i.e ∠1=∠5
Also, ∠5 and ∠12 are co-interior angles ∴these are supplementary
⇒ m∠5+m∠12=180°
⇒ 6x+4x=180°
⇒ 10x = 180° ⇒ x=18
∴ m∠5 = 6x = 6(18) = 108°
Hence, measure of ∠5 is 108°
Which of the following expressions shows the number of 8 character passwords that can be formed using letters and digits if the password must begin with a letter?
a. 21x36 to the 7th power
b.26x36 to the 7th power
c.26 to the 5th power x 10 to the 3rd
d.36 to the 8th power
Answers
Answer:
26x36 to the 7th power
Step-by-step explanation:
Answer:
its B my boys!!!!!!!
Step-by-step explanation:
The ratio of boys to girls is 4:3 .T here are 35 students in a class how many students are boys ?? Pleaseee help
Answers
It takes a ship 4 hours to cover 420 km with the current and 6 hours against the current. Find the speed of the ship in still water and the speed of the current.
Answers
We know that distance equals speed multiplied by the time.
With the current
[tex]\left(s+c\right)\left(4\:hours\right)=420\:km[/tex]
or
[tex]s+c=105[/tex]
This is our first equation.
Now, against the current, we have
[tex]\left(s-c\right)\left(6\right)=420[/tex]
or
[tex]s-c=70[/tex]
This is our second equation.
We solve the equations simultaneously, by adding them together
[tex]2s-0c=175[/tex]
[tex]s=87.5\:km/hr[/tex]
Substitute s=87.5 to the first equation to solve for c.
[tex]87.5+c=105[/tex]
[tex]c=17.5\:km/hr[/tex]
The speed of the ship is 87.5 km/hr and the speed of the current is 17.5 km/hr.
The speed of the ship is 87.5 km/hr and the speed of the current is 17.5 km/hr.
What is the speed of the current?The actual propagation speed depends on the size of the wire and electrical parameters such as its inductance, but it is normally approximately 90% of the speed of light—about 270,000 km/s.
Let x be the speed of the ship, and y be the speed of the current.
It takes a ship 4 hours to cover 420 km with the current and 6 hours against the current.
[tex]\rm (x+y)4=420\\\\x+y=\dfrac{420}{4}\\\\x+y=105[/tex]
And the second equation is;
[tex]\rm (x-y)6=420\\\\x-y=\dfrac{420}{6}\\\\x-y=70[/tex]
Adding both the equation
[tex]\rm x+y+x-y=105+70\\\\2x=175\\\\x=\dfrac{175}{2}\\\\x=87.5[/tex]
Substitute the value of x in equation 1
[tex]\rm x+y=105\\\\87.5 +y=105\\\\y=105-87.5 \\\\y= 17.5[/tex]
Hence, the speed of the ship is 87.5 km/hr and the speed of the current is 17.5 km/hr.
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If the factors of a polynomial are x + 7 and x – 3, what values of x make that polynomial 0?
A. 7 and 3
B. –7 and –3
C. 7 and –3
D. –7 and 3
Answers
Apply the zero product property:
(x + 7) = 0 or (x - 3) = 0
x = - 7 or x = 3
Answer: x = - 7 or x = 3
The values of x make that polynomial 0 is x=-7 and x=3. Therefore, option D is the correct answer.
Given that, the factors of a polynomial are x + 7 and x – 3.
We need to find what values of x make that polynomial 0.
What is polynomial?An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Now, x+7=0 and x-3=0
⇒x=-7 and x=3
The values of x make that polynomial 0 is x=-7 and x=3. Therefore, option D is the correct answer.
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Jim surveys a random sample of 150 students from his middle school to learn what type of school lunch is preferred. The table below shows the result of the survey: Lunch Type Sandwich Hot Dog Salad Number of students 70 69 11 Which of the following inferences can be made based on the data? (5 points) A hot dog is two times more popular than salad. A salad is two times more popular than sandwich. Sandwiches and hot dogs are almost equally preferred. A hot dog is the most preferred lunch.
Answers
Answer:
Sandwiches and hot dogs are almost equally preferred.
Step-by-step explanation:
Given data is :
Lunch Type Sandwich Hot Dog Salad
Number of students 70 69 11
The total students are = [tex]70+69+11=150[/tex]
The following inferences can be made:
A hot dog is two times more popular than salad. False. It is more than 5 times popular.
A salad is two times more popular than sandwich. False. Salad is only 11 and sandwiches are 70.
Sandwiches and hot dogs are almost equally preferred. True. Both have almost same number.
A hot dog is the most preferred lunch. False. Sandwich is most preferred.
Therefore, the correct answer is - Sandwiches and hot dogs are almost equally preferred.
Help please!!! I would appreciate it.
Answers
Explanation:
In this case you need to find the area under the curve, which would essentially be the probability of a value in the sample space.
Now the range given is from 60 to 80; it contains the mini-triangle.
The area of the triangle = (1/2)*base*height
Since base = 80 - 60 = 20
Height = 0.025 - 0 = 0.025 (see in the graph)
Therefore
The area of the triangle = 1/4 = 0.25 which is essentially be the probability.
In percentage form, it is 25% (Option D)
Please help asap!!!!!!!!!!!!!
Answers
What is the total amount and the amount of interest earned on $6,500 at 6% for 25 years?
Answers
Final answer:
The total amount on a $6,500 investment at a 6% annual interest rate compounded yearly for 25 years is approximately $28,353.25, with the interest earned being about $21,853.25.
Explanation:
To calculate the total amount and the amount of interest earned on $6,500 at 6% for 25 years, we need to identify whether the interest is compounded annually, monthly, or if it's simple interest. Since the type of interest isn't specified, we'll assume it's compounded annually, which is common in savings accounts. The formula for compound interest is:
A = P(1 + r/n)nt
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount ($6,500)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the time the money is invested or borrowed for, in years (25 years)
For this example, since the interest is compounded annually, n = 1. First, convert the interest rate from a percentage to a decimal by dividing by 100: 6% / 100 = 0.06. Now, plug the values into the formula:
A = $6,500(1 + 0.06/1)1*25
A = $6,500(1 + 0.06)25
A = $6,500(1.06)25
Calculating this out, A, which is the total future amount, comes out to:
A = $28,353.25 approximately
To find out the total interest earned, subtract the principal from the final amount:
Total interest = A - P
Total interest = $28,353.25 - $6,500
Total interest = $21,853.25 approximately
The total amount is about $28,353.25 and the amount of interest earned is about $21,853.25.
Complete the steps in order to write the inverse of f(x) = x + 5 1. = x + 5 2. = y + 5 3. = y 4. = x – 5
Answers
f(x) = x + 5
1. y = x + 5
Switch places from x and y.
2. x = y + 5
Subtract 5.
3. x - 5 = y
Switch sides.
4. y = x – 5
Now we've found our answer.
~ Hope this helps you!
The inverse of the given function y =x+5 is y =x-5.
The given function is:
y =x+5
What is an invertible function?
A function is invertible only if each input has a unique output and each element of the codomain should be mapped.
Step1: check function is invertible i.e. bijective i.e. one-one and onto or not
f(x₁)=x₁+5
f(x₂) = x₂+5
f(x₁) = f(x₂)
x₁+5 =x₁+5
x₁=x₂
So, the function is one-one or injective.
For onto range and codomain should be same.
Step 2: interchange x and y
x=y+5
Step3: bring 5 to the left side of the equation.
x-5 = y
Step4: write in y=f(x) form
y = x-5
Therefore, the inverse of the given function y =x+5 is y =x-5.
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If you round off 23 to nearest ten.what is the answer
Answers
What is the smallest integer greater than the square root of 148?
A.)13
B.)15
C.)14
D.)12
E.)11
Answers
The square root of 148 is 12.1655250606.
Now with the options, you provided which are 13, 15, 14, 12, and 11, we can represent our known numbers on a number line.
----- | ----- | ----- | -*---- | ----- | ----- | ----- | -----
10 11 12 13 14 15 16
* = 12.1655250606
Now we can use the process of elimination to determine the smallest integer greater than the square root of 148.
15 is obviously too far to be the smallest integer greater than the square root of 148, so that is not the correct answer.
14 is pretty close to 12.1655250606, but we have three other options that are even closer so that cannot be the correct answer.
11 is also incredibly close to our square root, but there are two other options that are even closer. Therefore 11 will not be our answer.
Now that we have eliminated all other options but two of them, we have made it easy to determine which is correct.
12 is closer to 12.1655250606 than the number 13, so this would be the correct answer if the question was asking for the smallest integer of the square root of 148. But it is not.
The question asks for the smallest integer GREATER THAN the square root of 148. Therefore the correct answer to this question is 13.
QUICK ANSWER:
The smallest integer greater than the square root of 148 is 13.
The square root of 148 is 12.16. The smallest integer greater than the square root of 148 is 13.
The given number is 148.
What is the square root?The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.
The square root of the given number is
√148 = 12.16
12<12.16>13
The square root of 148 is 12.16. The smallest integer greater than the square root of 148 is 13.
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A square sheet of paper is folded diagonally. What is the length of LN?
A) 10 in.
B) 20 in.
C) 10√2
D) 5√4
Answers
Length of side of the paper is 10 in
LN is the diagonal of the square
Answer:
C) 10√2 in
Explanation:
A square has 4 equal sides. This means that all four sides of the square are equal to 10 in.
Now, folding a square diagonally will lead to the formation of two congruent right-angled triangles (check the attachment).
We need to get the length of LN which is the hypotenuse of the right-angled triangle.
This means that we can apply the Pythagorean theorem:
(hypotenuse)² = (side1)² + (side2)²
(LN)² = (10)² + (10)²
(LN)² = 200
LN = √200
LN = 10√2 in
Hope this helps :)
The amounts of 8 charitable contributions are $100, $80, $250, $100, $500, $1000, $100, and $150.
The mean, median, and mode of the amounts are given below. mean = $285 median = $125 mode = $100
Which value describes the amount received most often?
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Use the formula aequals upper p left parenthesis 1 plus r right parenthesis superscript t to find the rate r at which $3000 compounded annually grows to $4320 in 2 years.
Answers
The answer is 7,323 because i dont know i asked my friend
The weekly fee for staying at the Pleasant Lake Campground is $20 per vehicle and $10 per person. Last year, weekly fees were paid for v vehicles and p persons. Which of the following expressions gives the total amount, in dollars, collected for weekly fees last year?
Answers
v: number of vehicles
p: number of people
Writing the equation of the weekly rates we have:
20v + 10p
Answer:
An expressions that gives the total amount, in dollars, collected for weekly fees last year is:
20v + 10p
When figuring out how much concrete is needed to build a pool what do you multiply the price of the concrete by?
Answers
Multiply the price ($/ft²) by the surface area to calculate the total cost.
Give the next 3 numbers in the sequence: 3, 7, 11, 15, ___, ___, ___
Answers
7 + 4 = 11
11 + 4 = 15
So from the pattern here, we need to add 4 to term to get the next term.
So the remaining three terms would be
15 + 4 = 19
19 + 4 = 23
23 + 4 = 27
Hope this helps.
do banks always charge the same interest rate on new and used cars
Answers
even a house home everything yeah cars espically if your taking out a loan or that type of thing yeah your gonna have a big hard time with i would try this app offer up for a car its really cheap
Banks generally do not charge the same interest rate on new and used cars. Factors including the type of car, borrower's credit history, length of the loan, and market conditions can affect the interest rate. Usually, used cars have higher interest rates due to higher perceived risk.
Explanation:No, banks do not always charge the same interest rate on new and used cars. Interest rates are based on numerous factors that include the type of car (new or used), the borrower's credit history, the length of the loan, and the existing market conditions. Typically, used cars have higher interest rates because they are viewed by the bank as more risky investments. Therefore, individuals with excellent credit may secure lower interest rates on both new and used car loans, but the rate for used cars will still generally be higher than for new cars.
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