With some rewriting, you get
[tex]\displaystyle \sum_{k=0}^\infty (-1)^k\frac{x^{k+2}}{4^k} = x^2 \sum_{k=0}^\infty \left(-\frac x4\right)^k[/tex]
Recall that for |x| < 1, you have
[tex]\displaystyle \frac1{1-x} = \sum_{k=0}^\infty x^k[/tex]
So as long as |-x/4| = |x/4| < 1, or |x| < 4, your series converges to
[tex]\displaystyle x^2 \sum_{k=0}^\infty \left(-\frac x4\right)^k = \frac{x^2}{1-\left(-\frac x4\right)} = \frac{x^2}{1+\frac x4} = \boxed{\frac{4x^2}{4+x}}[/tex]
Based on known expressions from Taylor series, the power series [tex]\sum \limits_{k = 0}^{\infty} (-1)^{k}\cdot \frac{x^{k+2}}{4^{k}}[/tex]Taylor series-derived formula of the rational function [tex]\frac{4\cdot x^{2}}{4+x}[/tex].
How to derive a function behind the approximated formula by Taylor seriesTaylor series are polynomic approximations used to estimate values both from trascendental and non-trascendental functions. It is commonly used in trigonometric, potential, logarithmic and even rational functions.
In this question we must use series properties and common Taylor series-derived formulas to infer the expression behind the given series. Now we proceed to find the expression:
[tex]\sum \limits_{k = 0}^{\infty} (-1)^{k}\cdot \frac{x^{k+2}}{4^{k}}[/tex]
[tex]x^{2}\cdot \sum\limits_{k = 0}^{\infty} \left(-\frac{x}{4} \right)^{k}[/tex]
[tex]x^{2}\cdot \left(\frac{1}{1+\frac{x}{4} } \right)[/tex]
[tex]\frac{4\cdot x^{2}}{4+x}[/tex]
Based on power and series properties and most common Taylor series- derived formulas, the power series [tex]\sum \limits_{k = 0}^{\infty} (-1)^{k}\cdot \frac{x^{k+2}}{4^{k}}[/tex] represents a Taylor series-derived formula of the rational function [tex]\frac{4\cdot x^{2}}{4+x}[/tex]. [tex]\blacksquare[/tex]
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A medicine bottle contains 8 grams of medicine. One dose is 400 milligrams. How many milligrams does the bottle contain?
Answer:
8×1000 milligrams
8000 milligrams
URGENT HELP!!!!
Picture included
Answer:
Length (L) = 72 feet
Step-by-step explanation:
From the question given above, the following data were obtained:
Period (T) = 9.42 s
Pi (π) = 3.14
Length (L) =?
The length of the pendulum can be obtained as follow:
T = 2π √(L/32)
9.42 = (2 × 3.14) √(L/32)
9.42 = 6.28 √(L/32)
Divide both side by 6.28
√(L/32) = 9.42 / 6.28
Take the square of both side
L/32 = (9.42 / 6.28)²
Cross multiply
L = 32 × (9.42 / 6.28)²
L = 72 feet
Thus, the Lenght is 72 feet
Which of the following is equivalent to a real number?
A. (-46)^1/2
B. (-10596)^1/8
C. (-4099)^1/5
D. (-5403)^1/6
Answer:
C. (-4099)^1/5
Step-by-step explanation:
[tex]x^{\frac{1}{2} } = \sqrt{x}[/tex]
you can not take roots (real roots) of a negative number if the exponent is
even ... A,B,D have even exponents (in the denominator of the exponent.. in other words the index of the radical is even)...
the only odd index is in "B" (the 5 in the 1/5)
[tex]i^0 +i^1+i^2+i^3+............+i^{2021} = ?[/tex]
Include work.
Answer:
1+i
Step-by-step explanation:
I do believe i to be the imaginary unit.
Let's write out some partial sums from power=0 to power=7 or whatever we need to see a pattern.
i^0=1
i^0+i^1=1+i
i^0+i^1+i^2=1+i+-1=i
i^0+i^1+i^2+i^3=i+i^3=i+-i=0
i^0+i^1+i^2+i^3+i^4=0+i^4=0+1=1
Hmmm.... we might see 1+i, then i, then 0 again.... let's see.
i^0+i^1+i^2+i^3+i^4+i^5=1+i
Coolness so we should see a pattern
Sum from power=0 to power=multiples of 4 will give us 1.
Sum from power=0 to power=remainder of 1 when final power is divided by 4 gives us 1+i.
Sum from power=0 to power=remainder of 2 when final power is divided by 4 gives us i.
Sum from power=0 to power=remainder of 3 when final power is divided by 4 gives us 1
0.
So 2021 divided by 4....
Since 2020 is a multiple of 4, then 2021 has a remainder of 1 when divided by 4.
So the answer is 1+i.
Can someone please help me thank you !!!!!
A chemist has three different acid solutions.
The first solution contains 25% acid, the second contains 35%acid, and the third contains 55% acid.
She created 120 liters of a 40% acid mixture, using all three solutions. The number of liters of 55% solution used is 3 times the number of liters of 35% solution used.
How many liters of each solution was used?
Let x, y, and z be the amounts (in liters, L) of the 25%, 35%, and 55% solutions that the chemist used.
She ended up with 120 L of solution, so
x + y + z = 120 … … … [1]
x L of 25% acid solution contains 0.25x L of acid. Similarly, y L of 35% solution contains 0.35y L of acid, and z L of 55% solution contains 0.55z L of acid. The concentration of the new solution is 40%, so that it contains 0.40 (120 L) = 48 L of acid, which means
0.25x + 0.35y + 0.55z = 48 … … … [2]
Lastly,
z = 3y … … … [3]
since the chemist used 3 times as much of the 55% solution as she did the 35% solution.
Substitute equation [3] into equations [1] and [2] to eliminate z :
x + y + 3y = 120
x + 4y = 120 … … … [4]
0.25x + 0.35y + 0.55 (3y) = 48
0.25x + 2y = 48 … … … [5]
Multiply through equation [5] by -2 and add that to [4] to eliminate y and solve for x :
(x + 4y) - 2 (0.25x + 2y) = 120 - 2 (48)
0.5x = 24
x = 48
Solve for y :
x + 4y = 120
4y = 72
y = 18
Solve for z :
z = 3y
z = 54
If the cost of a 2.5 meter cloth is $30.5. What will be the cost of 22 meters ?
Answer:
268.40
Step-by-step explanation:
We can write a ratio to solve
2.5 meters 22 meters
----------------- = --------------
30.5 dollars x dollars
Using cross products
2.5 * x = 30.5 * 22
2.5x =671
Divide each side by 2.5
2.5x / 2.5 = 671/2.5
x =268.4
(3) If a tire rotates at 400 revolutions per minute when the car is traveling 72km/h, what is the circumference of the tire?
Show all your steps.
Answer:
3 meters.
Step-by-step explanation:
400 rev / minute = 400 × 60 rev / 60 minutes
= 24,000 rev / hour
24,000 × C = 72,000 m : C is the circumference
C = 3 meters
Answer:
3 meters
Step-by-step explanation:
72 km / hour * 1 hour/ 60 min * 1000m/ 1 km
72000 meters /60 minute
1200 meters / minute
velocity = radius * w
Where w is 2*pi * the revolutions per minute
1200 = r * 2 * pi *400
1200 / 800 pi = r
1.5 /pi = r meters
We want to find the circumference
C = 2 * pi *r
C = 2* pi ( 1.5 / pi)
C = 3 meters
How many subsets of at least one element does a set of seven elements have?
[tex]\boxed{\large{\bold{\blue{ANSWER~:) }}}}[/tex]
For each subset it can either contain or not contain an element. For each element, there are 2 possibilities. Multiplying these together we get 27 or 128 subsets. For generalisation the total number of subsets of a set containing n elements is 2 to the power n.
total subsets
2^n2⁷128Two angles of a triangle have the same measure and the third one is 36 degrees greater than the measure of each of the other two. Find the measure of the LARGEST angle in the triangle.
Answer:
Largest angle is 84
Step-by-step explanation:
Let the smallest angle be x, ATQ, x+x+x+36=180, 3x+36=180, x=48
A life insurance policy cost $8.52 for every $1000 of insurance at this rate what is the cost for 20,000 worth of life insurance
Answer:
$170.40
Step-by-step explanation:
cost of policy=$8.52
Groups of $1000=20,00÷1000
=20
20,000 worth=$8.52×20
$170.40
The Wowo Novelty Company makes three basic types of noisemakers: Toot, Wheet and honk. A toot can be made in 30 minutes and has a feather attached to it. A wheet requires 15 minutes, has two feathers and is sprinkled with 0.5 oz of sequin powder. A honk requires 30 minutes, has 3 feathers and 1 oz of sequin powder. The net profit is P0.40 per toot, P0.5 per wheet and P0.80 per honk. The following resources are available: 80 hours of labor, 360 feathers and 90 oz of sequin powder. Determine the quantity of each type of noisemakers that maximizes profit.
Answer:
P104
Step-by-step explanation:
Let x represent the number of toot, y represent the number of wheet and z represent the number of honk.
Since a toot is made in 30 minutes (0.5 hours), wheet in 15 minutes (0.25 hour), honk in 30 min (0.5 hr). There is 80 hours of labor, hence:
0.5x + 0.25y + 0.5z ≤ 80 (1)
There are 360 feathers, hence:
x + 2y + 3z ≤ 360 (2)
There is 90 oz of powder, hence:
0.5y + z ≤ 90 (3)
solving equation 1, 2 and 3, gives:
x = 70, y= 40, z = 70
The profit is given by:
Profit = 0.4x + 0.5y + 0.8z
substituting x, y and z gives:
Profit = 0.4(70) + 0.5(40) + 0.8(70) = P104
-28=7(x-7) what does x equal
Answer:
x=3
Step-by-step explanation:
7(x - 7) = -28
x - 7 = -4
x = 3
Answer:
x = 3
Step-by-step explanation:
Your goal is to isolate the x from the other numbers.
-28 = 7(x - 7)
Distribute the 7 to the (x - 7)
You will end up with:
-28 = 7x - 49
Add 49 to both sides of the equation to further isolate the x
21 = 7x
Finally, divide both sides by 7 so x is by itself
x = 3
What is the common difference between successive terms in the sequence?
0.36, 0.26, 0.16, 0.06, –0.04, –0.14,
HELP ASAP PLEASE! I tried inputting the numbers into the standard deviation equation but I did not get the right answer to find z. Can someone please help me? Thank you for your time!
Answer:
Z = -1.60
it is low ... it appears that for this problem 2 standard deviations below must be reached to be considered "unusual"
Step-by-step explanation:
Simplify this expression 3^-3
ASAPPPP PLSSSS
Step-by-step explanation:
-27 okay 3^-3 its same as 3^3
Answer: A)
[tex]3^{-3}[/tex]
[tex]3^{-3}=\frac{1}{3^3}[/tex]
[tex]=\frac{1}{3^3}[/tex]
[tex]3^3=27[/tex]
[tex]=\frac{1}{27}[/tex]
OAmalOHopeO
(3.5 x 10 ^ -4) ÷ (5 x 10 ^ 5) in standard form PLZZ ANSWER QUICK
Answer:
7x10 ^-10
Step-by-step explanation:
There were 2,300 applicants for enrollment to the freshman class at a small college in the year 2010. The number of applicants has risen linearly by roughly 170 per year. The number of applications f(x) is given by f(x) = 2,300 + 170x, where x is the number of years since 2010. a. Determine if the function g(x) = * = 2,300 is the inverse of f. 170 b. Interpret the meaning of function g in the context of the problem.
a. No
b. The value g(x) represents the number of years since the year 2010 based on the number of applicants to the freshman class, x.
a. Yes
b. The value 8(x) represents the number of applicants to the freshman class based on the number of years since 2010,
a. No
b. The value slx) represents the number of applicants to the freshman class based on the number of years since 2010,
a. Yes
b. The value six) represents the number of years since the year 2010 based on the number of applicants to the freshman class x
Answer:
The inverse function is [tex]g(x) = \frac{x - 2300}{170}[/tex]
The value of g(x) represents the number of applicants to the freshman class based on the number of years since 2010.
Step-by-step explanation:
Number of applicants in x years after 2010:
Is given by the following function:
[tex]f(x) = 2300 + 170x[/tex]
Inverse function:
We exchange the values of y = f(x) and x in the original function, and then find y. So
[tex]x = 2300 + 170y[/tex]
[tex]170y = x - 2300[/tex]
[tex]y = \frac{x - 2300}{170}[/tex]
[tex]g(x) = \frac{x - 2300}{170}[/tex]
The inverse function is [tex]g(x) = \frac{x - 2300}{170}[/tex]
Meaning of g:
f(x): Number of students in x years:
g(x): Inverse of f(x), is the number of years it takes for there to be x applicants, so the answer is:
The value of g(x) represents the number of applicants to the freshman class based on the number of years since 2010.
Seventeen individuals are scheduled to take a driving test at a particular DMV office on a certain day, nine of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let X be the number among the six who are taking the test for the first time. (a) What kind of a distribution does X have (name and values of al parameters)? 17 hx;6, 9, 17) O h(x; 6,? 17 bx; 6, 9,17) (x; 6, 9, 17) 17 (b) Compute P(X = 4), P(X S 4), and P(X PLX = 4) 0.2851 PX S 4)-13946X RX24) -0.1096 X 4). (Round your answers to four decimal places.) (c) Calculaethe mean value and standard deviation of X. (Round your answers to three decimal places.)
Answer:
a) h(x; 6, 9, 17).
b) P[X=2] = 0.2036
P[X ≤ 2] = 0.2466
P[X ≥ 2] = 0.9570.
c) Mean = 3.176.
Variance = 1.028.
Standard deviation = 1.014.
Step-by-step explanation:
From the given details K=6, n=9, N=-17.
We conclude that it is the hypergeometric distribution:
a) h(x; 6, 9, 17).
b)
[tex]P[X=2]=\frac{(^{g}C_{2})^{17-9}C_{6-2}}{^{17}C_{6}\textrm{}}[/tex]
P[X=2] = 0.2036
P[X ≤ 2] = P(x=0)+ P(x=1) + P(x=2)
P[X ≤ 2] = 0.2466
P[X ≥ 2] = 1-[P(x=0)+P(x=1)]
P[X ≥ 2] = 0.9570.
c)
Mean= [tex]n\frac{K}{N}[/tex]
= 3.176.
Variance = [tex]n\frac{K}{N}( \frac{N-K}{N})(\frac{N-n}{n-1} )[/tex]
= 2.824 x 0.6471 x 0.5625
= 1.028.
Standard deviation = [tex]\sqrt{1.028}[/tex] = 1.014.
A line has a slope of 7 and passes through the point (-2,-1) What is its equation in point-slope form?
Answer:
y + 1 = 7(x + 2).
Step-by-step explanation:
Point slope form:
y - y1 =m(x - x1).
Here m = 7 and (x1, y1) = (-2, -1)
So the answer is;
y - (-1) = 7(x - (-2))
[(2021-Y)-5]*X-X=XX cho biết X,Y,XX là gì?
plzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz help i will give
brainliest
Answer:
55
Step-by-step explanation:
55 appears 3 times, which is the most repetition in the data set
Answer:
55
Step-by-step explanation:
Mode = number that appears most often
The number 55 appears 3 times which is the most out of the other numbers
Hence mode = 55
What is the area of the circle in terms of [tex]\pi[/tex]?
a. 3.4225[tex]\pi[/tex] m²
b. 6.845[tex]\pi[/tex] m²
c. 7.4[tex]\pi[/tex] m²
d. 13.69[tex]\pi[/tex] m²
[tex] \sf \: d \: = 3.7m \\ \sf \: r \: = \frac{3.7}{2} = 1.85 \: m\\ \\ \sf \: c \: = \pi {r}^{2} \\ \\ \sf \: c \: = \pi ({1.85})^{2} \\ \sf c = 1.85 \times 1.85 \times \pi \\ \sf \: c = \boxed {\underline{ \bf a. \: 3.4225\pi \: m ^{2} }}[/tex]
Which property was used to simplify the expression 4(b+2)=4b+8
Answer: distributive property
Step-by-step explanation: the 4 is multiplied by everting in the parenthesis
3.52 A coin is tossed twice. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a head has a 40% chance of occurring, find (a) the joint probability distribution of W and Z; (b) the marginal distribution of W; (c) the marginal distribution of Z
Answer:
a) The joint probability distribution
P(0,0) = 0.36, P(1,0) = 0.24, P(2,0) = 0, P(0,1) = 0, P(1,1) = 0.24, P(2,1)= 0.16
b) P( W = 0 ) = 0.36, P(W = 1 ) = 0.48, P(W = 2 ) = 0.16
c) P ( z = 0 ) = 0.6
P ( z = 1 ) = 0.4
Step-by-step explanation:
Number of head on first toss = Z
Total Number of heads on 2 tosses = W
% of head occurring = 40%
% of tail occurring = 60%
P ( head ) = 2/5 , P( tail ) = 3/5
a) Determine the joint probability distribution of W and Z
P( W =0 |Z = 0 ) = 0.6 P( W = 0 | Z = 1 ) = 0
P( W = 1 | Z = 0 ) = 0.4 P( W = 1 | Z = 1 ) = 0.6
P( W = 1 | Z = 0 ) = 0 P( W = 2 | Z = 1 ) = 0.4
The joint probability distribution
P(0,0) = 0.36, P(1,0) = 0.24, P(2,0) = 0, P(0,1) = 0, P(1,1) = 0.24, P(2,1)= 0.16
B) Marginal distribution of W
P( W = 0 ) = 0.36, P(W = 1 ) = 0.48, P(W = 2 ) = 0.16
C) Marginal distribution of Z ( pmf of Z )
P ( z = 0 ) = 0.6
P ( z = 1 ) = 0.4
Part(a): The required joint probability of W and Z is ,
[tex]P(0,0)=0.36,P(1,0)=0.24,P(2,0)=0,P(0,1)=0,P(1,1)=0.24,\\\\P(2,1)=0.16[/tex]
Part(b): The pmf (marginal distribution) of W is,
[tex]P(w=0)=0.36,P(w=1)=0.48,P(w=2)=0.16[/tex]
Part(c): The pmf (marginal distribution) of Z is,
[tex]P(z=0)=0.6,P(z=1)=0.4[/tex]
Part(a):
The joint distribution is,
[tex]P(w=0\z=0)=0.6,P(w=1|z=0)=0.4,P(w=2|z=0)=0[/tex]
Also,
[tex]P(w=0\z=1)=0,P(w=1|z=1)=0.6,P(w=2|z=1)=0.4[/tex]
Therefore,
[tex]P(0,0)=0.36,P(1,0)=0.24,P(2,0)=0,P(0,1)=0,P(1,1)=0.24,\\\\P(2,1)=0.16[/tex]
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A bus driver makes roughly $3280 every month. How much does he make in one week at this rate.
Answer:
I think around $36
Hope it helps!
Answer:
It depends...
Step-by-step explanation:
It depends how much weeks are in the month if there are three weeks and no extra days then you would have an answer of about 1093 (exact: 1093.33333333). just divide the number of weeks by the number of money.
A wire 9 meters long is cut into two pieces. One piece is bent into a equilateral triangle for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each: For the equilateral triangle:
The length of wire used for the equilateral triangle is approximately 5.61 meters.
The remaining length of wire used for the circle will be 9 - 5.61 ≈ 3.39 meters.
Here,
To minimize the total area of both figures, we need to find the optimal cut point for the wire.
Let's assume the length of the wire used for the equilateral triangle is x meters, and the remaining length of the wire used for the circle is (9 - x) meters.
For the equilateral triangle:
An equilateral triangle has all three sides equal in length.
Let's call each side of the triangle s meters. Since the total length of the wire is x meters, each side will be x/3 meters.
The formula to find the area of an equilateral triangle with side length s is:
Area = (√(3)/4) * s²
Substitute s = x/3 into the area formula:
Area = (√(3)/4) * (x/3)²
Area = (√(3)/4) * (x²/9)
Now, for the circle:
The circumference (perimeter) of a circle is given by the formula:
Circumference = 2 * π * r
Since the remaining length of wire is (9 - x) meters, the circumference of the circle will be 2π(9 - x) meters.
The formula to find the area of a circle with radius r is:
Area = π * r²
To find the area of the circle, we need to find the radius.
Since the circumference is equal to 2πr, we can set up the equation:
2πr = 2π(9 - x)
Now, solve for r:
r = (9 - x)
Now, substitute r = (9 - x) into the area formula for the circle:
Area = π * (9 - x)²
Now, we want to minimize the total area, which is the sum of the areas of the triangle and the circle:
Total Area = (√(3)/4) * (x²/9) + π * (9 - x)²
To find the optimal value of x that minimizes the total area, we can take the derivative of the total area with respect to x, set it to zero, and solve for x.
d(Total Area)/dx = 0
Now, find the critical points and determine which one yields the minimum area.
Taking the derivative and setting it to zero:
d(Total Area)/dx = (√(3)/4) * (2x/9) - 2π * (9 - x)
Setting it to zero:
(√(3)/4) * (2x/9) - 2π * (9 - x) = 0
Now, solve for x:
(√(3)/4) * (2x/9) = 2π * (9 - x)
x/9 = (8π - 2πx) / (√(3))
Now, isolate x:
x = 9 * (8π - 2πx) / (√(3))
x(√(3)) = 9 * (8π - 2πx)
x(√(3) + 2π) = 9 * 8π
x = (9 * 8π) / (√(3) + 2π)
Now, we can calculate the value of x:
x ≈ 5.61 meters
So, the length of wire used for the equilateral triangle is approximately 5.61 meters.
The remaining length of wire used for the circle will be 9 - 5.61 ≈ 3.39 meters.
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At the Fidelity Credit Union, a mean of 3.5 customers arrive hourly at the drive-through window. What is the probability that, in any hour, more than 5 customers will arrive? Round your answer to four decimal places.
Answer:
0.1423 = 14.23% probability that, in any hour, more than 5 customers will arrive.
Step-by-step explanation:
We have the mean, which means that the Poisson distribution is used to solve this question.
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given interval.
A mean of 3.5 customers arrive hourly at the drive-through window.
This means that [tex]\mu = 3.5[/tex]
What is the probability that, in any hour, more than 5 customers will arrive?
This is:
[tex]P(X > 5) = 1 - P(X \leq 5)[/tex]
In which
[tex]P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)[/tex]
Then
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-3.5}*3.5^{0}}{(0)!} = 0.0302[/tex]
[tex]P(X = 1) = \frac{e^{-3.5}*3.5^{1}}{(1)!} = 0.1057[/tex]
[tex]P(X = 2) = \frac{e^{-3.5}*3.5^{2}}{(2)!} = 0.1850[/tex]
[tex]P(X = 3) = \frac{e^{-3.5}*3.5^{3}}{(3)!} = 0.2158[/tex]
[tex]P(X = 4) = \frac{e^{-3.5}*3.5^{4}}{(4)!} = 0.1888[/tex]
[tex]P(X = 5) = \frac{e^{-3.5}*3.5^{5}}{(5)!} = 0.1322[/tex]
Finally
[tex]P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) = 0.0302 + 0.1057 + 0.1850 + 0.2158 + 0.1888 + 0.1322 = 0.8577[/tex]
[tex]P(X > 5) = 1 - P(X \leq 5) = 1 - 0.8577 = 0.1423[/tex]
0.1423 = 14.23% probability that, in any hour, more than 5 customers will arrive.
Hello, please help ASAP. Thank you!
Answer:
23) No
24) No
25) Yes
Step-by-step explanation:
Question 23)
We want to determine if a zero exists between 1 and 2 for the function:
[tex]f(x)=x^2-4x-5[/tex]
Find the zeros of the function. We can factor:
[tex]\displaystyle 0 = (x-5)(x+1)[/tex]
Zero Product Property:
[tex]x-5=0\text{ or } x+1=0[/tex]
Solve for each case. Hence:
[tex]\displaystyle x = 5\text{ or } x=-1[/tex]
Therefore, our zeros are at x = 5 and x = -1.
In conclusion, a zero does not exist between 1 and 2.
Question 24)
We have the function:
[tex]f(x)=2x^2-7x+3[/tex]
And we want to determine if a zero exists between 1 and 2.
Factor. We want to find two numbers that multiply to (2)(3) = 6 and that add to -7.
-6 and -1 suffice. Hence:
[tex]\displaystyle \begin{aligned} 0 & = 2x^2-7x + 3 \\ & = 2x^2 -6x -x + 3 \\ &= 2x(x-3) - (x-3) \\ &= (2x-1)(x-3) \end{aligned}[/tex]
By the Zero Product Property:
[tex]2x-1=0\text{ or } x-3=0[/tex]
Solve for each case:
[tex]\displaystyle x=\frac{1}{2} \text{ or } x=3[/tex]
Therefore, our zeros are at x = 1/2 and x = 3.
In conclusion, a zero does not exist between 1 and 2.
Question 25)
We have the function:
[tex]f(x)=3x^2-2x-5[/tex]
And we want to determine if a zero exists between -2 and 3.
Factor. Again, we want to find two numbers that multiply to 3(-5) = -15 and that add to -2.
-5 and 3 works perfectly. Hence:
[tex]\displaystyle \begin{aligned} 0&= 3x^2 -2x -5 \\ &= 3x^2 +3x - 5x -5 \\ &= 3x(x+1)-5(x+1) \\ &= (3x-5)(x+1)\end{aligned}[/tex]
By the Zero Product Property:
[tex]\displaystyle 3x-5=0\text{ or } x+1=0[/tex]
Solve for each case:
[tex]\displaystyle x = \frac{5}{3}\text{ or } x=-1[/tex]
In conclusion, there indeed exists a zero between -2 and 3.
Please help with this question
9514 1404 393
Answer:
(d) -1/32
Step-by-step explanation:
It may be easier to rearrange the expression so it has positive exponents.
[tex]\dfrac{1}{2^{-2}x^{-3}y^5}=\dfrac{2^2x^3}{y^5}=\dfrac{4(2)^3}{(-4)^5}=-\dfrac{4\cdot8}{1024}=\boxed{-\dfrac{1}{32}}[/tex]
