Problem 1 (a) Use the bisection method to solve the equation x3 – x-1= 0 on the interval [0, 2]. Take three steps only and record the final interval. Use [0,2) as your initial interval. 1 (b) Does the equation cos(3x) = x have a solution on [0,1/3]? Apply the hypotheses of the Intermediate Value Theorem to show that there is a solution in [0,1/3]. Problem 2 (a) Evaluate the following limit. Show the steps and the limit laws used. sinx cosx lim 2 +0 (3 - x) ** (b) Suppose the inequality S1-cosxs is valid for x values close to zero. Show how this inequality can be used to evaluate the following limit. State any theorem that you use. 1 - COS X lim x2 Problem 3 (a) Let f(x) = V2 - x. Use the formal definition of derivative to find f'(x). (b) Find the equation of the line tangent to the graph of f(x) at x-2. (c) Graph both the function and the tangent line on the same coordinate system 2 24 X-70
Answers
The final interval containing the root is [1, 1.25). The equation has a solution on the given interval. The limit of the given expression is sin(4)/2. The derivative of f(x) is f'(x) = -1 / (2 * √(2 - x)). The equation of the tangent line to the graph of f(x) at x=2 is y = -x/√2 + 2/√2.
Using the bisection method, we start with the interval [0, 2) and take three steps to find the root of the equation x³ – x - 1 = 0.
f(1) = 1³ - 1 - 1 = -1
We evaluate the function at the midpoint of the interval [1, 2), which is 1.5.
f(1.5) = 1.5³ - 1.5 - 1 = 1.375
Since f(1.5) is positive, the root must be in the interval [1, 1.5).
We evaluate the function at the midpoint of the interval [1, 1.5), which is 1.25.
f(1.25) = 1.25³ - 1.25 - 1 = 0.265625
Since f(1.25) is positive, the root must be in the interval [1, 1.25).
Next, we evaluate f(0) and f(1/3):
f(0) = cos(0) - 0 = 1 > 0
f(1/3) = cos(1) - 1/3 < 0
Since f(0) is positive and f(1/3) is negative, by the Intermediate Value Theorem, there exists a solution to the equation cos(3x) = x on [0, 1/3].
We are given the limit:
sin(x)cos(x)/(3 - x)
Since sin(x) and cos(x) are both continuous functions, limits as x approaches 2 from the right:
sin(2) ≈ 0.9093
cos(2) ≈ -0.4161
Using these individual limits,
sin(x)cos(x) = (sin(x))(cos(x)) = (1/2)sin(2x)
(1/2)sin(2x)/(3 - x)
We can now use the limit laws,
(1/2)sin(2x)/(3 - x) = (1/2) sin(2x)/(3 - x)
= (1/2)sin(4)/(1)
= sin(4)/2
Suppose the inequality 1 - cos(x) < x is valid for x values close to zero.
(1 - cos(x))/x²
1/x < 1 + cos(x)/x
Taking the reciprocal of both sides,
x < 1/(1 + cos(x)/x)
1 - cos(x) < (1 - cos(x))/[x²(1 + cos(x)/x)²]
To find the derivative of f(x) using the formal definition,
f'(x) = [√(2 - (x + h)) - √(2 - x)] / h
f'(x) = [(√(2 - (x + h)) - √(2 - x)) * (√(2 - (x + h)) + √(2 - x))] / [h * (√(2 - (x + h)) + √(2 - x))]
Simplifying the numerator,
f'(x) = [-1 / (√(2 - (x + h)) + √(2 - x)) * (√(2 - (x + h)) - √(2 - x))] / h
Taking the limit as h approaches 0,
f'(x) = -1 / (2 * √(2 - x))
To find the equation of the tangent line to the graph of f(x) at x=2, we first find the value of f(2),
f(2) = √(2-2) = √0 = 0
Now, we use the point-slope form of the equation of a line, with the point (2,0) and the slope f'(2) = -1/√2,
y - 0 = (-1/√2)(x - 2)
y = -x/√2 + 2/√2
To graph both the function and the tangent line on the same coordinate system, we plot the function f(x) = √(2-x) and the tangent line y = -x/√2 + 2/√2 on the same set of axes. The dashed green-white line is the graph of f(x) = √(2-x), and the white line is the tangent line y = -x/√2 + 2/√2 passing through the point (2,0).
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Related Questions
1) At the supermarket you can fill your own honey bear container. A customer buys 12 oz of honey for $5.40.
Answers
From the problem, the cost of 12 oz honey is $5.40
a. the cost of honey per ounce will be :
\(\frac{\$5.40}{12oz}=\$0.45\text{per oz}\)The answer is $0.45 per ounce.
b. the amount of honey per dollar will be :
\(\frac{12oz}{\$5.40}=2.22\)The answer is 2.22 oz per dollar
c. when h = ounces of honey and c = cost in dollar
The equation will be :
For the cost of honey per ounce :
\(c=0.45h\)For the amount of honey per dollar :
\(h=0.22c\)d. Graphing one equation, we will graph c = 0.45h
A line with points (0, 0) and (12, 5.40)
x-axis will be the amount of honey
y-axis will be the cost in dollar.
Help ill give brainlist
Answers
Answer:
x=2
Step-by-step explanation:
Match the addition and subtraction problems to solutions
Answers
Answer: 1B. 2C 3E 4D 5E
Step-by-step explanation:
Answer:
\(\frac{5}{7} - \frac{3}{7} =\) B \(\frac{2}{7}\)
\(\frac{1}{6} + \frac{3}{6} =\) C \(\frac{2}{3}\)
\(\frac{1}{2} + \frac{1}{4} =\) E \(\frac{3}{4}\)
\(\frac{1}{5} + \frac{1}{2} =\) D \(\frac{7}{10}\)
\(\frac{1}{6} - \frac{1}{8} =\) A \(\frac{1}{24}\)
a function is translated from f(x)=2⋅3x 4 to g(x)=2⋅3x−1. what is the effect on f(x)?
Answers
The effect on f(x) is that it is shifted one unit to the left on the x-axis to obtain g(x).
Here,
In mathematics, a graph is a visual representation of a set of objects, called vertices or nodes, and the connections between them, called edges or links.
The function f(x) is translated to g(x) by subtracting 1 from the exponent of 3.
This means that g(x) is obtained by shifting f(x) one unit to the right on the x-axis.
The coefficient of 2 remains unchanged, which means that the overall scaling of the function remains the same.
Therefore, the effect on f(x) is that it is shifted one unit to the left on the x-axis to obtain g(x).
In other words, the graph of g(x) is obtained by taking the graph of f(x) and shifting it to the right by one unit.
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Can someone answer this for me?
Answers
Answer: as per question statement
we need to set up an equation first
p(t)=1200e(0.052*t)
they have given that it is relative to 1200 that means it starts to increase from 1200 at t=0 initially 1200 bacteria were present
we need to find population at t=6
we need to plug t=6 in p(t).
P(6)=1200e(0.052*6)=1639.38
1638.38 bacteria were present at that time t=6
Step-by-step explanation: I hope this helps.
Nate had 300 cookies for class. He dropped 145 on his way to school. His mom brought him 123 more cookies. How many cookies does Nate have now?
423
278
350
155
Answers
Answer: Nate have 278 cookies now.
Step-by-step explanation:
the original amount of cookies for Nate is 300, and he dropped 145, which means we need to subtract 145 from 300. And his mom brought him 123 more, which means we need to add 123.
300 - 145 + 123 = 278
The amplitude of a periodic function is 8.5 and its minimum value is 0. What is the functions
maximum value?
Answers
Answer:
maximum value is 17.
Step-by-step explanation:
8.5=1/2(x-0)
8.5=1/2x
8.5=x/2
8.5×2=x
17=x
x=17
please somebody help
Answers
The factored forms of the equations and the intercepts would be:
g(x) = x² - 4x - 21:
Factored ⇒ g(x) = (x - 7)(x + 3) x - intercept ⇒ x = 7 or x = -3y - intercept ⇒ (0,- 21)h(x) = x² + 8x + 12:
Factored ⇒ h(x) = (x + 2)(x + 6) x - intercept ⇒ (-2,0) and (-6,0) y - intercept ⇒ (0 ,12 ) How to find the factored versions and intercepts ?For g(x) = x² - 4x - 21:
To find the factored form, we need to factor the quadratic expression:
g(x) = (x - 7)(x + 3)
To find the x-intercepts, we set g(x) = 0 and solve for x:
(x - 7)(x + 3) = 0
x = 7 or x = -3
To find the y-intercept, we set x = 0 and evaluate g(x):
g(0) = (0 - 7)(0 + 3) = -21
Therefore, the y-intercept of g(x) is (0,-21).
For h(x) = x² + 8x + 12:
h(x) = (x + 2)(x + 6)
To find the x-intercepts, we set h(x) = 0 and solve for x:
(x + 2)(x + 6) = 0
x = -2 or x = -6
To find the y-intercept, we set x = 0 and evaluate h(x):
h(0) = (0 + 2)(0 + 6) = 12
Therefore, the y-intercept of h(x) is (0,12).
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PLS HELP WILL MARK BRAINLIEST, PLS HURRY
Answers
Answer:
C IS THE ANSWER
Step-by-step explanation:
plz tell me if I am right plZ
Answer:
c
Step-by-step explanation:
Tell whether the angles are complementary or supplementary. Then find the value of x
Answers
Answer:
complementaryx = 55
Step-by-step explanation:
if the sum of two angles is 90 degrees, then they are said to be complementary angles, and they form a right angle together.
Find xknowing that the sum is 90° we solve with a proportionx - 20 + x = 90
2x = 110
x = 55
-----------------------------
check
55 - 20 + 55 = 90
90 = 90
the answer is good
help me please, I been trying to get help for hours now.
Answers
We can conclude that -
General exponential growth equation is : \($f(x)=a(1+r)^{x}\)General exponential decay equation is : \($\frac {dN}{dt}= -\lambda N\)Option {A} and option {E} represent the exponential decay.Option {B}, {E} and {D} represent exponential growth.What is exponential function?The exponential function is of the form -
f(x) = eˣ
Given is to write the exponential growth and exponential decay equation.
{ 1 } -
The general exponential growth equation is -
\($f(x)=a(1+r)^{x}\)
{ 2 } -
The general exponential decay equation is -
\($\frac {dN}{dt}= -\lambda N\)
Option {A} and option {E} represent the exponential decay. Option {B}, {E} and {D} represent exponential growth.
Therefore, we can conclude that -
General exponential growth equation is : \($f(x)=a(1+r)^{x}\)General exponential decay equation is : \($\frac {dN}{dt}= -\lambda N\)Option {A} and option {E} represent the exponential decay.Option {B}, {E} and {D} represent exponential growth.To solve more questions on functions, visit the link-
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Solve system of linear equation with substitution
Answers
Answer:
x=1, y=2
Step-by-step explanation:
solve by substitution so substitute x= 6y-11 in the next equation
3 (6y-11) - 2y = -1
18y -33-2y = -1
16y = 33-1
y= 32/16 =2
x= 6y -11 = 6*2 -11 = 12 -11 = 1
What is the total surface area of the shape below?
Length) Line A-B = 4cm
Width) Line B-C = 2cm
Height) A-E = 3cm
Answers
Answer:
24
Step-by-step explanation:
What is the volume of a sphere with a radius of 18 units?
A. 77767 units
B. 1296 units
C. 5832 units
D. 1944 units
It's C
Answers
There was an activity planned outdoors, but it started to rain. They took a vote and 45 people of the 60 people wanted to stay inside. The other 15 people voted to go outside in the rain. What percent of the class wants to go outside? Round to the nearest percent if needed.
Answers
Answer:
15/60×100=25%
Step-by-step explanation:
the number of students wanted to go outside/the whole number of students×100
15/60×100=25%
A club has 200 members, 45 of whom are lawyers, 38 of the memebres are liars, while 132 are neither lawyers nor liars. What is the probability that if a random person is randomly chosen from the group of lawyers, the person will be a liar?
Answers
The probability that if a random person is chosen from the group of lawyers, the person will be a liar is 38/45, or 0.84.
As a fraction: The probability is given as 38/45, which means that out of 45 people chosen randomly from the group of lawyers, 38 of them are expected to be liars.
As a decimal: To express the probability as a decimal, we divide the numerator (38) by the denominator (45):
38 ÷ 45 ≈ 0.8444444444444444
Rounded to two decimal places, this would be approximately 0.84.
As a percentage: To express the probability as a percentage, we multiply the decimal form by 100:
0.8444444444444444 * 100 ≈ 84.44%
Rounded to two decimal places, this would also be approximately 84.44%.
So, the probability that if a random person is chosen from the group of lawyers, the person will be a liar can be expressed as 38/45 as a fraction, approximately 0.84 as a decimal, or approximately 84.44% as a percentage.
This can be expressed as a fraction, decimal, or percentage, whichever is more helpful.
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I thought of a number, divided it by 1 1/2 , then added 2 1/2 to the result, then multiplied the sum by 1 2/3 . Then I subtracted 1 2/3 and got the result 2 5/6 . What was my number?
Answers
Answer: The number is \(\dfrac3{10}\) or 0.3.
Step-by-step explanation:
Let x be the number.
Then,
\(((x\div 1\dfrac12)+2\dfrac12)\times1\dfrac23-1\dfrac23=2\dfrac56\)
First simplify each mixed fraction into improper fraction, we get
\(((x\div \dfrac32)+\dfrac52)\times\dfrac53-\dfrac53=\dfrac{17}6\\\\\text{Now, }((x\div \dfrac32)+\dfrac52)\times\dfrac53=\dfrac{17}{6}+\dfrac53\\\\\Rightarrow\ ((x\div \dfrac32)+\dfrac52)\times\dfrac53=\dfrac92\\\\\Rightarrow\ ((x\div \dfrac32)+\dfrac52)=\dfrac92\times\dfrac35\\\\\Rightarrow\ ((x\div \dfrac32)+\dfrac52)=\dfrac{27}{10}\\\\\Rightarrow\ (x\div \dfrac32)=\dfrac{27}{10}-\dfrac52\\\\\Rightarrow\ (x\div \dfrac32)=\dfrac2{10}\\\\\Rightarrow\ x=\dfrac2{10}\times\dfrac32\\\\\Rightarrow\ x=\dfrac3{10}\)
Hence, the number is \(\dfrac3{10}\) or 0.3.
Which side of the triangle in the diagram is the hypotenuse
Answers
Answer:
X side of the triangle in the diagram is the hypotenuse because it's side is long than other side
evaluate 11-(-8)-(-3)
Answers
Answer:
22
Step-by-step explanation:
11-(-8)-(-3)
19-(3)
19+3
22
hope i helped:)
numeric prefixes are used to name select one compounds. they are not used for select one compounds, but the charges must select one .
Answers
Greek number prefixes are employed in the naming of molecular (covalent) substances.
Greek number prefixes are employed in the naming of molecular (covalent) substances. Prefixes are always combined in order of size (left-to-right in the table below). For instance, "heptadecatetracta" (7 + 10 + 400) would be the prefix for the number 417.
We utilise prefixes (mono-, di-, tri-, etc.) at the beginning of each element in molecular compounds to denote the element's subscript, but we only use the prefix mono- for the compound's second element.
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if you drive at 55 mph on dry, level pavement, your average stopping distance will be
Answers
The average stopping distance when driving at 55 mph on dry, level pavement can vary depending on several factors such as vehicle condition, road conditions, and driver reaction time.
Therefore, it is not possible to provide an exact average stopping distance without more specific information.
However, as a general guideline, it is commonly recommended to use the "3-second rule" for following distance. This means that you should maintain a distance from the vehicle in front of you that would allow you to come to a complete stop within three seconds. At higher speeds like 55 mph, the stopping distance will be longer compared to lower speeds.
To estimate the average stopping distance, you can consider the total stopping distance, which is the sum of the perception distance, reaction distance, and braking distance. The perception distance is the distance your vehicle travels while you recognize a need to stop, the reaction distance is the distance traveled during your reaction time, and the braking distance is the distance covered while your vehicle comes to a stop after applying the brakes.
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Keisha has $585.43 in her checking account. She must maintain a balance of $300 to avoid a fee. She wrote a check for $302.05 today. Write an inequality, using x for the variable, that would be used for the least amount of money she needs to deposit to avoid a fee.
Answers
The inequality that would be used for the least amount of money she needs to deposit to avoid a fee will be 585.43 + x - 302.05 ≥ 300.
How to calculate the value?From the information, it was stated that Keisha has $585.43 in her checking account and that she must maintain a balance of $300 to avoid a fee and then she wrote a check for $302.05 today.
The inequality that would be used for the least amount of money she needs to deposit to avoid a fee will be:
585.43 + x - 302.05 ≥ 300
Collect like terms
x ≥ 300 - 585.43 + 302.05
x ≥ $16.62
She's needs to deposit at least $16.62.
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a patient is admitted to the hospital because of vomiting that has lasted 3 days despite minimal intake and severe abdominal distention and pain. which intervention should the nurse perform first?
Answers
In this case, the nurse should perform a complete physical exam to determine the cause of the patient’s symptoms. The nurse should also assess the patient's vital signs, including heart rate, blood pressure, and temperature.
Once the assessment is completed, the nurse should immediately initiate fluid resuscitation to prevent dehydration and hypovolemic shock. The nurse should also administer pain medication as ordered and elevate the head of the bed to relieve abdominal distention.In this case, the patient is exhibiting symptoms of dehydration, including vomiting and minimal intake, which may result in hypovolemic shock. The patient is also experiencing severe abdominal distention and pain, which may indicate a gastrointestinal obstruction or perforation. These symptoms can be life-threatening and require immediate intervention by the nurse. Therefore, the nurse should perform a complete physical examination to determine the cause of the patient's symptoms. This examination should include a detailed abdominal assessment, such as auscultation for bowel sounds, palpation for tenderness and rigidity, and percussion for the presence of ascites. The nurse should also assess the patient's vital signs, including heart rate, blood pressure, and temperature. The nurse should also monitor the patient's urine output to ensure adequate fluid replacement and to detect any signs of renal failure. The nurse should administer pain medication as ordered and elevate the head of the bed to relieve abdominal distention.
In conclusion, the nurse should perform a complete physical exam to determine the cause of the patient’s symptoms. The nurse should also assess the patient's vital signs, including heart rate, blood pressure, and temperature. Once the assessment is completed, the nurse should immediately initiate fluid resuscitation to prevent dehydration and hypovolemic shock. The nurse should also administer pain medication as ordered and elevate the head of the bed to relieve abdominal distention.
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Given the inequality 3(n − 6) < 2(n + 12), determine which integer makes the inequality false.
S:{−5}
S:{3}
S:{12}
S:{42}
Answers
The solution of the inequality 3(n - 6) > 2(n + 12) will be less than 42. Then the correct option is D.
What is inequality?Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.
The inequality is given below.
3(n - 6) < 2(n + 12)
Simplify the inequality, then we have
3(n - 6) < 2(n + 12)
3n - 18 < 2n + 24
3n - 2n < 24 + 18
n < 42
The solution of the inequality 3(n - 6) > 2(n + 12) will be less than 42. Then the correct option is D.
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Are you reticular prism has a base of 6 m x 3.5 m in a prism is 9 m high what is the surface area of the prism
Answers
The surface area of the rectangular prism is 213 square meters.
To calculate the surface area of a rectangular prism, we need to find the areas of all its faces and then sum them up.
A rectangular prism has six faces: top, bottom, front, back, left, and right.
The top and bottom faces are identical and have the same dimensions as the base of the prism. So, the area of each of these faces is 6 m * 3.5 m = 21 m².
The front and back faces have dimensions of 6 m (width) and 9 m (height). So, the area of each of these faces is 6 m * 9 m = 54 m².
The left and right faces have dimensions of 3.5 m (width) and 9 m (height). So, the area of each of these faces is 3.5 m * 9 m = 31.5 m².
To find the total surface area, we sum up the areas of all the faces:
Total Surface Area = 2 * (Area of top face) + 2 * (Area of front face) + 2 * (Area of left face)
Total Surface Area = 2 * 21 m² + 2 * 54 m² + 2 * 31.5 m²
Total Surface Area = 42 m² + 108 m² + 63 m²
Total Surface Area = 213 m²
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Which is the simplified form of r Superscript negative 7 Baseline + s Superscript negative 12?
StartFraction 1 Over r Superscript 7 Baseline s Superscript 12 EndFraction
Negative r Superscript 7 Baseline minus s Superscript 12
StartFraction r Superscript 7 Over s Superscript 12 EndFraction
StartFraction 1 Over r Superscript 7 Baseline EndFraction + StartFraction 1 Over s Superscript 12 EndFraction
Answers
Answer:
im pretty sure its D
Step-by-step explanation:
A large planetarium in the shape of a dome is being built. When it is complete, the base of the dome will have a circumference of 870 meters. How many square meters of land were required for this planetarium?
Answers
The land required for the planetarium is approximately 189,225π square meters.
To find the area of land required for the planetarium, we need to determine the radius of the base of the dome and then calculate the area of the circle.
The circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius.
In this case, we are given that the circumference of the base of the dome is 870 meters. We can set up the equation:
870 = 2πr
To find the radius, we divide both sides of the equation by 2π:
r = 870 / (2π)
Now, we can calculate the area of the circle using the formula A = πr², where A is the area and r is the radius:
A = π(870 / (2π))²
Simplifying further:
A = π(435)²
A = 189,225π
Therefore, the land required for the planetarium is approximately 189,225π square meters.
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4 (– 2а – 6b + 5а — 1)
Answers
Answer:
Evaluate: 12a - 24b - 4
refer to table 13-12. firm 4's efficient scale occurs at what quantity?
Answers
The firm 4's efficient scale occurs at 4
In this question, we have been given a table of long-run total cost and the quantity.
We need to find the quantity at which the firm 4's efficient scale occurs.
We know that the efficient scale occurs at the quantity of output where average total cost is minimum.
For the firm 4 the average total cost would be,
(210 + 340 + 490 + 660 + 850 + 1060 + 1290) / 7
= 4900 / 7
= 700
This value is clsoe to 660.
Therefore, the firm 4's efficient scale occurs at quantity 4.
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Given question is incomplete.
researchers wish to determine if a new experimental medication will reduce the symptoms of allergy sufferers without the side effect of drowsiness. to investigate this question, the researchers randomly assigned 100 adult volunteers who suffer from allergies to two groups. they gave the new medication to the subjects in one group and an existing medication to the subjects in the other group. forty-four percent of those in the treatment group and 28% of those in the control group reported a significant reduction in their allergy symptoms without any drowsiness. the experimental units are the
Answers
This random assignment of participants and comparison of outcomes helps to establish a cause-and-effect relationship between the medication and the reduction in symptoms.
The experimental units in this study are the adult volunteers who suffer from allergies.
These volunteers were randomly assigned to two groups: the treatment group, which received the new experimental medication, and the control group, which received an existing medication.
The researchers then measured the percentage of participants in each group who reported a significant reduction in their allergy symptoms without experiencing drowsiness. The results showed that 44% of those in the treatment group and 28% of those in the control group experienced this improvement.
By comparing the outcomes between the two groups, the researchers can determine if the new medication effectively reduces allergy symptoms without causing drowsiness compared to the existing medication.
This random assignment of participants and comparison of outcomes helps to establish a cause-and-effect relationship between the medication and the reduction in symptoms.
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