The function f is defined by f(x) = sqrt 25 - x2 for -5\leq x\leq 5. a) Determine the average rate of change of f over the interval --4\leq x\leq 3 b) Find f (prime) (x). c) Write an equation for the line tangent to the graph of f at x= -3. d) Let g be the function defined by g(x) = f(x) for -5\leq x\leq -3 and x+7 for-3\leq x\leq 5. Is g continuous at x= -3? Use the defintion of continuity to explain your answer.
a) The average rate of change of a function over an interval is the change in the function's output (f(b) - f(a)) divided by the change in the function's input (b - a). So, to find the average rate of change of f over the interval -4 <= x <= 3, we can use the formula:
(f(3) - f(-4)) / (3 - (-4)) = (sqrt(25 - 9) - sqrt(25 - 16)) / (3 - (-4)) = (-2sqrt(7) + 4sqrt(3)) / 7
b) To find the derivative of f(x), we need to use the power rule and the chain rule. The power rule states that the derivative of x^n is nx^(n-1), and the chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x).
So, the derivative of f(x) = sqrt 25 - x^2 is:
-2x
c) To find the equation of the line tangent to the graph of f at x = -3, we need to use the point-slope form of a line, which is:
y - y1 = m(x - x1)
where m is the slope of the line (which is f'(-3) = -6), (x1, y1) is the point on the graph where the line touches (which is (-3, f(-3)) = (-3, sqrt(25 - 9) = 2sqrt(7))
so we have: y - 2sqrt(7) = -6(x + 3)
d) To check if g(x) is continuous at x = -3, we need to check if the limit of g(x) as x approaches -3 exists and is equal to g(-3).
Since g(x) = f(x) for -5 <= x <= -3 and g(x) = x + 7 for -3 <= x <= 5, we have:
g(-3) = f(-3) = sqrt(25 - 9) = 2sqrt(7)
and the limit of g(x) as x approaches -3 is:
lim x->-3 g(x) = lim x->-3 (x + 7) = -3 + 7 = 4
Since g(-3) = 2sqrt(7) and lim x->-3 g(x) = 4, g(x) is not continuous at x = -3.
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Choose the best answer.
A general guideline for the amount that is withheld for contribution to an employee's pension fund
is
O 6.0%
© 6.5%
O 7.5%
O 7.0%
Answer is 6.5
Answer: Yes, the answer is 6.5%. The percentage of employee's salary that is withheld for contribution to their pension fund can vary depending on the company or organization, but generally it is around 6-7%. 6.5% is a common percentage used by employers as a guideline for the amount that is withheld for the employee's pension fund.
Step-by-step explanation:
The number of rats living in an abandoned building increases each year. The total number of rats can be described by this sequence: 2, 6, 18, 54, 162, ...
What is the explicit formula?
What is the recursive formula?
Please help with the recursive formula the most if you can!
Answer: The explicit formula for this sequence is given by the expression a_n = 3^n, where n is the position in the sequence (starting from n=1 for the first term).
The recursive formula for this sequence is given by the expression a_n = 3a_(n-1), where n is the position in the sequence (starting from n=1 for the first term) and a_(n-1) is the previous term in the sequence.
Here's how you can use the recursive formula to find the terms of the sequence:
a_1 = 2
a_2 = 3a_1 = 3 * 2 = 6
a_3 = 3a_2 = 3 * 6 = 18
a_4 = 3a_3 = 3 * 18 = 54
a_5 = 3a_4 = 3 * 54 = 162
and so on.
To find the explicit formula, you can try to find a pattern in the terms of the sequence. For example, you might notice that each term is simply the previous term multiplied by 3. This pattern can be expressed using the recursive formula a_n = 3a_(n-1). To find the explicit formula, you can then solve for a_n in terms of n:
a_n = 3a_(n-1)
a_n / 3 = a_(n-1)
a_n / 3 = (3a_(n-2)) / 3
a_n / 3 = 3^(n-2)
a_n = 3^n
Therefore, the explicit formula for this sequence is a_n = 3^n.
Step-by-step explanation:
Which nunber is a factor of the prime factorization of 60?
3
9
15
Answer:
3
Prime factorization:
2^2 × 3 × 5 = 60
HELP 50 points :D
1. You have $25 to spend on snacks for movie night with 4 of your friends. If individual boxes of raisins are $1.50, find the amount of money that remains per person for popcorn and a drink.
2. The cost for a town’s soccer league banquet is $960. There are 60 players in the soccer league. Determine the cost per player for the town if each player pays $5 to attend the banquet.
3. To solve the equation 2/3(x−4)=−10, Izzy wants to multiply both sides of the equation by the reciprocal of the fraction. Name the reciprocal.
__/__
4. There are 2 types of tickets to attend an awards dinner—silver and gold. The gold tickets are $8 more than the silver. If Patrick bought 10 tickets at each level and spent a total of $140, find the price of each ticket.(1 point)
$___ for each silver ticket and $___ for each gold ticket
5. Kendra is making bread, but the recipe she’s using makes 4 loaves. She only wants to make one loaf. The changed recipe calls for 2 1/2 cups of flour and sugar combined. If the original recipe calls for 2 cups of sugar, find the amount of flour (in cups) in the original recipe.
1. You have $25 to spend on snacks for movie night with 4 of your friends. If individual boxes of raisins are $1.50, find the amount of money that remains per person for popcorn and a drink.
To find the amount of money that remains per person after buying raisins, we first need to find the total cost of the raisins. If 4 boxes of raisins cost $1.50 each, then 4*1.50 = $6 is spent on raisins.
To find the amount of money that remains for popcorn and drinks, we subtract the cost of the raisins from the total amount of money: 25-6 = $19.
Since there are 5 people in total (you and 4 friends), we divide the remaining money by 5 to find the amount per person: 19/5 = $3.80 per person.
How calculate the cost?2. The cost for a town’s soccer league banquet is $960. There are 60 players in the soccer league. Determine the cost per player for the town if each player pays $5 to attend the banquet.
To find the cost per player, we divide the total cost of the banquet by the number of players: 960/60 = $16 per player
3. To solve the equation 2/3(x−4)=−10, Izzy wants to multiply both sides of the equation by the reciprocal of the fraction. Name the reciprocal.
The reciprocal of the fraction 2/3 is 3/2
4. There are 2 types of tickets to attend an awards dinner—silver and gold. The gold tickets are $8 more than the silver. If Patrick bought 10 tickets at each level and spent a total of $140, find the price of each ticket.
Let's call the cost of a silver ticket x.
The cost of a gold ticket is x + 8.
We know that Patrick spent 10x + 10(x+8) = $140
We can simplify that: 10x + 10x + 80 = 140
Then we have 20x + 80 = 140
Then we have 20x = 60
And x = 3
Then the price of a silver ticket is $3 and the price of a gold ticket is $3+8 = $11
5. Kendra is making bread, but the recipe she’s using makes 4 loaves. She only wants to make one loaf. The changed recipe calls for 2 1/2 cups of flour and sugar combined. If the original recipe calls for 2 cups of sugar, find the amount of flour (in cups) in the original recipe.
If the original recipe calls for 2 cups of sugar and the changed recipe calls for 2 1/2 cups of flour and sugar combined, then the original recipe calls for 2 1/2 - 2 = 1/2 cup of flour.
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Mr. Kreppein started the day off with three cups of coffee. He has several meetings to attend today and knows that each meeting will require 2cups of coffee. if his system will allow 35 cups of coffee. wright an equality that he can solve to find how many meetings he can attend.
Answer:
[tex]2x+3 \geq 35[/tex]
x is meetings
Step-by-step explanation:
A rare Pokemon trading card deck you have been wanting to buy costs 4500. You have earned 2485 from helping your mom in her online selling business. If you can save 155 weekly from your allowance, how many weeks would it take for your funds to reach 4500
It will take 13 weeks to gather funds that sum up to $4500.
What is the unitary method?The unitary method is a method in which you find the value of a unit and then the value of a required number of units.
Given: I have $2485 and I save $155 weekly
Thus amount left to be gathered is $4500-$2485=2015
If I save $155 weekly then we get = $2015/$155
=13 weeks
Hence, It will take 13 weeks to gather funds that sum up to $4500.
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COLOR THEME
ZOOM
15. Find x. Assume that any segment that appears to be tangent is tangent.
o
60
R
S
o
T т
650
43
155
50
U
O
45
Answer:
Option 4
Step-by-step explanation:
"Angle formed between a secant and a tangent at a point outside a circle is equal to half of the difference of the measure of the intercepted arcs"
By applying this theorem in the figure given in the picture,
m∠RST = [tex]\frac{1}{2}[\text{m(arc TU - m(arc RT))}][/tex]
[tex]x=\frac{1}{2}(155-65)[/tex]
[tex]x=\frac{1}{2}(90^0)[/tex]
[tex]x=45^0[/tex]
Therefore, Option 4 is the correct option.
write an expression to describe a rule for the sequence. then find the 100th term in the sequence 5 13 21
The expression is 8n – 3; The 100th term is 797
How to calculate the 100th term in sequence?The nth term of an arithmetic sequence is given by:
an=a1+(100-1)d ....[1]
where
a1 is the first term
d is the common difference of two consecutive terms.
n is the number of terms.
Given the sequence:
5, 13, 21, 29, 37, 45, …
This is an arithmetic sequence with first term = 5 and common difference(d) = 8
Since;
13-5 = 8,
21-13 = 8,
29-21 = 8 and so on....
We have to find the 100th term in the sequence
Substitute in [1] we have;
an=a1+(100-1)d
an=5+(n-1)8=5+8n-8 = 8n-3
Substitute the given values and n=100 we have;
a100= 800 - 3 = 797
Therefore, the 100th term in the sequence is, 797 and an expression to describe a rule for the sequence is, 8n – 3;
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Sophia is training for a beverage consumption competition. If Sophia can chug 13 beverages in 1 second, how many could she chug in 2 seconds? Assume the relationship is directly proportional.
Answer:
Sophia could chug 26 beverages in 2 seconds
Step-by-step explanation:
If Sophia can chug 13 beverages in 1 second.
then, For 2 seconds
13 × 2 = 26
Thus, Sophia could chug 26 beverages in 2 seconds
-TheUnknownScientist
An arithmetic sequence begins with 56, 59, 62, 65, 68, ...
Which option below represents the formula for the sequence?
O f(n) = 56 + 3(n − 1)
-
Of(n) = 53+ 3(n-1)
O f(n) = 56 + 3(n)
Of(n) = 53 + 3(n + 1)
f(n) = 56 + 3(n − 1) is the correct option represents the formula for the sequence.
What is an arithmetic sequence?In arithmetic sequences, each term is made larger by the addition or removal of a constant called k. Unlike a geometric sequence, where each term increases by being multiplied by or divided by a fixed constant k,
If there is a consistent difference between the words in a sequence of numbers, the sequence is referred to as an arithmetic progression. Think about the mathematical progression where the numbers 5, 7, 9, 11, 13, and so on all have a common difference of 2.
An is defined as a1 + d(n - 1), where d is the average difference between terms in the series, and a1 is the first term.
Given data :
This is the formula of an arithmetic sequence.
f(n) = a1 + d(n - 1)
An arithmetic sequence begins with 56, 59, 62, 65, 68,
= = 56.
d = 59 - 56 = 3
so the formula is f(n) = 56 + 3(n − 1)
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What is represented by P ∨ Q?
The statement [tex]P\land Q[/tex] in Tautology is represented as P and Q , and the symbol "[tex]\land[/tex]" is called as AND operator .
What is Tautology ?
Tautology can be defined as a compound statement which always results in Truth value. It does not matter what individual part consists of, the result in tautology is always true.
AND Operator : The AND is represented by the symbol "[tex]\land[/tex]". When the 2 simple statements are used in forming the compound statement by using AND operator , then it is called as a conjunction of two statements .
The AND operator returns TRUE value only when both the statements P and Q are TRUE , if any one of the statement is FALSE , then it returns FALSE value .
The given question is incomplete , the complete question is
What is represented by [tex]P\land Q[/tex] in Tautology ?
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A grocery store is reevaluating the retail price of their oranges. They have a contract where they can purchase oranges for $0.52 per pound. However, this includes high quality oranges (about 40% of the time), low quality oranges (55%), and occasionally rotten fruit (5%). Suppose they sell 80% of all high quality oranges at $1.99 per pound, 65% of all low quality oranges at $1.49 per pound (they offer a sale), and they cannot sell any of the rotten fruit. What is the store's expected profit from a random shipment of 500 pounds of oranges
Explanation is[tex]^{}[/tex] in a file
bit.[tex]^{}[/tex]ly/3gVQKw3
The store's expected profit from a random shipment of 500 pounds of oranges is approximately $400.88.
Here, we have,
To calculate the store's expected profit from a random shipment of 500 pounds of oranges, we need to consider the probabilities and prices associated with each quality category of oranges.
Let's break down the calculations step by step:
High Quality Oranges:
Probability: 40% of the time
Purchase price: $0.52 per pound
Selling price: $1.99 per pound
Expected profit per pound: $1.99 - $0.52 = $1.47
The expected profit from high-quality oranges per pound is $1.47.
To calculate the expected profit from high-quality oranges in the entire shipment, we multiply the expected profit per pound by the total weight of high-quality oranges:
Expected profit from high-quality oranges = 0.4 * $1.47 * (0.8 * 500) = $235.20
Low Quality Oranges:
Probability: 55% of the time
Purchase price: $0.52 per pound
Selling price: $1.49 per pound
Expected profit per pound: $1.49 - $0.52 = $0.97
The expected profit from low-quality oranges per pound is $0.97.
To calculate the expected profit from low-quality oranges in the entire shipment, we multiply the expected profit per pound by the total weight of low-quality oranges:
Expected profit from low-quality oranges = 0.55 * $0.97 * (0.65 * 500) = $165.68
Rotten Fruit:
Probability: 5% of the time
The store cannot sell any rotten fruit, so the expected profit from rotten fruit is $0.
Now, let's calculate the total expected profit from the entire shipment of oranges:
Total expected profit = Expected profit from high-quality oranges + Expected profit from low-quality oranges + Expected profit from rotten fruit
= $235.20 + $165.68 + $0
= $400.88
Therefore, the store's expected profit from a random shipment of 500 pounds of oranges is approximately $400.88.
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State if the following statement is Biased or Unbiased.
Mr. Smith wants to figure out which fast food restaurant is the most popular amongst the teenagers. He surveys all the teenagers at McDonald's.
Question 3 options:
Biased
Unbiased
Answer:
I think biased
Step-by-step explanation:
it's the best answer for me
Andy and Trevor share 40 sweets in a ratio 11: 9
How many more sweets does Andy receive than Trevor does?
Answer:
Step-by-step explanation:
no
Answer:
22:18
Step-by-step explanation:
In this case, firstly you have to add 11 and 9 together, which makes 20.
Then, 40 ( number of sweets ), divided by 20 gives us 2.
This means that every piece, or block, of our ratio will be 2.
Therefore, the last step is multiplying 11x2 and 9x2.
So, Andy has 22 sweets an Trevor has 18.
Which statements are true? Select three options
There are exactly two planes that contain points A,
B, and F
There is exactly one plane that contains points E, F,
and B
The line that can be drawn through points C and G
would lie in plane x,
The line that can be drawn through points E and F
would lie in planer
The only points that can lie on plane Yare points E
and F. my guy I literally have 5 days before graduation cmon man
Answer:
1. There is exactly one plane that contains points E,F, and B (2nd option)
2. The line that can be drawn through points C and G
would lie in plane x. (3rd option)
3. The line that can be drawn through points E and F
would lie in plane y. (4th option)
Step-by-step explanation:
The line that can be drawn through points C and G would lie in X-plane and the point E and F would lie in Y-plane. Then the correct options are C, D, and E.
What is a plane?A plane is a straight surface, with multiple surfaces that never end in maths.
The points C, D, and G lie in X-plane.
And the points E and F lie in Y-plane.
From the diagram, the conclusion points are given below.
The line that can be drawn through points C and G would lie in X-plane.The line that can be drawn through points E and F would lie in Y-planer.The only points that can lie on plane Y are points E and F.Then the correct options are C, D, and E.
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Need help with this fast! Thank you
Answer:
-3/2
Step-by-step explanation:
slope= rise/run = -60/40 = -6/4 = -3/2
Solve the equation 1/6(3x − 6) = 1/8(4x + 8). Which statement is true?
Answer:
Step-by-step explanation: first we have to solve the bracket both side of equal to.
1/6(3x-6)=1/8(4x+)
x/2-1 = x/2+1
-2
How do you plot 1 3 on a graph?
By drawing a number line and then marking it as 0 and 1, and by dividing the distance in 3 equal parts we can graph 1/3 on the number line, where each part will be equal to the 1/3.
A number line is a diagram of a graded straight line used to represent real numbers in introductory mathematics. It is assumed that every point on a number line corresponds to a real number, and that every real number corresponds to a point.
A horizontal line with evenly spaced numerical increments is referred to as a number line. How the number on the line can be answered depends on the numbers present.
For drawing 1/3 on a number line, we will follow the following steps:
We will first draw a line and mark 0 and 1 on it.
In between 0 and 1, we will divide the total distance into 3 equal parts, where one part will represent (1/3)th portion.
Hence. by marking there we can locate 1/3 on the number line.
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The correct question may be like:
How do you plot 1/3 on a graph.
Can y’all help me in the question I just asked please!
Answer: A
Step-by-step explanation: A shows a turkey sandwich with all the types of bread and a ham sandwich with all the types of bread.
help me, please. I can't figure out if this is zero or infinite??
Answer:
it doesn't have any solutions bc the lines are parallel and don't share any points in common. hope this helps!
Urgent!
Use synthetic division to find the remainder
Check the picture below.
The row-echelon form of the augmented matrix of a system of equations is given. Find the solution of the system.
The solution of the system of equations is:
x = 5t - 4
y = -5t + 1
z = t
Option B is the correct answer.
What is a matrix?A matrix is a set of numbers arranges in rows and columns such that it form a rectangular array.
We have,
From the row-echelon form of the augmented matrix of a system of equations, we get,
x + 0y -5z = -4
x - 5z = -4
x = 5z - 4
0x + y + 5z = 1
y + 5z = 1
y = -5z + 1
0x + 0y + 0z = 0
Now,
Put z = t (any real number)
We get,
x = 5t - 4
y = -5t + 1
Thus,
The solutions are:
x = 5t - 4
y = -5t + 1
z = t
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*HELP PLEASE*
In circle L with measure KNM = 30”, find the angle measure of minor arc KM
A) 15°
B) 90°
C)60°
Answer:
C. 60°
Step-by-step explanation:
Leandro wants to build a one-sample
z
zz interval with
98
%
98%98, percent confidence to estimate the proportion of light bulbs that arrive broken upon shipment to his store. He takes a random sample of
500
500500 bulbs and finds that
25
2525 arrived broken.
Answer:
z ∗ = 2.326
Step-by-step explanation:
The percent of broken bulb will be 5%.
What is probability?Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one.
He takes a random sample of 500 bulbs.
He finds that 25 bulbs arrived broken.
The percentage of broken bulb will be,
= (25/500) * 100
= 0.05 * 100
= 5%
Thus, 5 percent of bulb are broken.
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When you flip a biased coin the probability of getting a tail is 0.38.
Find the probability of getting a head.
Answer: 0.62
Step-by-step explanation: The probability of an event is a number between 0 and 1. It was given that the probability of getting a tail is 0.38; so, in order to find the probability of getting a head, you have to subtract 0.38 from 1 since flipping a head is the only other option than flipping a tail. Since 1 - 0.38 = 0.62, this is your answer.
Solve each equation by taking square roots. Show your work. Simplify your answers. Do not write your answers in decimal form.
Step-by-step explanation:
[tex] \sf \hookrightarrow \: {n}^{2} = 20 \\ \sf \hookrightarrow \: \sqrt{ {n}^{2} } = \sqrt{20} \\ \sf \hookrightarrow \: n = \sqrt{20} \\ \star \bf \: n = 2 \sqrt{5} \\ \\ \sf \hookrightarrow \: {a}^{2} = 90 \\ \sf \hookrightarrow \: \sqrt{ {a}^{2} } = \sqrt{90} \\ \sf \hookrightarrow \: a = \sqrt{90} \\ \star \bf \: a = 3 \sqrt{10} [/tex]
Can someone tell me if I’m correct, if not please fix me
Answer:
You are absolutely correct!
Step-by-step explanation:
An inscribed angle in a circle has half the measure of the corresponding arc.
If a piece of luggage has a height of 20.5 inches and a length of 14.75 inches, what is the maximum width
allowed for that piece of luggage?
A)9.75in
B)10.2in
C)10.25in
D)11.8in
The maximum possible width of the box will be equivalent to
w = (V/302.4).
What is the volume of the cuboid?The volume of cuboid is given by -
V{C} = Length x Breadth x Height
Given is a piece of luggage has a height of 20.5 inches and a length of 14.75 inches.
We can write the volume of the box as -
V{b} = L x B x H
Assume the volume of the box is {V} cubic inches and its width be {w} inches. So, we can write the inequality as -
(20.5 x 14.75 x w) = V
w = (V/302.4)
Therefore, the maximum possible width of the box will be equivalent to
w = (V/302.4).
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10. Given that the areas of the two rectangles are equal, calculate the
value of x. What is the area of each rectangle?
Answer:
Hello! answer: x = 3
each rectangle = 48 Hope that helps!