Step-by-step explanation:
The limit of (sin(x)) / x as x approaches 0 is a well-known limit in calculus, known as the limit definition of the derivative of the sine function.
lim (x -> 0) (sin(x)) / x = 1
This limit can be proven using L'Hopital's rule or by considering the Taylor series expansion of the sine function about x = 0:
sin(x) = x - (x^3) / 6 + (x^5) / 120 + ...
As x approaches 0, the higher order terms in the Taylor series become arbitrarily small, and the expression (sin(x)) / x approaches 1. This shows that the limit of (sin(x)) / x as x approaches 0 is equal to 1.
Kelly can run 1/2 mile in 3 minutes. How far does
Kelly run in 1 minute if he runs at the same pace the
entire time?
Kelly can run 1/2 mile in 3 minutes. 1/6 mile Kelly run in 1 minute if he runs at the same pace the entire time.
What makes anything a mile?The word "mile" is derived from the Latin "mille passus," which means "one thousand paces." The English Parliament established the Mile at eight furlongs (one furlong is 660 feet) in 1592. A mile is a unit of measurement that is standardised as exactly 1609.344 metres by the International agreement in 1959. A mile is exactly equivalent to 5280 feet, or 1760 yards.
Kelly can run 1/2 mile in 3 minutes so the ratio is 1/2 mile per 3 minutes
= [tex]\frac{\frac{1}{2} }{3}[/tex]
For transforming 3 to 1 we need to divide by 3 because we have to keep the ratio value, we have to divide also the numerator by 3 :
= [tex]\frac{\frac{1}{2} }{3} \times \frac{\frac{1}{3} }{\frac{1}{3}}[/tex]
[tex]= \frac{\frac{1}{2} \times \frac{1}{3} }{3 \times \frac{1}{3} }[/tex]
= 1/6 / 1
= 1/6
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Solve the formula for the indicated variable.
Z=m+mwg for m
m=
(Simplify your answer.)
The value for variable m in equation Z = m + mwg is m = Z - mwg.
What are variables?
The alphabetic character that expresses a numerical value or a number is known as a variable in mathematics. A variable is used to represent an unknown quantity in algebraic equations.
The equation is given as -
Z = m + mwg
To get the value for variable m, subtract mwg from Z and take it to left-hand side.
Z - mwg = m
Reverse the equation and get it in right form -
The equation obtained is -
m = Z - mwg
Therefore, the value is m = Z - mwg .
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A business development manager discusses international steel production and economic development at a conference. He passes out this graphic and asks for comments. Which comment about the data is most accurate? Japan’s production has increased and shows no signs of decreasing. China’s steel production is increasing while its major competitors’ production is declining. By 2009, China’s tons of steel produced per US million ($) GDP was 50, about 10 tons less than Japan’s all-time high. At 50 tons of steel produced per US million ($) GDP, China has the highest steel production of all producers shown as of 2009.
The most accurate comment about the graph data is: "China's steel production is increasing while its major competitors' production is declining."
A graph is what?Academics are using graphs to logically chart or graphically depict events or quantities. A graph point is a common way to depict how different things interact. The non-linear storage structure known as a graph is made up of nodes, also referred to as vertices, and lines. Glue ought to be used to connect the edges, also known as nodes , respectively. (4.5).
Here,
The most accurate comment about the data is: "China's steel production is increasing while its major competitors' production is declining."
This is because the graph shows that China's steel production has been steadily increasing since the early 2000s, while the steel production of other major producers such as Japan, the United States, and Germany has been declining or stagnant. This trend is particularly evident after the global financial crisis of 2008-2009.
While it is true that Japan's steel production has increased and shows no signs of decreasing, this comment does not address the overall trend of declining steel production among major producers.
The other two comments are not accurate. The statement "By 2009, China’s tons of steel produced per US million ($) GDP was 50, about 10 tons less than Japan’s all-time high" is not supported by the graph, as there is no information about tons of steel produced per US million ($) GDP on the graph.
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Prove that the value of the expression (3/5x^2-0.4cy-1.5y+1)-(y^2-2/5xy+0.6x^2) does not depend on x
Therefore, we have proved that the value of the expression
(3/5x²-0.4cy-1.5y+1)-(y²-2/5xy+0.6x²) doesn't depend on x.
Expression: What is it?In mathematics, addition, multiplication, and division are necessary. They result in the following when combined: An equation, some data, and a mathematical function Values, parameters, and mathematical operations like additions, subtractions, multiplications, and divisions can all be found in a declaration of fact. Different phrases and words can be compared and evaluated.
Here,
To prove that the value of the expression doesn't depend on x, we need to show that the x-term disappears when we simplify the expression.
Starting with:
(3/5x² - 0.4cy - 1.5y + 1) - (y² - 2/5xy + 0.6x²)
Distributing the negative sign in the second set of parentheses:
3/5x² - 0.4cy - 1.5y + 1 - y² + 2/5xy - 0.6x²
Rearranging the terms by collecting like terms, we get:
(3/5 - 0.6)x² + (2/5y - 0.4c)y - y²- 1.5y + 1
Simplifying further, we get:
(-0.1)x² + (0.4y - 0.4c)y - (y² + 1.5y - 1)
Notice that there is no x-term in the final expression, which means that the value of the expression doesn't depend on x.
Therefore, we have proved that the value of the expression (3/5x²-0.4cy-1.5y+1)-(y²-2/5xy+0.6x²) doesn't depend on x.
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Richard and Teo have a combined age of 36. Richard is 15 years older than twice Teo's age. How old are Richard and Teo?
Teo is 10.5 years old and Richard is 25.5 years old.
How old are Richard and Teo?Represent Teo's age with x.
Using the above as a guide, we have the following:
Richard = x + 15
Their combined age is 36
So, we have the following representation
x + (x + 15) = 36
Evaluate the like terms
2x + 15 = 36
2x = 21
x = 10.5
Recall that
Richard = x + 15
So, we have
Richard = 10.5 + 15
Richard = 25.5
So Teo is 10.5 years old
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Mark and his three friends ate dinner out last night. Their bill totaled $52.35, the tax was 8.5% and they left their server an 18% tip. If they split the bill evenly, how much did each person pay? Round to the nearest cent.
Answer:
$56.84
Step-by-step explanation:
52.35 + 8.5% + 18% ÷ 4 = $56.84475
Lourdes can buy 3-packs of paper for $10.50, or she can buy 4-packs of paper for $13.50. She wants to buy 12 packs of paper. How much more will she spend if she buys only 3-packs of paper
Lourdes will spend an amount of $1.50 more if she buys only 3-packs of paper.
What are Arithmetic operations?Arithmetic operations can also be specified by adding, subtracting, dividing, and multiplying built-in functions.
If Lourdes buys only 3-packs of paper, she will need to buy 12 / 3 = 4 sets of paper.
At $10.50 per set, this will cost her 4 x $10.50 = $42.
If Lourdes buys only 4-packs of paper, she will need to buy 12 / 4 = 3 sets of paper.
At $13.50 per set, this will cost her 3 x $13.50 = $40.50.
So, Lourdes will spend $42 - $40.50 = $1.50 more if she buys only 3-packs of paper.
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100-40+10-100-20+20-+10
Answer:
-40
Step-by-step explanation:
100-40+10-100-20+20-(+10)
=60+10-100-20+20-10
=70-100-20+20-10
=-30-20+20-10
=-50+20-10
=-30-10
=-40
Why is [tex]0,3^{-1}[/tex] = [tex]\frac{10}{3}[/tex] ?
By the inverse of a number, 0.3⁻¹ = 10/3
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ 0.3⁻¹ = 10/3
Now, We can simplify as;
⇒ 0.3⁻¹
⇒ (3/10)⁻¹
⇒ 10/3
Thus, By the inverse of a number, 0.3⁻¹ = 10/3
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What is the inverse of a function?
A function for which the domain remains the same, and reciprocal values of the range are used
A function for which the values or the domain and the range are interchanged
A function for which the range remains the same, and reciprocal values of the domain are used
The requried, inverse of a function is a function for which the values or the domain and the range are interchanged. Option C is correct.
What is an inverse function?The inverse of a function is a new function that results from interchanging the input (domain) and output (range) of the original function.
Here,
if f(x) is a function, its inverse function is denoted by f⁻¹(x), and it is defined such that f⁻¹(y) = x, where y = f(x) is the original function.
For example, if f(x) = 2x + 3, the inverse function f⁻¹(x) is found by interchanging x and y, and solving for y:
x = 2y + 3
2y = x - 3
y = (x - 3)/2
Therefore, f⁻¹(x) = (x - 3)/2, and it is the inverse of the original function f(x). Note that the domain and range of the inverse function are interchanged with respect to the original function.
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A salesperson earned a commission of
$806.59
last week on gross sales of
$5880
. If returns and allowances were 11.5% of gross sales, what is his rate of commission based on net sales?
Using percentages we know that the rate of commission of the salesperson based on the net sales is 15.5%.
What are percentages?A percentage is a figure or ratio stated as a fraction of 100 in mathematics. Although the abbreviations "pct.", "pct.", and sometimes "pc" is also used, the percent symbol, "%," is frequently used to indicate it.
A % is a number without boundaries and without a standard measurement.
So, calculate as follows:
5880/100 * 11.5 = 676.2
Now,
5880 - 676.2 = 5,203.8
Now, calculate the percentage as follows:
806.59/5,203.8 * 100
15.5%
Therefore, using percentages we know that the rate of commission of the salesperson based on the net sales is 15.5%.
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Caleb has 12 grams of tea. He bought 5/4 of that amount for his mom. How many grams did Cameron get his mom?
Answer: Caleb bought 15 grams for his mom
Step-by-step explanation:
5/4 is equal to 1.25, so Caleb bought 1.25 x 12 grams.
1.25 x 12 = 15 grams.
Determine the domain for the interval of decrease.
d
Distance (km)
60
50
40
30
20
10
8 9 10 11 12 1 2 3 4
am
noon
p.m.
Time of the day
A2≤x≤5
B8≤x≤ 10
3≤x≤5
Answer:
The domain for the interval of decrease is 3 ≤ x ≤ 5
Find the GCF of 35x^5y and 42x^5y^3.
Answer:
x^5y
Step-by-step explanation:
To find the greatest common factor (GCF) of 35x^5y and 42x^5y^3, we need to find the largest expression that divides into both of these polynomials.
The common factor in both of these polynomials is x^5y, which is the GCF. So, the GCF of 35x^5y and 42x^5y^3 is x^5y.
please help me with this problem
The images of the functions are listed below:
Case 1: f(x) = √(- x) (Correct choice: A)
Case 2: f(x) = √(x - 4) (Correct choice: D)
Case 3: f(x) = √x - 3 (Correct choice: A)
Case 4: f(x) = √(x - 4) - 4 (Correct choice: A)
How to use rigid transformations on functions
In this problem we find four cases of functions that must be transformed by rigid transformations. Rigid transformations are transformations applied on functions such that its features are not altered. We should apply, if possible, the following rigid transformations:
Reflection across the y-axis
f(x) → f(- x)
Translation to the right
f(x) → f(x - k), where k > 0
Translation downward
f(x) → f(x) - k, where k > 0
Now we proceed to determine the image of each function:
Case 1
f(x) = √x
Reflection across the y-axis
f(x) = √(- x)
Case 2
Translation 4 units right
f(x) = √(x - 4)
Case 3
Translation 3 units down
f(x) = √x - 3
Case 4
Translation 4 units right and 4 units down
f(x) = √(x - 4) - 4
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write an exponential function that has a y-intercept of -4 and passes through the points (1, -1.6) and (2, -0.64)
The function can be used to model the exponential growth or decay that occurs in real-life situations that fit this particular pattern is y = 4(-0.4)ˣ + -4
An exponential function is a mathematical function that has the form
=> y = abˣ,
where a is a constant and b is the base of the exponential.
To start, we will use the first point, (1, -1.6), to write an equation that relates x and y. Let's call the constant a in our exponential function "k". Using the point (1, -1.6), we can write the equation:
=> -1.6 = k * b¹
Next, we will use the second point, (2, -0.64), to write another equation that relates x and y. Using the point (2, -0.64), we can write the equation:
=> -0.64 = k * b²
Now that we have two equations, we can solve for the two unknowns, k and b, by dividing the two equations:
=> (-0.64) / (-1.6) = k * b² / k * b¹
b = (0.64 / -1.6)¹
b = -0.4
With b determined, we can substitute it back into one of the equations and solve for k:
=> -1.6 = k * (-0.4)¹
=> k = -1.6 / (-0.4)
=> k = 4
So our exponential function that passes through the points (1, -1.6) and (2, -0.64) and has a y-intercept of -4 is:
y = 4 * (-0.4)ˣ + -4
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HELP FAST WILL GIVE 60 POINTS For THE RIGHT ANSWER
A repeating decimal like 0.2222222.... is a rational number and can be written as the fraction 2/9.
A stopping decimal like 0.22 is a rational number and can be written as the fraction 22/100.
A non-stopping and non-repeating decimal like 0.1234567891011... or 0.121121112... is an irrational number that can be written as the fraction 1514013912/5000000
The square root of a perfect square is a rational number and also a whole number.
The square root of a prime number is an irrational number.
What are rational and irrational numbers?
The definition of a rational number is one that may be stated as a fraction or as a portion of a whole number. For example -7, 2/3, 3.75 Numbers that cannot be stated as a fraction or ratio of two integers are referred to be irrational numbers. There is no limit to how one might convey them. For example √2, √3, e
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Please help me out! Just put these in order from the picture below.
Answer:
1) {2,0,-2}
2) {1,-3,1}
3) {1,3,5}
4) {0,1,2}
Mike buys 3 oranges and 1 lemon at the
farmers market. What fraction of his
purchase were oranges?
Mike buys 3 oranges and 1 lemon at the
farmers market. What fraction of his
purchase were oranges?
answer :
3/4
Answer: 3/4
Step-by-step explanation:
there are a total of four purchases and three of them were purchases we can represent this with the fraction 3/4
find the dilated coordinates with the scale factor of 1/5
The dilated coordinates:
D(-7, 0) → D' (-7/5, 0)
E(-7, -5) → E' (-7/5, -1)
F(-2, -5) → F' (-2/5, -1)
What is dilation?Resizing an item uses a transformation called dilation. Dilation is used to enlarge or shorten the structures. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. The initial form should be stretched or contracted during a dilatation.
Given:
Coordinates: D(-7, 0), E(-7, -5), F(-2, -5).
The shape is dilated by the scale factor of 1/5.
The dilation rule:
D(-7, 0) → D' (-7/5, 0)
E(-7, -5) → E' (-7/5, -1)
F(-2, -5) → F' (-2/5, -1)
Hence, the dilated coordinates are given above.
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The complete question:
Find the dilated coordinates with the scale factor of 1/5,
D(-7, 0), E(-7, -5), F(-2, -5).
Determine when the function is positive, negative, increasing, or decreasing: y = -9ˣ
brian is splitting 4 quarts of ice cream with members of the team. If the ice cream is split evenly, how many cups will each person get
Each person will get 1 [tex]\frac{7}{9}[/tex] cups of ice cream.
What is Division?Division is one of the operation in mathematics where number is divided into equal parts as that of a definite number.
Given that, Brian is splitting 4 quarts of ice cream with members of the team.
Assume the number of members of the team = 9
We have to find how many cups each member gets.
First we have to convert the measurement quarts to cups.
1 quart = 2 pints
1 pint = 2 cups
1 quart = 2 × 2 cups = 4 cups
We have 4 quarts of ice cream.
4 quarts = 4 × 4 = 16 cups
Number of cups each member gets = 16/9 cups = 1 [tex]\frac{7}{9}[/tex] cups
Hence there will be 1 [tex]\frac{7}{9}[/tex] cups for each member.
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Simplify (3x2+9x-1)+(-5x2+6x-10)
Andrew wins 4 fencing bouts for every 3 that he loses. If he had 35 bouts during the last year, how many bouts did he win?
The number of bouts that Andrew won during the last would be = 20 bouts.
How to calculate the number of bouts won?The number of fencing bouts that Andrew won = 4
The number of bouts he losses every 4 wins = 3 bouts
The total number of bouts he is having this year = 35 bouts.
The number of bouts he won amongst the number that he lost would be calculated as;
= 4+3 = 7
= 4/7 × 35/1
= 4× 5 = 20 bouts.
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suppose 5x^{2} +5x+xy=4 and y(4)=-24. Find y^'(4) by implicit differentiation.
Your answer:
if 5x² +5x+xy=4 then the value of y'(4) is -21/4
What is Differential equation?A differential equation is an equation that contains one or more functions with its derivatives.
The given expression is 5x² +5x+xy=4
Five times of x square plus five times of x plus x times of y equal to four.
Differentiate with respect to x.
10x+5+y(x)+xy'(x)
if x=4
10(4)+5+y(4)+4y'(4)=0
40+5-24+4y'(4)=0
21+4y'(4)=0
4y'(4)=-21
Divide both sides by 4
y'(4)=-21/4
Hence, if 5x² +5x+xy=4 then the value of y'(4) is -21/4
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What would 73,412 be rounded to the nearest thousand
HELP ASAP QUESTION 2!
The area of the circular section is equal to 3π square meters. (Correct choice: C)
How to find the area of a circular section
Herein we find the case of a circular section whose area must be found by the following formula:
A = (θ / 360) · π · R²
Where:
θ - Central angle, in degrees.R - Radius, in meters.A - Area, in square meters.If we know that θ = 270° and R = 2 m, then the area of the circular section is:
A = (270 / 360) · π · (2 m)²
A = (3π / 4) · (2 m)²
A = 3π m²
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What is the weighted average length of the nails in the drawer?
A. 350 cm
B. 333 cm
C. 5.00 cm
D. 3.50 cm
Let a and b be a pair of vectors, and let θ be the angle between them. Show that ||a ×b||= ||a||||b||sin θ|.
You are not allowed to use the cross product formula directly!
how to get the volume of a rectangular pyramid
Answer:
length x width x height divided by 3