The required answers are:
a. The linear approximating polynomial at a = 1: P₁(x) = x - 2
b. The quadratic approximating polynomial at a = 1: P₂(x) = -3 + 3x - x²
c. Appromating the given quantity -1/1.06 for P₁(x) and P₂(x): -3.06 and -7.30 respectively.
Using Taylor polynomial series, the required approximating polynomials are calculated.
What is the Taylor polynomial series?The Taylor polynomial series is
Pₙ(x) = ∑ [tex]\frac{f^n(a)}{n!}[/tex](x - a)ⁿ; limits from 0 to n
Where fⁿ(a) is the nth derivation of f(x) at a
The first order(linear) Taylor polynomial series is
P₁(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a)¹
The second order(quadratic) Taylor polynomial series is
P₂(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex](x - a)²
Calculation:The given function is f(x) = -1/x at a = 1
f(a) = f(1) = -1
f'(x) = -(-1/x²) = 1/x²; f'(a) = f'(1) = 1
f''(x) = -2/x³; f''(a) = f''(1) = -2
a. Linear approximating polynomial:
From the Taylor series, we have
P₁(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a)¹
On substituting,
P₁(x) = -1 + (1/1!)(x - 1)¹
= -1 + (x - 1)
= x - 2
b. Quadratic approximating polynomial:
From the Taylor series, we have
P₂(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex](x - a)²
On substituting,
P₂(x) = f(1) + (1/1!)(x - 1) + (-2/2!)(x - 1)²
= -1 + (x - 1) - (x - 1)²
= -1 + x - 1 - (x² - 2x + 1)
= x - 2 - x² + 2x - 1
= -3 + 3x - x²
c. Approximating the given quantity -1/1.06:
Substituting x = -1.06 in P₁(x) and P₂(x),
P₁(x) = x - 2
⇒ P₁(-1.06) = (-1.06) - 2 = -3.06
P₂(x) = -3 + 3x - x²
⇒ P₂(-1.06) = -3 + 3(-1.06) - (-1.06)² = -7.30
Therefore, the approximating polynomials are calculated by using Taylor polynomial series.
Disclaimer: The given question is incomplete. Here is the complete question.
Question:
a. Find the linear approximating polynomial for the following function centered at the given point a.
b. Find the quadratic approximating polynomial for the following function centered at the given point a.
c. Use the polynomials obtained in parts a. and b. to approximate the given quantity.
f(x) = -1/x , a = 1; approximate -1/1.06
Learn more about approximating polynomials here:
https://brainly.com/question/15540322
#SPJ1
find the volume of these rctangular prisms length=6.4mm width=2.5mm hight=7mm
step by step pls
Answer:
The volume of this rectangular prism is 112mm.
Step-by-step explanation:
[tex]volume \: = \: length \times width \times height \\ v = 6.4mm \times 2.5mm \times 7mm \\ v = 112mm[/tex]
What are relative advantages and disadvantages of the algebraic notations of viete and harriot?
Generally speaking, Harriot's notation was clearer than Viète's, which made it simple for him to connect Viète's identities.
Harriot made significant contributions to the theory of equations and developed a more straightforward notation for algebra.
The employment of special notation is algebra's most evident characteristic. Variables are the symbols that are used to represent numbers.
A symbolic system known as notation is used to represent mathematical concepts and objects.
The primary components of notation are letters, symbols, figures, and signs. Notation can be made using symbols, characters, numbers, or a combination of these, such as the factorial symbol.
Learn more about notations here:
https://brainly.com/question/1767229
#SPJ1
Given that tanx=14 and sinx is negative, determine sin(2x), cos(2x), and tan(2x). Write the exact answer. Do not round.
By applying trigonometric identities, the value of tan(x) = 14, gives;
[tex]sin(2 \cdot x) = \frac{28}{197} [/tex]
[tex]cos(2 \cdot x) = -\frac{195}{197} [/tex]
[tex]tan(2 \cdot x) = -\frac{28}{195} [/tex]
Which method can be used to find the value of the trigonometric expressions?Given;
tan(x) = 14
sin(x) is negative
By definition, we have;
[tex]sin(2 \cdot x) = \mathbf{\frac{2 \cdot \: tan(x)}{1 + {tan(x)}^{2} }} [/tex]
Which gives;
[tex]sin(2 \cdot x) = \frac{2 \times 14}{1 + {14}^{2} } = \frac{28}{197} [/tex]
Similarly, we have;
[tex]cos(2 \cdot x) = \mathbf{\frac{1- tan(x)^2}{1 + {tan(x)}^{2} }} [/tex]
Which gives;
[tex]cos(2 \cdot x) = \frac{1- 14^2}{1 + {14}^{2} } = -\frac{195}{197} [/tex]
[tex]tan(2 \cdot x) = \mathbf{\frac{sin(2 \cdot x) }{cos(2 \cdot x) } } [/tex]
Which gives;
[tex]tan(2 \cdot x) = \frac{ \frac{28}{197}}{-\frac{195}{197} } = -\frac{28}{195} [/tex]
Learn more about trigonometric identities and functions here:
https://brainly.com/question/7331447
#SPJ1
Consider this quadratic equation.
x2 + 2x + 7 = 21
The number of positive solutions to this equation is
.
The approximate value of the greatest solution to the equation, rounded to the nearest hundredth, is
Answer:
1 positive solution (and 1 negative)
the greatest solution is about 2.87
Step-by-step explanation:
so, it is
x² + 2x + 7 = 21
x² + 2x - 14 = 0
the general solution to such a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = 2
c = -14
x = (-2 ± sqrt(2² - 4×1×-14))/(2×1) =
= (-2 ± sqrt(4 + 56))/2 = (-2 ± sqrt(60))/2 =
= (-2 ± sqrt(4×15))/2 = (-2 ± 2×sqrt(15))/2 =
= -1 ± sqrt(15)
x1 = -1 + sqrt(15) = 2.872983346... ≈ 2.87
x2 = -1 - sqrt(15) = -4.872983346...
A credit card company charges a late fee of 10% on all late payments. last month, jamal was charged a late fee. the total amount paid, including the late fee, was $275. what was the original amount on the credit card before the late fee?
Answer:
265
Step-by-step explanation:
You subtract 275 minus 10
$247.5 was the original amount on the credit card
Lets simplify the problem,
Total amount after late fee = $275
Late fee deduction = 10%
Original amount = 275 - 10% of 275
= 275 - 27.5
= $247.5
So $247.5 was the original amount on the credit card before the late fee.
Percentage:
In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol "%" or simply as "percent" or "pct." For example, 35% is equivalent to the decimal 0.35, or the fraction.
Learn more about Percentage on:
https://brainly.com/question/2236179
#SPJ4
Solve for x in the diagram below.
Answer:
x = 20
Step-by-step explanation:
x°
2x°
(x + 10)°
little box on the bottom left means this is a right angle which is 90°
add up all the angles and make it equal to 90°
( x° + 2x° + (x + 10)° ) = 90°
4x + 10 = 90
4x = 80
x = 20
Since we can see the "square" at the bottom left corner of the angle, the square indicates that the angle is a right angle (which measures 90°).
We can also see three smaller angles forming the right angle. Therefore, the sum of the measure of the smaller angles = 90°.
According to the diagram, the measures of the smaller angles are x°, 2x°, and (x + 10)° respectively. Then we get the following equation:
[tex]\implies x + 2x + (x + 10) = 90\°[/tex]
Step-2: Solving the equation obtained in step-1Here, we had the equation: [tex]\underline{x + 2x + (x + 10) = 90\°}[/tex]
[tex]\implies x + 2x + x + 10 = 90\°[/tex] [tex]\text{(Ope} \text{ning the parentheses)}[/tex]
[tex]\implies 4x + 10 = 90\°[/tex] [tex]\text{(Combining like terms)}[/tex]
[tex]\implies 4x + 10 - 10 = 90 - 10[/tex] [tex]\text{(Subtracting 10 on both sides)}[/tex]
[tex]\implies 4x = 80[/tex] [tex]\text{(Simplifying both sides)}[/tex]
[tex]\implies \dfrac{4x}{4} = \dfrac{80}{4} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(Dividing 4 on both sides)}[/tex]
[tex]\implies \boxed{x = 20}[/tex] [tex]\text{(Simplifying both sides)}[/tex]
Therefore, the value of x, in the diagram provided, is 20.
write each as the sum of two terms in the form t_n,r. t_5,3 , t_11,2 , t_15,13
The sum of two terms are:
t₅, ₃ = 8t₁₁, ₂ = 13 t₁₅,₁₃ = 28.What is the sum about?The sum of two numbers is one that tells you to find the sum of two or more numbers, and as such, you have to add the numbers altogether.
t₅, ₃ = 5 + 3
= 8
t₁₁, ₂ = 11 + 2
= 13
t₁₅,₁₃ = 15 + 13
= 28.
Hence the correct sum of two terms are:
t₅, ₃ = 8t₁₁, ₂ = 13 t₁₅,₁₃ = 28.Learn more about sum from
https://brainly.com/question/17695139
#SPJ1
terrell watched 300 dogs being groomed. He found that 0.35 of the groomed dogs tried to bite the groomer. what percent of dogs did not try to bite the groomer?
The percentage of dogs who didn't try to bite the groomer is; 65%.
What is the percentage of dogs that did not try to bite the groomer?If follows from the task content that the number of dogs is 300 of which, 0.35 faction of the whole tried to bite the groomer.
Hence, the faction that did not try to bite the groomer is; 0.65 which corresponds to 65% when expressed as a percentage.
Read more on percentages;
https://brainly.com/question/24304697
#SPJ1
Ms. drake invests $1000 at a simple interest rate of 8.5% for 2 years. how much interest will she earn in this time?
Ms. Drake will earn [tex]\$ 170[/tex] in this time from the investment of [tex]\$ 1000[/tex] at a simple interest with rate of interest [tex]8.5\%[/tex]
What is the definition of simple interest?Simple interest is a way to figure out how much interest will be charged on a sum of money at a specific rate and for a specific duration of time. Conversely, for compound interest, where we add the interest of one year's principal to the next year's principal to compute interest, the principal amount under simple interest remains constant.
Formula for finding simple interest is [tex]\frac{principal \times rate \ of \ interest \times time}{100}[/tex]
Given principal of Ms. Drake is [tex]\$1000[/tex]
Rate of interest is [tex]8.5 \%[/tex]
Time is [tex]2[/tex] years.
Interest [tex]=\frac{1000 \times 8.5 \times 2}{100}[/tex]
[tex]=\frac{17000}{100}\\=170[/tex]
Therefore, Ms. Drake will earn [tex]\$ 170[/tex] in this time.
To learn more about simple interest from the given link
https://brainly.com/question/7639734
#SPJ4
In the figure shown, line AB is parallel to line CD. Part A: What is the measure of angle x? Show your work. (5 points)
Part B: Explain how you found the measure of angle x by identifying the angle relationships that you used along the transversal. (5 points) A 65⁰ P 120⁰ R B O
a) ∠PQR=65° (alternate interior angles theorem)
∠PRQ = 60° (linear pair)
x = 55° (angles in a triangle add to 180°)
b) ∠APQ and ∠PQR are congruent alternate interior angles.
A
B
C
D
LO
-5
Y
A'
В'
O
D'
Which rule describes the translation?
(x, y)
(x-8, y - 3)
(x, y)
(x-3, y + 8)
C
(x, y) → (x + 8, y − 3)
O (x, y) → (x + 3, y + 8)
Answer:
C. (x,y) ⇒ (x + 8, y − 3)
Step-by-step explanation:
See attached image.
If you rolled two dice, what is the probability
that you would roll a sum of 9?
Give your answer as a simplified fraction. Enter
the number that belongs in the green box.
second
first
1 2 3 4 5 6
1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2 2,3 2,4 2,5 2,6
3 3,1 3,2 3,3 3,4 3,5 3,6
4 4,1 4,2 4,3 4,4 4,5 4,6
5 5,1 5,2 5,3
5,4 5,5 5,6
6 6,1 6,2 6,3 6,4 6,5 6,6
The probability you will roll a sum of 9 is 1/4.
What is the probability?Probability determines the chances that an event would happen. The probability the event occurs is one and the probability that the event does not occur is 0. The more likely the event is to happen, the closer the probability value would be to 1. The less likely it is for the event not to happen, the closer the probability value would be to zero.
The probability you will roll a sum of 9 = total times a sum of 9 is derived / total sample spaces
total times a sum of 9 is derived = (3 + 6) , (6 + 3) , (4 + 5) , (5 + 4) Total sample spaces = 36The probability you will roll a sum of 9 = 4/36 = 1/9
To learn more about probability, please check: https://brainly.com/question/13234031
#SPJ1
Write as a decimal the number that is exactly halfway between 1 100 and 1 10
Answer:
605
Step-by-step explanation:
We are starting at 110 and not zero, so I need to find have the distance between 110 and 110 and add that to 110.
110 + 1/2(1100 - 110) Subtract what is in the parentheses
110 + 1/2(990) Find 1/2 of 990
110 + 495
604
Can someone help me with these problems pleaseeee!!
Answer:
I have wrote just answer.
HELPPPPPP MATHHHH pleaseee
Answer:
A
Step-by-step explanation:
A tub contains red, blue, green and white counters. 40% of the counters are red or white. The ratio of blue to green counters is 3:2.The ratio of red to white counters is 3: 5. What is the probability of picking a red or green counter?
The probability of randomly getting a red or green counter is:
P = 39%/100% = 0.39
How to get the probability?We know that 40% of the counters are red or white. Then 60% of the counters are blue or green.
Now we know that the ratio of red to white is 3:5
Then the percentage of red counters is:
(3/8)*40% = 15%
And the ratio of blue to green is 3:2
Then the percentage of red counters is:
(2/5)*60% = 24%
Then the percentage of counters that are either red or green is:
15% + 24% = 39%
Then the probability of randomly getting a red or green counter is:
P = 39%/100% = 0.39
If you want to learn more about probability:
https://brainly.com/question/25870256
#SPJ1
a^4 b^2÷4a^3 b^-2
[tex] {a}^{4} {b}^{2} \div 4 {a}^{3} {b}^{ - 2} [/tex]
WILL MAKE BRAINLIEST!!
Find the correct angle measure for angle b in each figure.
_1. 52°
_2. 27°
_3. 32°
_4. 98°
An auditorium with 30 rows of seats has 10 seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, what is the maximum number of students that can be seated for an exam
Answer:
Step-by-step explanation:
380 rows in total
bcs
use sense
Based on an 8-hour day, the number of hours worked in a hospital food service department was 55,267/yr., and the total number of hours paid was 59,995/yr. The actual number of productive FTEs was:
That year FTE was 26.
According to statement
Labor hours worked that year = 55,267
Total number of hours paid that year = 59,995
The total number of hours during the year
8 hours × 5 days × 52 weeks = 2080 hours
Now we find the FTE per year
So, FTE/YEAR = Labour works on that year / total number of hours in year
Substitute the values in it then
FTE/YEAR = 55267/2080
FTE/YEAR = 26
So, That year FTE was 26.
Learn more about AVERAGE here https://brainly.com/question/8728504
#SPJ4
what is the length of side s of the square shown below
Hello,
an other method
Pythagorean theorem :
x² + x² = 6²
2x² = 36
x² = 36/2
x² = 18
x = √18 (and not -√18 because x ≥ 0)
x = √(9 × 2)
x = √(3² × 2)
x = 3√2
Answer A
i dont understand this at all
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{What are we solving for?}[/tex]
[tex]\mathsf{56 = w(w + 10) }[/tex]
[tex]\huge\textbf{How are we going to solve for the}\\\\\huge\textbf{equation?}[/tex]
[tex]\mathsf{56 = w(w + 10)}[/tex]
[tex]\mathsf{w(w + 10) = 56}[/tex]
[tex]\mathsf{w(w) + w(10) = 56}[/tex]
[tex]\mathsf{w(w) + 10(w) = 56}[/tex]
[tex]\mathsf{w^2 + 10w = 56}[/tex]
[tex]\huge\textbf{Subtract \boxed{\bf 56} to both sides:}[/tex]
[tex]\mathsf{w^2 + 10w - 56 = 56 - 56}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{w^2 + 10w - 56 = 0}[/tex]
[tex]\huge\textbf{Factor the left side of the equation:}[/tex]
[tex]\mathsf{(w - 4)(w + 14) = 0}[/tex]
[tex]\mathsf{(w - 4)\times(w + 14) = 0}[/tex]
[tex]\huge\textbf{Simplify it:}[/tex]
[tex]\mathsf{w - 4 = 0\ or\ w + 14 = 0}[/tex]
[tex]\mathsf{w = 4\ or \ w = -14}[/tex]
[tex]\huge\textbf{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{\frak{\mathsf w = 4\ or \ \mathsf w = -14}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
Noah has a summer tree-trimming business. Based on experience, Noah knows that his profit, P, in dollars, can be modelled by = −32 + 150 − 1200, where x is the amount he charges per tree.
1. How much does he need to charge if he wants to break even?
2. How much does he need to charge if he wants to make a profit of $600?
Answer:
Addition form: x = 75 + √3 (675 − P) / 3
or
Subtract form: x = 75 − √3 (675 − P) / 3
The probability of event X is 30%. The probability of event Y is 40%. If events X and Y are mutually exclusive, what is the probability of event X or Y
The probability of event X or Y is P = 0.7, or, in percentage form, the probability is 70%.
How the get the probability of event X or Y?
First, two events are mutually exclusive if these events can't happen at the same time.
This means that if happens X, with a 30% of probability, then Y can't happen.
If happens Y, with a 40% of probability, then X can't happen.
Then the probability of X or Y (the probability that one of these two events happens, not both) is just the addition of the two individual probabilities.
Then we have:
P(X or Y) = 0.3 + 0.4 = 0.7
Or, in percent form, we get:
P(X or Y) = 70%
The probability of event X or Y is just 70%.
If you want to learn more about probability:
https://brainly.com/question/25870256
#SPJ1
Which equation is equivalent to
f(x) = 16x4 -81 = 0?
✓(4x² +9)(4x² - 9) = 0 ✓
COMPLETE
Select all of the zeroes of the function.
O
2/3
DONE
The equivalent equation is (4x² + 9)(4x² - 9) and the zero's are x =± 3 / 2 and x = ± 3 / 2i
How to find equivalent equation?The equivalent equation can be found by factoring. Factoring of a polynomial is the method of breaking the polynomial into a product of its factors.
Therefore,
f(x) = 16x⁴ - 81 = 0
Hence,
9 × 9 = 81
4x² × 4x² = 16x⁴
Therefore,
16x⁴ - 81 = (4x² + 9)(4x² - 9)
Hence,
The zeros of a function, also referred to as roots or x-intercepts, occur at x-values where the value of the function is 0 (f(x) = 0).
The zero's of the function are as follows:
4x² = 9
x² = 9 / 4
x =± 3 / 2
4x² = -9
x² = - 9 / 4
x =√-9 / 2
x = ± 3 / 2i
learn more on function here: https://brainly.com/question/11670913
#SPJ1
PLEASE HELP IM STUCK PLEASE HELP
Answer:
y = 3x + 5
Step-by-step explanation:
x = number of years
• bush grows 3cm per year
• the initial height is 5cm
3cm per year times x number of years: y = 3x
Add the initial height of 5cm: y = 3x + 5
A square sandbox has sides of length 9 ft and a depth of 1 ft. If sand costs $5 / cubic foot, how much will it cost to fill the sandbox with sand?
The cost spent on fill the box with sand is 405$.
According to the statement
we have given a square sandbox with length 9 ft and depth of 1 ft.
and the rupees of sand per cubic ft is $5
then
Now, we have to find the volume of sandbox with the given length and depth.
So, for this purpose we use Volume formula
V = (a)^3
V = 9*9*1
V = 81 cubic feet
And the cost of sand per cubic feet is $5.
then
The cost of sand for 81 cubic per feet is calculate by multiplication formula then
Total cost of sand = 81*5
Total cost of sand = 405$
So, The cost spent on fill the box with sand is 405$.
Learn more about VOLUME here https://brainly.com/question/11408596
#SPJ4
A board of directors consists of fourteen men and six women. A four-member search committee is randomly chosen to recommend a new company president. What is the probability that all four members of the search committee will be women
Answer:
probability of all the four women chosen over people needed in a committee=4 over 4 =1
My fruit basket contain apples and oranges.The ratio apples to oranges is 3:8.When I remove one apple the ratio changes to 1:3 .How many apples are in the basket
The number of apples in the basket are; 24 apples
How to simplify simultaneous equations?Let there be X number of apples and Y number of oranges.
Thus by the given ratio we can say that;
x/y = 3/8 -----(1)
(x — 1)/y = 1/3
3x — 3 = y ----(2)
Substitute 3x - 3 for y in eq 1 to get;
x/(3x - 3) = 3/8
8x = 9x - 9
9x - 8x = 9
x = 9
Thus;
y = 3(9) - 3
y = 24 apples
Read more about Simultaneous Equations at; https://brainly.com/question/148035
#SPJ1
Simplify the expression
The solution to the expression [tex]\frac{x^2+7x+12}{x-3} .\frac{x^2-6x+9}{2x^2-18}[/tex] gives (x + 4)/2
What is an equation?An equation is an expression that shows the relationship between two or more number and variables.
[tex]\frac{x^2+7x+12}{x-3} .\frac{x^2-6x+9}{2x^2-18} \\\\=\frac{(x +3)(x +4)}{x-3} .\frac{(x-3)(x-3)}{2(x+3)(x-3)} =\frac{x+4}{2}[/tex]
The solution to the expression [tex]\frac{x^2+7x+12}{x-3} .\frac{x^2-6x+9}{2x^2-18}[/tex] gives (x + 4)/2
Find out more on equation at: https://brainly.com/question/2972832
#SPJ1