find the average rate of change of f from pi to 11pi/3. f(x) = cos(x/2) ...?
Answers
The average rate of change of the function f(x) = cos(x/2) from pi to 11pi/3 is (3\sqrt{3}) / (16pi).
Explanation:The average rate of change of a function is the change in the function's value divided by the change in the independent variable. For the function f(x) = cos(x/2), we want to find the average rate of change from pi to 11pi/3. This can be calculated using the following formula:
Average rate of change = (f(b) - f(a)) / (b - a)
Let's calculate f(pi) and f(11pi/3):
f(pi) = cos(pi/2) = 0f(11pi/3) = cos((11pi/3)/2) = cos(11pi/6) = cos(pi/6) since cosine is periodic with period 2picos(pi/6) = \(\sqrt{3}/2\)Now we can substitute these values back into the average rate of change formula:
Average rate of change = (\(\sqrt{3}/2\) - 0) / ((11pi/3) - pi) = \(\sqrt{3}/2\) / (8pi/3) = (3\sqrt{3}) / (16pi)
The average rate of change of [tex]\(f(x) = \cos(\frac{x}{2})\)[/tex] from [tex]\(\pi\)[/tex] to [tex]\(\frac{11\pi}{3}\)[/tex] is [tex]\(-\frac{3\sqrt{3}}{16\pi}\)[/tex]. This represents the slope of the secant line over the given interval.
Let's find the average rate of change of f from π to 11π/3.
[tex]$$f(x)=\cos(\frac{x}{2})$$[/tex]
The average rate of change of a function f over the interval [a, b] is the slope of the secant line that intersects the graph of f at the points (a, f(a)) and (b, f(b)).
In other words, it's the change in f divided by the change in x.
[tex]$$\text{Average rate of change} = \dfrac{f(b) - f(a)}{b - a}$$[/tex]
We are given that [tex]f(x) = \cos(\frac{x}{2}), $a = \pi, and b = \frac{11\pi}{3}.[/tex]
Let's find f(a) and f(b).
[tex]\begin{aligned} f(a) &= f(\pi) \ \ and= \cos(\frac{\pi}{2}) \ \ and= 0 \end{aligned}[/tex]
[tex]$\begin{aligned} f(b) &= f\left(\frac{11\pi}{3}\right) \ \ &= \cos\left(\frac{11\pi}{6}\right) \ \ &= -\frac{\sqrt{3}}{2} \end{aligned}$[/tex]
Now we can plug these values into the formula for the average rate of change.
[tex]$\begin{aligned} \text{Average rate of change} &= \dfrac{f(b) - f(a)}{b - a} \ \ &= \dfrac{-\frac{\sqrt{3}}{2} - 0}{\frac{11\pi}{3} - \pi} \ \ &= \dfrac{-\frac{\sqrt{3}}{2}}{\frac{8\pi}{3}} \ \ &= -\dfrac{3\sqrt{3}}{16\pi} \end{aligned}$[/tex]
Therefore, the average rate of change of f from π to 11π/3 is [tex]-\dfrac{3\sqrt{3}}{16\pi}.[/tex]
Related Questions
Which of these statements is punctuated properly?
"You're kidding she said". You are not going in there.
"You're kidding" she said "You are not going in there."
"You're kidding," she said. "You are not going in there."
"You're kidding, she said. You are not going in there."
Answers
7/8,0.98,8/9 least to greatest
Answers
8/9
.98
hope this helps good luck
An object with a mass of 6.3 kg has a mass of 7.1newtons applied to it, what is the resulting acceleration of the object
Answers
Therefore a = 7.1 / 6.3 = 1.13 m/sec^2
The resulting acceleration of the object is approximately [tex]\( 1.127 \, \text{m/s}^2 \).[/tex]
To find the resulting acceleration of an object when a force is applied to it, you can use Newton's second law of motion, which states:
F = ma
Given that the force F = 7.1 Newtons and the mass m = 6.3 kg, we can rearrange the formula to solve for acceleration:
[tex]\[ a = \frac{F}{m} \]Substitute the given values:\[ a = \frac{7.1 \, \text{N}}{6.3 \, \text{kg}} \]Now, divide to find the acceleration:\[ a = \frac{7.1}{6.3} \, \text{m/s}^2 \]\[ a = 1.127 \, \text{m/s}^2 \][/tex]
So, the resulting acceleration of the object is approximately [tex]\( 1.127 \, \text{m/s}^2 \).[/tex]
Complete question:
An object with a mass of 6.3 kg has a force of 7.1 Newtons applied to it. What is the resulting acceleration of the object?
Determine the x values that cause the polynomial function to be positive:
f(x)=(x+2)(x+1)(x-5) Determine the x values that cause the polynomial function to be positive:
f(x)=(x+2)(x+1)(x-5)
Answers
Set each one greater than zero:x+2>0x+1>0x−5>0
Then, solve for x on all three:
x > -2 x > -1 x > 5
so in order for the polynomial to be positive the answer is x > 5.
Hope this is the answer that you are looking for.
Amounts of trade discounts are not fixed? True or False ...?
Answers
Trade discounts are not fixed; they are true to vary based on several factors including order volume, buyer-seller relationships, and industry norms. They are a dynamic component of economic transactions within the complex trade environment and are influenced by market interactions and broader economic principles.
The statement "Amounts of trade discounts are not fixed" is true. Trade discounts can vary greatly and are typically influenced by a variety of factors including the volume of the goods ordered, the relationship between the buyer and seller, and the industry standards.
For instance, when businesses deliver larger lots, volume discounts may be offered, which reflect the lower costs of providing goods in bulk rather than differences in willingness to pay. This variation in trade discounts ensures that each transaction is adjusted to meet the precise needs of both the buyer and seller, depending on the specific circumstances of the trade.
Moreover, the trade environment is complex and the interactions between buyers and sellers determine the gains from trade in an economy. These interactions depend on market structure, negotiation possibilities, and the mode of sale, such as retail or auction.
Consequently, trade discounts are part of a dynamic economic system where various elements such as free trade agreements, industry adjustments, transaction costs, and the unmeasured benefits of competition and product variety all play a role in shaping the final outcome.
Express the function \[\frac{n^3}{1000} - 100n^2 - 100n + 3\]in terms of \(\huge{\Theta}\)notation ...?
Answers
I hope my answer has come to your help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead!
Let θ (in radians) be an acute angle in a right triangle, and let x and y, respectively, be the lengths of the sides adjacent and opposite θ. Suppose also that x and y vary with time.
a. How are dθ/dt, dx/dt and dy/dt related?
Please give steps and explain!
Answers
tan(theta(t))=y(t)/x(t)
differentiate
sec^2(theta(t))*theta'(t)=y'(t)x(t)-y(t)x'(t)/x^2(t)
thus
theta'(t)=(y'(t)x(t)-y(t)x'(t))
divided by (x^2(t)*sec^2(θ(t))
Answer:
dθ/dt = [(cos^2 θ)*(dy/dt * x - y * dx/dt)]/(x^2)
Step-by-step explanation:
Given that x and y are the lengths of the sides adjacent and opposite θ, then they are related by:
tan θ = y/x
Differentiating respect to t, we get:
sec^2 θ * dθ/dt = (dy/dt * x - y * dx/dt)/(x^2)
dθ/dt = [(cos^2 θ)*(dy/dt * x - y * dx/dt)]/(x^2)
Choose the method of pay that would result in the most earnings for one month on sales of $73,620. a. Straight commission of 6% on all sales. b. Monthly salary of $3,000 plus 2% commission on all sales. c. Graduated commission of 4% on the first $50,000 in sales and 9% on anything over that. d. Graduated commission of 5% on the first $35,000 in sales and 7% on anything over that.
Answers
b. 3000 + 0.02(73,620) = 3000 + 1472.40 = 4472.40 (salary + 2% com)
c. 0.04(50,000) + 0.09(73620 - 50,000) = 2000 + 2125.80 = 4125.80
d. 0.05(35,000) + 0.07(73620 - 35,000) = 1750 + 2703.40 = 4453.40
most earnings would be : B. monthly salary + 2% commission
The method of pay that results in the most earnings is the second method of pay
In order to determine which method of pay would result in the most earnings , the value of each method would be determined
Value of the first option
6% commission on all sales = 6% x $73,620.
0.06 x $73,620. = $4,417.20
Value of the second option
Monthly salary of $3,000 plus 2% commission on all sales = $3,000 + (2% x $73,620)
$3,000 + (0.02% x $73,620)
= $3000 + $1472.40
= $4472.40
Value of the third option
Graduated commission of 4% on the first $50,000 in sales and 9% on anything over that.
= (4% x $50,000) + (9% x $23,620)
= $2000 + $2125.80
= $4125.80
Value of the fourth option
Graduated commission of 5% on the first $35,000 in sales and 7% on anything over that.
= (5% x$35,000) + (7% x $38,620)
= $1750 + $2703.40
= $4,453.40
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Elsie pays $21.75 for 5 student tickets to the fair what is the cost of each student ticket
Answers
2 ÷ 5 = 0
21 ÷ 5 = 4 (remainder 1)
17 ÷ 5 = 3 (remainder 2)
25 ÷ 5 = 5
So it would cost $4.35 each, hope this helps!
What polynomial must be added to x^2-2x+6 so that the sun is 3x^2+7x?
Answers
For example, what do you add to 5 to get 13?
Subtract 13 - 5 = 8
The answer is you add 8 to 5 to get 13.
Now you need to do the same with your polynomials.
What do you add to x^2 - 2x + 6 to get 3x^2 + 7x?
To find the answer, subtract
(3x^2 + 7x) - (x^2 - 2x + 6)
Find the product.
(-2x ^2) ^3 -3x
Answers
Final answer:
To find the product of (-2x^2)^3 and -3x, first calculate the cube of -2x^2 (-8x^6), then multiply it by -3x to obtain 24x^7.
Explanation:
The student has asked to find the product of two expressions, (-2x^2)^3 and -3x. To solve this, we first calculate the cube of -2x^2, which is -8x^6, then we multiply this result by -3x to find the final product.
So, let's perform these two steps:
Cube of -2x^2: (-2x^2)^3 = (-2)^3 × (x^2)^3 = -8 × x^6 = -8x^6
Multiplication by -3x: -8x^6 × -3x = 24x^7
Therefore, the product is 24x^7.
[6.01] Systems of equations with different slopes and different y-intercepts have more than one solution.
Always
Sometimes
Never
Answers
if a polynomial is divided by (x-a) and the remainder equals zero then (x -a) is a factor of the polynomial. true or false
Answers
The Factor theorem states that for any polynomial f(x) if f(c)=0 then x-c is a factor of the polynomial f(x).
If any polynomial f(x) is divided by x-a and remainder is 0 that means f(a)= 0 .In other words we can say x-a is a factor of the polynomial f(x).So the statement :If a polynomial is divided by (x-a) and the remainder equals zero then (x -a) is a factor of the polynomial is True.
It is true that (x -a) is a factor of the polynomial.
How to determine the true statement?Let the polynomial function be f(x)
When divided by x -a, we have:
f(x)/(x - a) = Some polynomial remainder 0
The above can be represented as
f(a) = 0
This means that it is true that (x -a) is a factor of the polynomial.
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Explain how to find the distance between -2 and 3 on a number line
Answers
|x-y|
|-2-3|=|-5|=5
the distance is 5
or
it take 2 units to get to 0 from -2
it takes 3 units to get from 0 to 3
2+3=5
what is another way to express 48 + 32?
Answers
Answer:16( 3+2)
Step-by-step explanation:
Another way of expressing 48 + 32 are:
32 + 48 and 80.
Addition is a basic mathematical operation that combines two numbers to give a total or sum.
The given expression is:
48 + 32
The commutative property is:
A+B = B+A
Since Addition always holds the commutative property
Therefore, we can write the given expression as,
32 + 48
We can also express it by direct sum,
3 2
+ 4 8
_______
8 0
_______
So, the sum of 32 and 48 is 80.
Hence,
We can express the given expression as 32 + 48 and 80.
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Find the value or values of p in the quadratic equation p2 + 13p – 30 = 0. A. p = 15, p = 2 B. p = –10, p = –3 C. p = 10, p = 3 D. p = –15, p = 2
Answers
p^2+13p-30=0
p +15
p -2
(p+15)(p-2)=0
p= -15
p=2
Answer:
The correct option is D.
Step-by-step explanation:
The given quadratic equation is
[tex]p^2+13p-30=0[/tex]
First find two numbers whose sum is 13 and whose product is -30.
The two numbers are 15 and -2.
[tex]p^2+(15-2)p-30=0[/tex]
[tex]p^2+15p-2p-30=0[/tex]
[tex]p(p+15)-2(p+15)=0[/tex]
[tex](p+15)(p-2)=0[/tex]
By using zero product property, equate each factor equal to 0.
[tex]p+15=0[/tex]
[tex]p=-15[/tex]
[tex]p-2=0[/tex]
[tex]p=2[/tex]
Therefore the values of p are -15 and 2. Option D is correct.
The vertices of ∆ABC are A(-2, 2), B(6, 2), and C(0, 8). What is the perimeter of ∆ABC in units?
Answers
L(AB) = √[(6 - -2)² + (2 - 2)²] = 8
L(BC) = √[(0 - 6)² + (8 - 2)²] = 8.4853
L(AC) = √[(0 - -2)² + (8 - 2)²] = 6.3246
8 + 8.4853 + 6.3246 = 14.8099
I hope this is what you were looking for.
During a manufacturing process, a metal part in a machine is exposed to varying temperature conditions. The manufacturer of the machine recommends that the temperature of the machine part remain below 132 ° F. The temperature T in degrees Fahrenheit x minutes after the machine is put into operation is modeled by T = -0.005x^2 + 0.45x + 125. Will the temperature of the part ever reach or exceed 132 ° F? Use the discriminant of a quadratic equation to decide.
A. no
B. yes
Answers
Answer: The answer saying YES, is WRONG, the CORRECT ANSWER IS NO.
Answer:
The answer is no
0.45^2-4(-.005)(125)=2.7025
so no it do not exceed
Step-by-step explanation:
What is the equation of the line that passes through (4, -1) and (-2, 3)?
2x - 3y - 5 = 0
2x + 3y - 5 = 0
-2x + 3y - 5 = 0
I know for sure that the answer is 2x + 3y - 5 = 0 I just need somone to explain it!
Answers
substitute (4,-1) where x=4 and y=-1
2*4+3*(-1)-5 = 0
8-3-5=0
0=0
correct!
this point belongs to this line
test point (-2,3) where x = -2 and y=3
2x+3y-5 = 0
substitute
2*(-2) + 3*3 - 5 = 0
-4+9-5=0
0=0
belongs!
Find the missing length indicated.
Answers
generate the first five terms in the sequence using the explicit formula. yn=-5n-5
Answers
y1 is y on position 1 and so on. Sequence goes: y1,y2,y3,y4... and so on
So all you have to do is to implement values 1,2,3,4,5 for n and get 5 members of sequence. (start from 0 if your n starts from 0)
y0 = -5
y1 = -5-5 = -10
y2 = -15
y3 = -20
y4 = -25
y5 = -5*5 - 5 = -30
The first five terms in the sequence generated by the explicit formula [tex]y_n[/tex] = -5n - 5 are: -10, -15, -20, -25, -30.
To generate the first five terms in the sequence using the explicit formula [tex]y_n[/tex] = -5n - 5,
we substitute the values of n from 1 to 5 into the formula.
For n = 1:
[tex]y_1[/tex]= -5(1) - 5 = -10
For n = 2:
[tex]y_2[/tex] = -5(2) - 5 = -15
For n = 3:
[tex]y_3[/tex] = -5(3) - 5 = -20
For n = 4:
[tex]y_4[/tex] = -5(4) - 5 = -25
For n = 5:
[tex]y_5[/tex] = -5(5) - 5 = -30
Therefore, the first five terms in the sequence generated by the explicit formula [tex]y_n[/tex] = -5n - 5 are:
-10, -15, -20, -25, -30.
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what is 1626 divided by 12 as a mixed number
Answers
So, your final answer is 135 1/2
Hope this helps!
Answer:
1626 divided by 12 is written as [tex]\frac{271}{2}=135\frac{1}{2}[/tex]
Step-by-step explanation:
To find : What is 1626 divided by 12 as a mixed number ?
Solution :
Writing 1626 divided by 12 in a fraction from,
[tex]\frac{1626}{12}[/tex]
Writing fraction in simpler form by reducing with common factors,
[tex]\frac{1626}{12}=\frac{2\times 3\times 271}{2\times 2\times 3}[/tex]
[tex]\frac{1626}{12}=\frac{271}{2}[/tex]
Writing [tex]\frac{271}{2}[/tex] in mixed fraction by dividing 271 by 2,
[tex]271=135\times 2+1[/tex]
i.e. [tex]\frac{271}{2}=135\frac{1}{2}[/tex]
Therefore, 1626 divided by 12 is written as [tex]\frac{271}{2}=135\frac{1}{2}[/tex]
If Zelda has a 300 inch roll of binding for a quilt and the quilt must be 84 inches long how wide can she make it
Answers
how to solve a quadratic equation by using the quadratic formula?
Answers
you can check this question
Final answer:
To solve a quadratic equation using the quadratic formula, put the equation in ax² + bx + c = 0 form, identify a, b, and c, plug them into the formula x = (-b ± √(b² - 4ac)) / (2a), calculate the discriminant, and find the two possible values for x.
Explanation:
To solve a quadratic equation by using the quadratic formula, you should follow these steps:
First, make sure that the equation is in the standard quadratic form, which is ax² + bx + c = 0.Identify the coefficients a, b, and c from the equation.Plug these values into the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a).Calculate the discriminant D, which is the part under the square root: b² - 4ac.The discriminant tells you the nature of the roots:If D is positive, there are two real and distinct solutions.If D is zero, there is one real solution.If D is negative, there are two complex solutions.Finally, calculate the two possible values for x using the plus and minus signs in the formula and simplify as needed.
For example, if we have the equation t² + 10t - 200 = 0, by identifying a=1, b=10, and c=-200, and plugging them into the quadratic formula, we can solve for t.
write the statement using absolute value notation:
the distance between x and 4 is 3
Answers
The supreme esteem condition above implies that x-4 is equivalent to 3, OR, that x-4 is equivalent to - 3.
On the off chance that x-4=3, then x=7. On the off chance that x-4=-3, then x=1.
Presently envision the number line with the numbers 1 and 7 set apart with a dab.
The number 4 is in the focal point of that fragment. What's more, the numbers 1 and 7 are at a separation of precisely 3 units of the number 4 which is at the middle.
That is precisely what the outright esteem condition is requesting that you discover. Which numbers are inside a separation of 3 units of 4. Also, your answer is x=1 and x=7.
What is the determinant of an identity matrix?
Answers
Answer:
The determinant of an identity matrix is always 1.
Step-by-step explanation:
Given : An identity matrix.
We have to find the determinant of an identity matrix.
Consider an identity matrix,
Identity matrix is a matrix having entry one in its diagonal and rest all entries are zero.
Let us consider a 2 × 2 identity matrix,
[tex]I=\left[\begin{array}{cc}1&0\\0&1\end{array}\right][/tex]
We know determinant of a 2 × 2 matrix
[tex]I=\left[\begin{array}{cc}a&b\\c&d\end{array}\right][/tex]
is given by
D = ad - bc
Thus, here a = 1 , b = 0 , c= 0 d= 1
Thus, D = 1 - 0 = 1
Thus, The determinant of an identity matrix is always 1.
Which of the following is the most profitable investment for a candy shop that earns $1 profit per pound of candy?
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Worker at $10 per hour, producing eight pounds of candy per hour
Worker at $12 per hour, producing 16 pounds of candy per hour
Machine with $5 per hour operating cost, producing 10 pounds of candy per hour
Machine with $8 per hour operating cost, producing 14 pounds of candy per hour
Answers
Answer:
Option 4). Machine with $8 per hour operating cost, producing 14 pounds of candy per hour.
Step-by-step explanation:
We have to find the most profitable investment for a candy shop that earns $1 per pound of candy.
We will take up each option one by one.
Option 1).
Producing eight pounds of candy per hours means profit = $1×8 = $8
But workers take $10 per hours so expenditure = $8 - $10 = -$2
There is a loss of $2 in this investment.
Option 2).
Production of 16 pounds candy per hour will make the profit = $1 × 16 = $16
But workers take $12 per hour so profit earned = 16 - 12 = $4
Option 3).
Machine produces 10 pound of candy per hour so profit will be = $1 ×10 = $10
Operating cost of the machine = $5 per hour
So profit generated = profit - Operating cost of machine
= 10 - 5 = $5
Option 4).
Profit earned by production of 14 pounds of candy per hour = $1 × 14 = $14
Operating cost of the machine = $8 per hour
Therefore, Total profit per hour = 14 - 8 = $6
Finally we can say that the maximum profit will be generated in option 4.
Therefore, Option 4) will be the best option to invest.
Final answer:
The most profitable investment for a candy shop is the machine with an $8 per hour operating cost, producing 14 pounds of candy per hour, yielding a net profit of $6 per hour.
Explanation:
The task is to determine the most profitable investment for a candy shop that makes $1 profit per pound of candy. To calculate the profitability of each option, we need to figure out the net profit per hour for each.
Worker at $10 per hour, producing eight pounds of candy per hour:
Profit = 8 pounds * $1 profit per pound - $10 wage per hour = $8 - $10 = -$2 per hour.
Worker at $12 per hour, producing 16 pounds of candy per hour:
Profit = 16 pounds * $1 profit per pound - $12 wage per hour = $16 - $12 = $4 per hour.
Machine with $5 per hour operating cost, producing 10 pounds of candy per hour:
Profit = 10 pounds * $1 profit per pound - $5 operating cost per hour = $10 - $5 = $5 per hour.
Machine with $8 per hour operating cost, producing 14 pounds of candy per hour:
Profit = 14 pounds * $1 profit per pound - $8 operating cost per hour = $14 - $8 = $6 per hour.
Comparing the net profit per hour for each option, the Machine with $8 per hour operating cost, producing 14 pounds of candy per hour, is the most profitable investment.
What is the apparent solution to the system of equations graphed above?
(0,-1)
(0,3)
(1,2)
(2,1)
Answers
This method here is graphical method of solving system of equations. Other method is, of course, solving system of equations using math...
Answer is (1,2)
435 is 15% of what number?
Answers
Write a function Rule that represents the situation: A workers earnings e are a function of the number of hours n worked at a rate of $8.75 per hour
Answers
The function rule representing a worker's earnings e as a function of hours n worked at a rate of $8.75 per hour is e(n) = 8.75 × n. This formula is used to calculate the worker's pay based on their hours worked.
Explanation:To represent a worker's earnings e as a function of the number of hours n worked at a rate of $8.75 per hour, you would write the function rule as:
e(n) = 8.75 × n
Here, n represents the number of hours worked and e(n) represents the earnings for those hours.
For example, if a worker puts in 8 hours of work, their earnings would be calculated as:
e(8) = 8.75 × 8
= $70
This means the worker would earn $70 for an 8-hour workday.