find a polynomial function whose graph passes through (7,11) (11,-12) and (0,4)
The polynomial function whose graph passes through (7,11) (11,-12) and (0,4) will be y = -0.614x² + 5.295x + 4.
What is a function?A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.
Assume the polynomial function is quadratic. Then the equation is given as,
y = ax² + bx + c
At (0, 4), we have
4 = a(0)² + b(0) + c
c = 4
Then the equation is written as,
y = ax² + bx + 4
At (7, 11), we have
y = ax² + bx + 4
11 = 49a + 7b + 4 ...1
At (11, -12), we have
- 12 = 121a + 11b + 4 ...2
Equations 1 and 2 are solved by a calculator. Then we have
a = - 0.614 and b = 5.295
The polynomial capability whose diagram goes through (7,11) (11,- 12) and (0,4) will be y = - 0.614x² + 5.295x + 4.
More about the function link is given below.
https://brainly.com/question/5245372
#SPJ1
You buy 9 granola bars for a camping trip. Each bar costs $0.95. What
is the total cost of 9 granola bars before tax?
Answer:
Step-by-step explanation: 9 multiplied by .95 = $8.55 before tax
a) What is 10% of 40 ?
b) What number is 10% more than 40 ?
The expression 10% of 40 is 4 and 10% more than 40 is 44
(a) What is 10% of 40?From the question, we have the following parameters that can be used in our computation:
10% of 40
Express "of" as products
So, we have
10% of 40 = 10% * 40
Evaluate
10% of 40 = 4
b) What number is 10% more than 40?In this case, we have:
10% more than 40
This means that
10% more than 40 = 40 * (1 + 10%)
Evaluate
10% more than 40 = 44
Hence. 10% more than 40 is 44
Read more about percentage at
https://brainly.com/question/24877689
#SPJ1
What are the values of u and v?
u = ?°
v = ?°
a duck flew at 18 miles per hour for 3 hours than at 15 miles per hour for 2 ours how far did the duck fly in all
The distance the duck fly in all is 99 miles
How to determine how far the duck fly in allFrom the question, we have the following parameters that can be used in our computation:
Distance 1: 18 miles per hour for 3 hoursDistance 2: 15 miles per hour for 2 hoursThe distance covered in all can be calculated as
Distance = The sum of the product of speed and time
Substitute the known values in the above equation, so, we have the following representation
Distance = 18 * 3 + 15 * 2
Evaluate the products
This gives
Distance = 54 + 45
Evaluate the sum
Distance = 99 miles
Hence, the distance is 99 miles
Read more about speed at
https://brainly.com/question/14335655
#SPJ1
10. ANEW ACAR. Solve for x and CR. *
X=
Given: ANEW ACAR
NE = 11
EW = 12
I
AR = 4y - 12
CA = 4x + 3
NW = x + y
Applying the definition of congruent triangles, the values of x and y are:
x = 2; y = 6
Length of CR = 8 units.
How to Find the Sides of Congruent Triangles?If two triangles are congruent to each other, based on the CPCTC, all their corresponding angles, and their corresponding sides will be equal to each other.
Given that triangles NEW and CAR are congruent to each other, therefore:
NE = CA
NW = CR
EW = AR
Given the following measures:
NE = 11
EW = 12
AR = 4y - 12
CA = 4x + 3
NW = x + y
Therefore:
NE = CA
11 = 4x + 3
Solve for x:
11 - 3 = 4x
8 = 4x
8/4 = x
x = 2
EW = AR
12 = 4y - 12
12 + 12 = 4y
24 = 4y
24/4 = y
y = 6
NW = CR = x + y
Plug in the values of x and y:
CR = 2 + 6
CR = 8 units.
Learn more about congruent triangles on:
https://brainly.com/question/1675117
#SPJ1
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15
minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time.
Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84
shoppers per hour make a purchase and each of these shoppers spends an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers,
on average, are waiting in the checkout line to make a purchase at the Good Deals Store
Note that at any time during business hours, about 7 shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store. This is solved using Little Law.
What is littles Law?Little's Law asserts that the long-term average number of individuals in a stable system, L, is equal to the long-term average effective arrival rate,λ, multiplied by the average duration a customer spends in the system, W.
To compute,
First, let's make sure that all the variables use the same time unit.
Thus, if there are 84 shoppers who are making a purchase per house, then there will be: 84/60 minutes
= 1.4
This means that λ (rate) = 1.4 and the average time (W) is 5 minutes
Using Little's Law which states that the queuing formula is:
L = λW
Note that:
λ = 1.4 (computed)
W = 5 mintues (given)
Hence, the Average number of people in queue per time is:
L = 1.4 x 5
L = 7 shoppers.
Thus, it is correct to state that at any time during work hours, about 7 shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store.
Learn more about Little's Law:
https://brainly.com/question/29538196
#SPJ1
Full Question:
If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N=rT. This relationship is known as Little's law.
The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time.
Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?
If (x+2) is a factor of
Answer: a=3 and b=1
Step-by-step explanation:
Let p(x)=x 2 +ax+2b
If x+2 is a factor of p(x)
Then by factor theorem
p(−2)=0
⇒(−2)
2
+a(−2)+2b=0
⇒4−2a+2b=0
⇒a−b=2 ---(i)
Also given that a+b=4 ---(ii)
Adiing (i) and (ii) we get
2a=6
⇒a=3
putting a=3 in (i) we get
a−b=2
⇒3−b=2
⇒b=1
Therefore we get a=3 and b=1
NO LINKS!!
The expoential function given by f(x) = e^x called the (a. natural logarithmic, natural exponential, c. 1 to 1 exponential, d. 1 to 1 algebraic, e. transcendental algebraic) function and the base e is called the (a. algebraic, b. 1 to 1, c. natural, d. rational, e. transcendental)
Answer:
natural exponential
natural
quizlet
Answer:
b. natural exponential
c. natural
Step-by-step explanation:
Given function:
[tex]f(x)=e^x[/tex]
The given exponential function is called the:
natural exponential functionThe base e is called the:
natural baseThe number "e" occurs naturally in math and the physical sciences.
It is the base rate of growth shared by all continually growing processes, and so is called the natural base.
It is an irrational number and named after the 18th century Swiss mathematician, Leonhard Euler, and so is often referred to as "Euler's number".
Staci pays $32.70 for 5 cell phone cases. Each case costs the same amount. How much does each case cost?
Part A
Which expression represents the problem?
$32.70 × 5
$32.70 ÷ 5
$32.70 + 5
$32.70 – 5
Part B
Evaluate the expression from Part A.
$ ( ??? )
Answer:
$32.70 ÷ 5
Each case is $6.54.
Step-by-step explanation:
$3270 for 5 cases, meaning you would split $32.70 into 5.
$32.70 ÷ 5 phone cases = $6.54
This also means each case is $6.54. To prove this, multiply by 5.
Which of the following represents members of the domain of the graphed function?
See attached picture
Responses
{-4, 2, 3}
{-4, 0, 1}
{1, 2, 3, 4}
{1, 2, 3, 5}
The {-4, 0, 1} represents members of the domain of the graphed function.
What is the domain of the function?
A function is a mathematical object that accepts input, appears to apply a rule to it, and returns the result.
A function can be thought of as a machine that requires in a number, performs some operation(s), and then outputs the result.
The domain of a function is the collection of all its inputs. Its codomain is the set of possible outputs.
The range refers to the outputs which are actually used.
Domain: {-4, 0, 1}.
Simply list the domain as -4 < x < 2, which would imply ALL values between -4 and 2 inclusive.
Yes, this is a function. No x-values repeat, and it passes the Diagonal Line Test for functions.
Hence, the {-4, 0, 1} represents members of the domain of the graphed function.
To learn more about the domain of the function visit,
https://brainly.com/question/28934802
#SPJ1
Can someone please solve this problem for me.
The cost of one chair is [tex]x=54.75[/tex] $ and the cost of one table is [tex]y=49.5[/tex] $.
What is the total cost?
The total cost formula is used to combine the variable and fixed costs of providing goods to determine a total. The formula is:
Total cost = (Average fixed cost x average variable cost)
Let cost of one chair is [tex]x[/tex]
Let cost of one table is [tex]y[/tex]
So,
[tex]5x+3y=31[/tex] ...(1)
[tex]2x+6y=52[/tex] ......(2)
Solving equation (1) and (2) we get
[tex]a_1x+b_1y+x_1=0\\\\a_2x+b_2y+c_2=0[/tex]
[tex]\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}\\\\\frac{x}{3(31)-6(52)}=\frac{y}{31(2)-52(5)}=\frac{1}{5(2)-2(3)}\\\\\frac{x}{93-312}=\frac{y}{62-260}=\frac{1}{10-6}\\\\\frac{x}{-219}=\frac{y}{-198}=\frac{1}{-4}\\\\x=54.75,y=49.5[/tex]
Hence, the cost of one chair is [tex]54.75[/tex] $ and the cost of one table is [tex]49.5[/tex] $.
To know more about the total cost
https://brainly.com/question/15171617
#SPJ1
Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment. If his students performed the experiment 6 1/2 times, how much baking soda did they use?
Total no. of cups of baking soda used for the experiment in improper fraction is [tex]\frac{91}{8}[/tex].
How much baking soda was used?Initial no. of cups of baking soda used for an experiment = [tex]1\frac{3}{4}[/tex]
Converting mixed fraction to improper fraction,
[tex]1\frac{3}{4}=\frac{7}{4}[/tex]
No. of times the experiment performed by students = [tex]6\frac{1}{2}[/tex]
Converting mixed fraction to improper fraction,
[tex]6\frac{1}{2} = \frac{13}{2}[/tex]
Total no. of cups of baking soda =(Initial no. of cups)* (No. of times)
[tex]=\frac{7}{4} *\frac{13}{2} \\\\=\frac{91}{8} \\\\[/tex]
= 11.4 (decimal form)
What is a mixed fraction?A mixed fraction is one that has both its quotient and remainder represented. A mixed fraction like [tex]1\frac{3}{4}[/tex] is one where the remainder is 3 and the quotient is 1. Therefore, a mixed fraction is made up of both a full number and a correct fraction. A whole number and a legal fraction are both expressed together as a mixed number.A number between any two whole numbers is typically represented by it. Any fraction whose numerator exceeds or is equal to its denominator is considered improper. When the numerator value is less than the denominator, the fraction is said to be proper.To learn more about improper fraction, refer:
https://brainly.com/question/387381
#SPJ1
a persons weight varies directly with gravity if a person weights 180 pounds on earth they will weigh only 30 pounds on the moon if harsh weights 54 pounds on earth how much would he weight on the moon
The weight of harsh on the moon if he weighs 54 pounds on earth is 9 pounds
How to calculate direct variation?Weight varies directly with gravity
Let
Weight of a person = w
Gravity = g
So,
w = k × g
Where,
k = constant of proportionality
If w = 180 pounds and g = 30 pounds
w = k × g
180 = k × 30
180 = 30k
divide both sides by 30
k = 180/30
k = 6
If w = 54 pounds g = ?
w = k × g
54 = 6 × g
54 = 6g
divide both sides by 6
g = 54/6
g = 9 pounds
Therefore, harsh weighs 9 pounds on the moon.
Read more on direct variation:
https://brainly.com/question/6499629
#SPJ1
PLS HELP
Select the correct answer.
A museum curator estimates that 85% of people who attend the museum would return a second time. She randomly surveys
50 people and finds that only 75% indicate that they would return a second time. So she decides to randomly survey an additional
100 people.
if her model is valld, what could she expect from the 150 total survey results?
The difference between the data and the model will get larger.
The difference between the data and the model will stay the same.
O it is impossible to predict how the difference between the data and the model will change
The difference between the data and the model will get smaller.
O
Submit
The difference between the model and the data will get smaller.
What is the sample size?A sample is a percentage of the total population in statistics. You can use the data from a sample to make inferences about a population as a whole.
Given here, she 50 people and finds that only 75% and additional 100 people hence changing the sample size
As our sample size increases, the confidence in our estimate increases, our uncertainty decreases and we have greater precision. Thus with the increase in sample size, the data would move closer to the estimated probability.
Hence The difference between the model and the data will get smaller.
Learn more about sample size:
https://brainly.com/question/25894237
#SPJ1
Answer:
The difference between the data and the model will get smaller.
Step-by-step explanation:
Plato/Edmentum
Suppose that in a certain state, all automobile license plates have three uppercase letters followed by four digits. Use the method illustrated in Example 9.2.2 to answer the following questions. (a) How many different license plates are possible? To answer this question, think of creating a license plate as a 6-step process, where steps 1-3 are to choose the uppercase letters to put in positions 1-3 and the remaining steps are to choose the digits to put in the remaining positions. There are 17576 ways to perform steps 1-3, and there are 10000 ways to perform the remaining steps. Thus, the number of license plates is 175760000 (b) How many license plates could begin with A and end in 0? and the number of ways to place the 0 in the In this case, the number of ways to place the A in Step 1 is 1 final step is 1 . Thus, the answer is 676000 (c) How many license plates could begin with BWC? In this case, the number of ways to perform steps 1-3 is ___ Thus, the answer is ___(d) How many license plates are possible in which all the letters and digits are distinct? (e) How many license plates could begin with AB and have all letters and digits distinct? Enter an exact number
the total number of license plate is 17576000 and In this the number ways to place 'A' in step 1 is = '1'1, and the number of steps '0' in final steps id '1' thus the total answer is 67600
What is permutation and combination?In mathematics, there are two alternative methods for dividing up a collection of items into subsets: combinations and permutations. Any order may be used by a combination to list the subset's elements. An ordered list of a subset's components is called a permutation.
There are [tex]26^3[/tex] = 17576, ways to perform step 1-3, and there are [tex]10^3[/tex] = 1000, ways to perform remaining steps .
the total number of license plate is 17576000
In this the number ways to place 'A' in step 1 is = '1'1, and the number of steps '0' in final steps id '1'
thus the total answer is 67600
26 X 25 X 24 X 10 X 9 X 8 = 11232000
24 X 10 X 9 X 8 = 17280
To know more about permutation and combination visit:
https://brainly.com/question/29594894
#SPJ4
i literally don’t know
∠U is congruent ∠S because all right angles are congruent.
Define congruent.If it is possible to superimpose one geometric figure on the other so that their entire surface coincides, that geometric figure is said to be congruent, or to be in the relation of congruence. When two sides and their included angle in one triangle are equal to two sides and their included angle in another, two triangles are said to be congruent. This concept of congruence appears to be based on the idea of a "rigid body," which may be moved without affecting the internal relationships between its components.
Given
∠U ≅ ∠S
All right angles are congruent.
To learn more about congruent, visit:
https://brainly.com/question/7888063
#SPJ1
on their next training run, pepe averaged a speed of 2/3 of a mile in 5 minutes, while paula averaged 1/4 of a mile in 2 minutes. if pepe and paula each ran at their individual pace for 60 minutes, how many total miles did they cumulatively run?
Answer:
15.5
Step-by-step explanation:
Pepe= 2/3 x 12 = 8
Paula= 1/4 x 30 = 7.5
8+7.5=15.5
The two-way frequency table contains data about students' preferred exercise.
Enjoys swimming Enjoys cycling Row totals
Likes running 28 62 90
Does not like running 46 64 110
Column totals 74 126 200
What is the joint relative frequency of students who do not like to run but enjoy cycling?
64%
55%
32%
23%
The joint relative frequency for those students who don't like to run, but however enjoy cycling, is C. 32%.
How to find the joint frequency ?The joint frequency for students who like cycling, but do not like running, can be found by the formula :
= Number of students who enjoy cycling but don't enjoy running / Number of students in total
Number of students who enjoy cycling but don't enjoy running = 64
Number of students in total = 200
The joint frequency is :
= 64 / 200 x 100%
= 0.32 x 100 %
= 32 %
Find out more on joint frequency at https://brainly.com/question/3775864
#SPJ1
Does anyone know the answer for this?
The Venn diagram has been drawn and the product of LCM and GCF are the same as 24 x 36.
What is a Venn diagram?A Venn diagram is a pictorial representation of data represented in a circle.
The intersection parts of the circle represent the commonality of both sets.
As per the given, 24 and 36
The factors of 24 are as 2³ × 3
The factors of 36 are as 2² × 3²
The common factors 2² × 3
The LCM of 24 and 36 is 72 and the GCF is 12.
LCM x GCF = 24 x 12
72 x 12 = 24 x 12
864 = 864
Hence "After drawing a Venn diagram, it was discovered that the LCM and GCF product equals 24 x 36.".
To learn more about the Venn diagram,
https://brainly.com/question/29301560
#SPJ1
Write an equation for the line through the given point with the given slope in slope-intercept form. (10, –9); m = –2
Please explain.
Considering the definition of a line, the equation of the line that passes through the point (10, -9) and has a slope of -2 is y= -2x +11.
Linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin and represents the coordinate of the point where the line crosses the y axis.Line in this caseIn this case, you know:
The line has a slope of -2.The line passes through the point (10, -9).Substituting the value of the slope m and the value of the point y=mx+b, the value of the ordinate to the origin b is obtained as follow:
-9= -2×10 + b
-9= -20 + b
-9 + 20= b
11= b
Finally, the equation of the line is y= -2x +11.
Learn more about equation of a line:
brainly.com/question/20842634
#SPJ1
Question 12 options:
Nick determines the remainder of
, using the remainder theorem.
How does she proceed to the correct answer?
Fill in the blanks from the work bank below: (do not enter spaces)
x = 1 x = -1 x = 0 -23 -19 -15 -29
Nick evaluates the numerator of the rational expression when
. He concludes that the remainder of the division is
The input value for the expression is 1 and the remainder is -15.
What is the remainder theorem?The Remainder Theorem begins with an unnamed polynomial p(x), where "p(x)" simply means "some polynomial p with variable x". The Theorem then discusses dividing that polynomial by some linear factor x a, where an is simply a number.
Given polynomial is 5x⁴⁵ - 3x¹⁷ +2x⁴ - 19 and is divided by x + 1. Find the value of the polynomial at 1 and then calculate the remainder by synthetic solution.
F(x) = 5x⁴⁵ - 3x¹⁷ +2x⁴ - 19
F(1) = 5 - 3 + 2 - 19
F(1) = -19 + 4
F(1) = -15
The remainder will be calculated by synthetic calculation,
1 __5 ___-3___2_____-19
|
| ______5___ 2______4
5 2 4 -15
The remainder of the given expression is -15.
To know more about the remainder theorem follow
https://brainly.com/question/27749132
#SPJ1
What is as a fraction?
A fraction means a part of something or a number of parts of something. The number on the bottom shows how many parts something has been divided into ½ means 1 part of something that has been divided into 2 parts. We call this a half. If you add two halves together you get one.
Answer: A fraction is a numerical quantity that is not a whole number
Step-by-step explanation:
Use the Divergence Test to determine whether the following series diverges or state that the test is inconclusive. Select the correct answer below and fill in the answer box to complete your choice. O A. According to the Divergence Test, the series converges because lim ak = k- 00 (Simplify your answer.) OB. According to the Divergence Test, the series diverges because lim ax = kos (Simplify your answer.) OC. The Divergence Test is inconclusive because lim ak (Simplify your answer.) OD. The Divergence Test is inconclusive because lim ak does not exist.
The Divergence Test is inconclusive because lim ak = ∞
The Divergence Test is used to determine whether a given infinite series converges or diverges.
In order to use the Divergence Test, we must examine the limit of the terms of the series as n approaches infinity.
In the given series, the limit of the terms as n approaches infinity is not defined. Therefore, the Divergence Test is inconclusive.
The Divergence Test is inconclusive because lim ak = ∞
Learn more about test here
https://brainly.com/question/16944575
#SPJ4
Here we show that function defined on an interval value property cannot have (a; b) and satisfying the intermediate removable or (a) jump discontinuity. Suppose has & jump discontinuity at Xo € (a,b) and lim f (x) lim f (x) xx0 {ix0 Choose 0 such that lim f (x) < 0 < lim f (x) and 0 + f(xo) xI*o Xx0 In Exercise & we showed there is interval [xo 0,.Xo) such that f(x) < 0 if Xe [xo 6,xo): Likewise, there an interval (xo, Xo + 6] such that f(x) > 0 if xe(xo, Xo + 6]. Conclude that does not satisly the intermediate value property on [xo 6,xXo + 6]. (6) Suppose has a removable discontinuity at Xo € (a,b) and a = lim f(x) < f(xo) Show that there is an interval [xo = 6,Xo) such that f(x)< a+[f(xo) - &] if x e[xo 6,Xo]: Conclude that f does not satisfy the intermediate value property
f cannot have a jump discontinuity at [tex]$x_0 \in(a, b)$[/tex] and [tex]$$ \lim _{x \uparrow x_0} f(x) < \lim _{x \mid x_0} f(x) .$$[/tex]
f cannot have a removable discontinuity at [tex]$$x_0 \in(a, b) $$[/tex] and [tex]\alpha=\lim _{x \rightarrow x_0} f(x) < f\left(x_0\right)[/tex]
Let f be a function defined on (a, b) satisfies intermediate value property.
Claim: f ca not have removable on jump discontinuity.
Suppose f has a jump discontinuity at [tex]$x_0 \in(a, b)$[/tex]
We take [tex]$\theta$[/tex] such that
[tex]$$\lim _{x \rightarrow x_0} f(x) < \theta < \lim _{x \downarrow x_0} f(x) \text { and } \theta \neq f\left(x_0\right)$$[/tex]
Now there exist [tex]$\delta > 0$[/tex] such that [tex]$f(x) < \theta$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0\right)$[/tex] and [tex]$f(x) > \theta$[/tex] for all [tex]$x \in\left(x_0, x_0+\delta\right]$[/tex]
Now [tex]$f\left(x_0-\delta\right)[/tex][tex]< \theta < f\left(x_0+\delta\right)$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0+\delta\right] \backslash\left\{x_2\right\}$[/tex] and [tex]$f\left(x_0\right) \neq \theta$[/tex].
Therefore the point [tex]$\theta$[/tex] has no preimage under f
that is, there does not exists [tex]$y \in\left[x_0-\delta, x_0+\delta\right][/tex] for which
[tex]$$f(y)=\theta[/tex] because [tex]\left\{\begin{array}{l}y=x_0 \Rightarrow f(y) \neq \theta \\y > x_0 \Rightarrow f(y) > \theta \\y < x_0 \Rightarrow f(y) < \theta\end{array}\right.$$[/tex]
Therefore f does not satisfies intermediate value property on [tex]$\left[x_0-\delta, x_0+\delta\right]$[/tex],
Hence f does not satisfies IVP on (a, b) which is not possible because we assume f satisfies IVP on (a, b),
Therefore f can not have a jump discontinuity.
Suppose f has a removable point of discontinuity at [tex]$x_0 \in(a, b)$[/tex],
Let [tex]$\alpha=\lim _{\alpha \rightarrow x_0} f(x)$[/tex],
Let [tex]\alpha < f\left(x_0\right)$[/tex] so [tex]$f\left(x_0\right)-\alpha > 0$[/tex].
Now [tex]$\lim _{x \rightarrow x_0} f(x)=\alpha$[/tex] then [tex]\exists$ \delta > 0$[/tex] such that
[tex]$$\begin{aligned}& |f(x)-\alpha| < \frac{f\left(x_0\right)-\alpha}{2} \text { for all } x \in\left\{x_0-\delta, x_0-\alpha\right]-\left\{x_0\right\} \\& \Rightarrow \quad f(x) < \alpha+\frac{f\left(x_0\right)-\alpha}{2} \text { for all } x \in\left[x_0-\delta, x_0+\delta\right]-\left\{x_0\right\}\end{aligned}$$[/tex]
So [tex]$f(x) < \frac{f\left(x_0\right)+\alpha}{2}$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0\right]-\left\{x_0\right\}$[/tex]
Now [tex]$f\left(x_0\right) > \alpha$[/tex].
And [tex]$f(x) < \frac{f\left(x_0\right)+\alpha}{2} < f\left(x_0\right)$[/tex] for all [tex]$x \in\left[\left(x_0 \delta, x_0\right)\right.$[/tex]
Let [tex]$\mu=\frac{f\left(x_0\right)+\alpha}{2}$[/tex].
Then there does not exist [tex]$e \in\left[x_0-\delta, c\right]$[/tex] such that [tex]$f(c)=\mu$[/tex]
Because for [tex]$e=x_0 \quad f(e) > \mu$[/tex]
for [tex]$c < x_0 \quad f(c) < \mu$[/tex].
Therefore f does not satisfy IVP on [tex]$\left[x_0-\delta_1 x_0\right]$[/tex] which contradict our hypothesis,
therefore [tex]$\alpha \geqslant f\left(x_0\right)$[/tex]
Let [tex]$\alpha > f\left(x_0\right)$[/tex]. so [tex]$\alpha-f\left(x_0\right) > 0$[/tex]
[tex]$\lim _{x \rightarrow x_0} f(x)=\alpha$[/tex]
Then [tex]\exists $ \varepsilon > 0$[/tex] such that
[tex]$|f(x)-\alpha| < \frac{\alpha-f\left(x_0\right)}{2}$[/tex] for all [tex]$\left.x \in\left[x_0-\varepsilon_0 x_0+\varepsilon\right]\right\}\left\{x_i\right\}$[/tex]
[tex]$\Rightarrow f(x) > \alpha-\frac{\alpha-f\left(x_0\right)}{2}$[/tex] for all [tex]$x \in\left[x_0-\varepsilon_1, x_0+\varepsilon\right] \backslash\left\{x_0\right\}$[/tex]
[tex]$\Rightarrow f(x) > \frac{\alpha+f\left(x_0\right)}{2}$[/tex] for all [tex]$x \in\left[x_0-\varepsilon_1, x_0\right)$[/tex]
Now [tex]$f\left(x_0\right) < \alpha$[/tex]
The [tex]$f(x) > \frac{f\left(x_0\right)+\alpha}{2} > f\left(x_0\right)$[/tex].
So [tex]$f\left(x_0\right) < \frac{f\left(x_0\right)+\alpha}{2} < f(x)$[/tex] for all [tex]$x \in\left[x_0 \varepsilon, \varepsilon_0\right)$[/tex]
Let [tex]$\eta=\frac{f\left(x_e\right)+\alpha}{2}$[/tex]
Then there does not exist [tex]$d \in\left[x_0-\varepsilon, x_0\right]$[/tex] such that [tex]$f(d)=\xi$[/tex].
Because if [tex]$d=x_0, f(d)=f\left(x_0\right) < \eta$[/tex] if [tex]$d E\left[x_0-\varepsilon, x_0\right)$[/tex]
Then [tex]$f(d) > \eta$[/tex]
Therefore f does not satisfies IVP on [tex]$\left[x_0-\varepsilon, x_0\right]$[/tex] which contradict olio hypothesis.
Therefore [tex]$\alpha \leq f\left(x_0\right)$[/tex] (b) From (a) and (b) it follows [tex]$\alpha=f\left(x_0\right)=\lim _{x \rightarrow x_0} f(x)$[/tex]. Therefore f can not have a removable discontinuous
For more questions on jump discontinuity and removable discontinuous
https://brainly.com/question/20530286
#SPJ4
Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)
f ''(x) = 2x + 4ex
After integration, the required function f is (2x³ - sin (x) + Cx + D).
What is the integration of 'xⁿ' and 'sin (x)'?
[tex]\int {x^{n} } \, dx = \frac{x^{n+1} }{n+1} + C\\\\\\\int {sinx} \, dx = -cosx + C[/tex]
Given, f''(x) = 12x + sin x
Therefore,
[tex]\int {f''(x)} \, dx \\\\=\int {f'(x)} \, dx \\\\\\= \int{12x + sin x} \, dx + C\\\\= 6x^{2} - cosx + C\\[/tex]
Again, f'(x) = 6x - cos (x) + C
Therefore,
[tex]\int {f'(x)} \, dx\\ \\=\int {f(x)} \, dx \\\\= \int {6x^{2} - cosx + C } \, dx \\\\= 2x^{3} - sinx + Cx + D[/tex]
Therefore, the required function is (2x³ - sin (x) + Cx + D).
Learn more about integration at:
brainly.com/question/17072971
#SPJ4
A circle with center w is shown in the figure below.
Radius = TW
Diameter = TX
Chord = UV
The length of TX is 6 units.
What is a circle?A circle is a two-dimensional figure with a radius and circumference of 2 x pi x r.
The area of a circle is given as πr².
We have,
From the figure:
Diameter = TX
Radius = TW or WX or WY
Chord = UV
WY = 3 units
TX = TW + WX = 3 + 3 = 6 units.
Thus,
TX is 6 units.
Learn more about circle here:
https://brainly.com/question/11833983
#SPJ1
5x^2-13+6=0
i need helppp
Answer:
answer is 2.5
Step-by-step explanation:
The square of a number is equal to 72 more than the number. Find all such numbers.
Answer:
You just need to think about this a little. There is a number, that you don't know, x, and if you square it ([tex]x^2[/tex]), that value will be equal to 72 more than the unknown number itself, so you put that into an equation.
Step-by-step explanation:
[tex]x^2=x+72\\x^2-x-72=0\\[/tex]
Apply quadratic formula (you can look it up)
[tex]x_1=\frac{-(-1)+\sqrt{(-1)^2+(-4*1*-72} )}{2(1)} \\\\x_2=\frac{-(-1)-\sqrt{(-1)^2+(-4*1*-72} )}{2(1)} \\\\[/tex]
There are two answers to the quadratic formula.
The rest should be easy.
Based on the figure given below.
AC = 20cm
BC = 24Cm
AB = 16cm
CD = 15Cm and CE = 18cm then
a) Show that triangle ABC sim triangle DEC
b) How long is DC ?
Answer:
Step-by-step explanation:
A.) To show that triangle ABC is similar to triangle DEC, we need to prove that the ratios of the sides of the two triangles are equal.
First, we can write the ratios of the sides of triangle ABC as follows:
AC/AB = 20/16 = 5/4
BC/AB = 24/16 = 3/2
Now, we can write the ratios of the sides of triangle DEC as follows:
CE/CD = 18/15 = 6/5
AC/CD = 20/15 = 4/3
Since the ratios of the sides of the two triangles are equal, it follows that triangle ABC is similar to triangle DEC.
B.) To find the length of DC, we can use the fact that triangle ABC is similar to triangle DEC. Since the ratios of the sides of the two triangles are equal, we can set up a proportion to solve for DC.
First, we can write the ratio of the sides of triangle ABC as follows:
AC/AB = DC/CE
Then, we can substitute the known values for AC, AB, and CE:
20/16 = DC/18
Then, we can cross-multiply to solve for DC:
DC = (20/16) * 18
= (5/4) * 18
= 45/4
= 11.25 cm
Therefore, the length of DC is approximately 11.25 cm.
Answer:
Step-by-step explanation:
To show that triangle ABC is analogous to triangle DEC, we need to prove that the rates of the sides of the two triangles are equal.
First, we can write the rates of the sides of triangle ABC as follows
AC/ AB = 20/16 = 5/4
BC/ AB = 24/16 = 3/2
Now, we can write the rates of the sides of triangle DEC as follows
CE/ CD = 18/15 = 6/5
AC/ CD = 20/15 = 4/3
Since the rates of the sides of the two triangles are equal, it follows that triangle ABC is analogous to triangleDEC.
B.) To find the length of DC, we can use the fact that triangle ABC is analogous to triangle DEC. Since the rates of the sides of the two triangles are equal, we can set up a proportion to break for DC.
First, we can write the rate of the sides of triangle ABC as follows
AC/ AB = DC/ CE
also, we can substitute the known values for AC, AB, and CE
20/16 = DC/ 18
also, we cancross-multiply to break for DC
DC = (20/16) * 18
= (5/4) * 18
= 45/4
= 11.25 cm
Thus, the length of DC is roughly11.25 cm.
To know more about Triangle
https://brainly.in/question/3318722