For the given function, the critical point is (-1,2), relative extrema is is 2, and saddle points is zero.
To find the critical points of a function, we need to find where the partial derivatives of the function are equal to zero or do not exist. In other words, we need to find the points where the function is not changing with respect to x and y. The partial derivative of f(x,y) with respect to x is:
fx = 2x + 2
And the partial derivative of f(x,y) with respect to y is:
fy = 2y - 4
To find the critical points, we set both partial derivatives to zero and solve for x and y:
fx = 2x + 2 = 0
=> x = -1
fy = 2y - 4 = 0
=> y = 2
So, the critical point of the function is (-1, 2).
To determine whether this critical point is a relative maximum, minimum, or saddle point, we need to look at the second partial derivatives of the function. The second partial derivative of f(x,y) with respect to x is:
fx = 2
And the second partial derivative of f(x,y) with respect to y is:
fy = 2
The mixed partial derivative of f(x,y) with respect to x and y is:
fy = 0
To classify the critical point, we can use the second derivative test.
If fx and fy are both positive (or both negative) at the critical point, then the critical point is a relative minimum (or maximum), respectively.
If fx and fy have different signs, then the critical point is a saddle point. If the second derivative test is inconclusive, then we need to use additional methods to determine the nature of the critical point.
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Complete Question:
Find the critical points, relative extrema, and saddle points of the function. (If an answer does not exist, enter DNE.) f(x, y) = x² + y² + 2x - 4y + 2
The function rule for this graph is Y equals___ X + ___
The answer is below in case someone needs it.
The function rule for this graph is y = -1/2(x) + 2.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]
Where:
m represent the slope.x and y represent the points.At data point (0, 2), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 2 = \frac{(0- 2)}{(4 -0)}(x -0)[/tex]
y - 2 = -1/2(x)
y = -1/2(x) + 2.
In this context, we can reasonably infer and logically deduce that an equation of the line that represents this graph in slope-intercept form is y = -1/2(x) + 2.
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Assume that a piece of land is currently valued at $50,000. If this piece of land is expected to appreciate at an annual rate of 5% per year for the next 20 years, how much will the land be worth 20 years from now?
The value of the land 20 years after it appreciates at annual rate at 5% is $132676.47.
What is appreciation of assets?An asset's value increases over time through a process called appreciation. Depreciation, on the other hand, reduces an asset's value throughout its useful life. The rate at which an asset's value increases is known as the appreciation rate. An increase in the value of financial assets, such as stocks, is referred to as capital appreciation. When a currency appreciates, it means that its value increases when compared to other currencies on the foreign exchange markets.
The annual rate is given as 5%.
The new value after 20 years can be calculated using the formula:
[tex]A = P * (1 + r/n)^{(nt)}[/tex]
Substituting the values we have:
[tex]A = $50,000 * (1 + 0.05/1)^{(1*20)}\\A = $50,000 * 1.05^{20}\\A = $132,676.47[/tex]
Hence, the value of the land 20 years after it appreciates at annual rate at 5% is $132676.47.
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For a standard normal distribution, suppose the following is true:
P(z < c) = 0.0166
Find c.
Answer:
From the given information, we know that the area to the left of c under the standard normal distribution curve is 0.0166.
Using a standard normal distribution table or calculator, we can find the corresponding z-score for this area.
A z-score represents the number of standard deviations away from the mean. For a standard normal distribution, the mean is 0 and the standard deviation is 1.
Looking up the area of 0.0166 in the z-table, we find that the corresponding z-score is approximately -2.06.
Therefore, we have:
P(z < c) = 0.0166
P(z < -2.06) = 0.0166
So, c = -2.06.
Answer:
Using a standard normal distribution table, we can find the z-score corresponding to a probability of 0.0166:
z = -2.07
Therefore, c = -2.07.
Step-by-step explanation:
The definition of differentiable also defines an error term E(x,y). Find E(x,y) for the function f(x,y)=8x^2 − 8y at the point (−1,−7).E(x,y)=
The value of error term E(x,y) = 8x^2 - 8x - 56.
The definition of differentiability states that a function f(x,y) is differentiable at a point (a,b) if there exists a linear function L(x,y) such that:
f(x,y) - f(a,b) = L(x,y) + E(x,y)
where E(x,y) is an error term that approaches 0 as (x,y) approaches (a,b).
In the case of the function f(x,y) = 8x^2 - 8y, we want to find E(x,y) at the point (-1,-7).
First, we need to calculate f(-1,-7):
f(-1,-7) = 8(-1)^2 - 8(-7) = 56
Next, we need to find the linear function L(x,y) that approximates f(x,y) near (-1,-7). To do this, we can use the gradient of f(x,y) at (-1,-7):
∇f(-1,-7) = (16,-8)
The linear function L(x,y) is given by:
L(x,y) = f(-1,-7) + ∇f(-1,-7) · (x+1, y+7)
where · denotes the dot product.
Substituting the values, we get:
L(x,y) = 56 + (16,-8) · (x+1, y+7)
= 56 + 16(x+1) - 8(y+7)
= 8x - 8y
Finally, we can calculate the error term E(x,y) as:
E(x,y) = f(x,y) - L(x,y) - f(-1,-7)
= 8x^2 - 8y - (8x - 8y) - 56
= 8x^2 - 8x - 56
Therefore, the error term E(x,y) for the function f(x,y) = 8x^2 - 8y at the point (-1,-7) is E(x,y) = 8x^2 - 8x - 56.
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10 POINTS!!!NEED HELP ASAP PLEASE HELP FIND THE AREA AND THE PERIMETER!!
Answer: Area: 460.48 ft^2 Perimeter: 90.12 ft
Step-by-step explanation:
The area is 1/2 * 3.14 * (16 / 2)^2 (area of semicircle)
+ 10 * 12 / 2 (area of triangle)
+ 20 * 15 (area of rectangle)
= 460.48
The perimeter is 1/2 * 16 * 3.14 (perimeter of semicircle)
+ 10 (perimeter of triangle)
+ 20 + 15 + 20 (perimeter of rectangle)
= 90.12
Solve for h -110=13+3(4h-6)
Answer:
H= -35/4
Decimal form: -8.75
Explanation:
Subtract 13 from both sides. { -110 - 13 =3(4h - 6) }Simplify -110 -13 to -123 { -123 = 3 (4h - 6) }Divide both sides by 3 { -123/3 = 4h - 6 }simplify 123/3 to 41 { -41 = 4h - 6 }add 6 to both sides { -41 +6 = 4h }simplify -41 + 6 to -35 { -35 = 4h }divide both sides by 4 { - 35/4 = h }switch sides { h= - 35/4 }The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 385 grams and a standard deviation of 8 grams find the weight that corresponds to each event(use excel or appendix c to calculate the z value round your final answers to 2 decimal places
The weight that corresponds to the highest 5% is also approximately 398.12 grams.
What is Z-Score?
A score's connection to the mean within a group of scores is statistically measured by a Z-Score.
To find the weight that corresponds to each event, we need to use the standard normal distribution and convert each value to a z-score using the formula:
z = (x - μ) / σ
Here are the calculations for each event:
The weight that corresponds to the 25th percentile:
-0.68 = (x - 385) / 8
Solving for x gives:
x = 379.44 grams (rounded to two decimal places)
Therefore, the weight that corresponds to the 25th percentile is approximately 379.44 grams.
The weight that corresponds to the 95th percentile. we find that the z-score is approximately 1.64 (rounded to two decimal places). Then we can use the formula above to solve for x:
1.64 = (x - 385) / 8
x = 398.12 grams (rounded to two decimal places)
Therefore, the weight that corresponds to the 95th percentile is approximately 398.12 grams.
The weight that corresponds to the highest 5%:
1.64 = (x - 385) / 8
x = 398.12 grams (rounded to two decimal places)
Therefore, the weight that corresponds to the highest 5% is also approximately 398.12 grams.
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The variable s represents the number of students in one class in your school. What does 1/2s represent?
Answer: it represents half of the students in 1 class
Step-by-step explanation:
1/2 divided by s
Answer:
1/2s would then represent one half (or 50%) of the students in the singular class stated.
I will mark you brainiest!
Vertical angles are supplementary.
True
False
Answer:
True
Step-by-step explanation:
Vertical angles are right angle that is 90°
A supplementary angle is an angle that forms up by 2 angles with the sum of 180°.
It is true because 2 vertical angles form a supplementary angle.
Answer:
True. Vertical angles are angles that are opposite each other when two lines intersect, so they have the same measure. Sum of measures of two angles is 180 degrees, which makes them supplementary angles.name three angles that sum up to 180 degrees
The three angles are= angleMCD + angleCMD + angleGMF= 180.
What are angles?Two lines intersect at a location, creating an angle.
An "angle" is the term used to describe the width of the "opening" between these two rays. The character is used to represent it.
Angles are frequently expressed in degrees and radians, a unit of circularity or rotation.
In geometry, an angle is created by joining two rays at their ends. These rays are referred to as the angle's sides or arms.
An angle has two primary components: the arms and the vertex. T
he two rays' shared vertex serves as their common terminal.
According to our question-
angleM= 127
angleC=27
angleG=26
127+27+26
180
Hence, The three angles are= angleMCD + angleCMD + angleGMF= 180.
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Answer: <MCD, <CMD, and <GMF
Step-by-step explanation:
Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading less than 0.35°C.
Round your answer to 4 decimal places
The probability of obtaining a reading less than 0.35° C is approximately 35%.
What exactly is probability, and what is its formula?Accοrding tο the prοbability fοrmula, the likelihοοd οf an event οccurring is equal tο the ratiο οf the number οf favοurable οutcοmes tο the tοtal number οf οutcοmes. Prοbability οf an event οccurring P(E) = The number οf favοurable οutcοmes divided by the tοtal number οf οutcοmes.
The readings at freezing οn a set οf thermοmeters are nοrmally distributed, with a mean (x) οf 0°C and a standard deviatiοn (μ) οf 1.00°C. We want tο knοw hοw likely it is that we will get a reading that is less than 0.35°C.
To solve this problem, we must use the z-score formula to standardise the value:
[tex]$Z = \frac{x - \mu}{\sigma}[/tex]
Z = standard score
x = observed value
[tex]\mu[/tex] = mean of the sample
[tex]\sigma[/tex] = standard deviation of the sample
Here
x = 0.35° C
[tex]\mu[/tex] = 0° C
[tex]\sigma[/tex] = 1.00°C
Using the values on the formula:
[tex]$Z = \frac{0.35 - 0}{1}[/tex]
Z = 0.35
The probability of obtaining a reading less than 0.35° C is approximately 35%.
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an adjusted r-squared value of 0 represents no ability of the model to explain the dependent variable.
An adjusted R-squared value of 0 indicates that the model has no ability to explain the variation in the dependent variable using the independent variables included in the model.
In other words, the model does not fit the data well and cannot make accurate predictions. An adjusted R-squared value of 1 represents a perfect fit, where the model explains all of the variation in the dependent variable using the independent variables. However, it is important to consider other factors such as the sample size, the quality of the data, and the appropriateness of the model to make valid conclusions about the model's ability to explain the dependent variable.
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The area of shape A is 3cm2 what is the area of shape B?
28.5cm^2 is the area of shape B.
What is area?A solid object's surface area is a measurement of the total area that the surface of the object takes up.
The definition polyhedra of arc length for one-dimensional curves and the definition of surface area for (i.e., objects with flat polygonal faces), where the surface area is the sum of the areas of its faces, are both much simpler mathematical concepts than the definition of surface area when there are curved surfaces.
A smooth surface's surface area is determined using its representation as a parametric surface, such as a sphere.
This definition of surface area uses partial derivatives and double much simpler mathematical concepts than the definition of surface area integration and is based on techniques used in infinitesimal calculus.sought a general definition of surface area.
(3×7)+(1.5×5)
21+7.5
28.5cm^2
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Find the probability of landing on yellow, the probability of the complement, and the sum of the event and the complement. Type your answers without any spaces.
The probability of landing on yellow is 0.2, probability of component is 0.8, and sum of event and complement is 1.
On assuming that the pie is evenly divided into 5 parts,
So, the probability of landing on yellow is = 1/5 = 0.2,
The complement of landing on yellow is the probability of not landing on yellow, which is the probability of landing on any of the other 4 parts of the pie.
So, the probability of the complement is = 4/5 = 0.8,
The sum of the event (landing on yellow) and the complement (not landing on yellow) is equal to the probability of the entire sample space, which is 1.
⇒ P(Yellow) + P(Not Yellow) = 1
⇒ 0.2 + 0.8 = 1
So, the sum of the event and the complement is 1 or 100%.
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The given question is incomplete, the complete question is
A circular pie is divided in 5 parts , Green , Yellow, Blue Black and Red.
Find the probability of landing on yellow, the probability of complement, and the sum of the event and the complement.
The pens in a box are repackaged equally into 9 packs. Each pack has more than 15 pens.
1. Find an inequality to represent n, the possible number of pens in the box.
2. Explain why you chose this inequality.
Therefore, the possible number of pens in the box is p, where p is greater than 135.
What is inequality?Inequality refers to a situation in which there is a difference or disparity between two or more things, usually in terms of value, opportunity, or outcome. Inequality can take many forms, including social, economic, and political inequality.
Inequalities are mathematical expressions that compare two values using the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). To solve an inequality, you need to isolate the variable (the unknown quantity) on one side of the inequality symbol and determine the range of values for which the inequality holds true.
Here are some general steps to solve an inequality:
Simplify both sides of the inequality as much as possible. This may involve combining like terms, distributing terms, or factoring.Get all the variable terms on one side of the inequality symbol and all the constant terms on the other side. Remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.Solve for the variable by isolating it on one side of the inequality symbol. If the variable has a coefficient, divide both sides of the inequality by that coefficient.Write down the solution as an inequality. If you have solved for x, the solution will be in the form of x < a or x > b, where a and b are numbers.Check your solution by testing a value in the original inequality that is within the range of the solution. If the inequality holds true for that value, then the solution is correct. If not, then you may need to recheck your work or adjust your solutionby the question.
Let's say there are 'p' pens in the box. Each pack has more than 15 pens, so we can write the inequality:
p/9 > 15
Multiplying both sides by 9, we get:
p > 135
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Question 23 (2 points)
A standard deck of cards contains 4 suits of the same 13 cards. The contents of a
standard deck are shown below:
Standard deck of 52 cards
4 suits (CLUBS, SPADES, HEARTS, DIAMONDS)
13 CLUBS
13 SPADES
13 HEARTS
13 DIAMONDS
If two cards are drawn at random from the deck of cards, what is the probability both
are kings?
4/52
3/51
12/2652
16/2704
Answer:
12/2652
Step-by-step explanation:
First, the probability of drawing a king for the first time is 4/52. The chance of drawing another is 3/51. Multiplying, we get the 3rd answer choice, 12/2652
Sharon used 8 roses and 6 tulips to make a bouquet. The tape diagram below shows the relationship between the number of roses and the number of tulips in the bouquet.
Answer:
Step-by-step explanation:
its C
Jerry writes down all the odd numbers 1, 3, 5, 7, up to 999. How many numbers does he write down?
There are 500 odd numbers between 1 and 999.
We can solve this problem using the arithmetic sequence formula, which is
an = a1 + (n - 1)d
where
an is the nth term of the sequence
a1 is the first term of the sequence
n is the number of terms in the sequence
d is the common difference between consecutive terms
In this case, a1 = 1, the common difference is 2, and we want to find the value of n such that an = 999. So we have
999 = 1 + (n - 1)2
Simplifying this equation, we get
998 = 2(n - 1)
499 = n - 1
n = 500
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Annie is concerned over a report that "a woman over age 40 has a better chance of being killed by a terrorist than of getting married." A study found that the likelihood of marriage for a never-previously-wed, 40 -year-old university-educated American woman was 2.5% . To demonstrate that this percentage is too small, Annie uses her resources at the Baltimore Sun to conduct a simple random sample of 546 never-previously-wed, university-educated, American women who were single at the beginning of their 40 s and who are now 45 . Of these women, 20 report now being married. Does this evidence support Annie’s claim, at the 0.01 level of significance, that the chances of getting married for this group is greater than 2.5% ? Step 1 of 3 : State the null and alternative hypotheses for the test. Fill in the blank below. H0Ha: p=0.025: p⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯0.025
Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).
what is null hypothesis ?The null hypothesis, which is an assertion or assumption that there is no significant difference or association between two or more variables or populations, is used in statistical hypothesis testing. It is frequently indicated by the letter H0 and is typically the hypothesis that is tested against a competing hypothesis. The objective of the hypothesis test is to either reject or fail to reject the null hypothesis based on the evidence or data seen. The null hypothesis serves as the default or baseline assumption. If the alternative hypothesis is supported by evidence, the null hypothesis is likely to be rejected.
given
The test's null and alternate hypotheses are as follows:
H0: p 0.025 (The percentage of American women with university educations who had never previously been married at the start of their 40s and are now 45 and married is less than or equal to 2.5%)
Ha: p > 0.025 (More than 2.5% of American women with college degrees who were unmarried at the start of their 40s and are now 45 and married are never before married).
Due to the directed nature of the alternative hypothesis, a one-tailed test is being used (greater than).
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Give an example to show that the Monotone Convergence Theorem (3.11) can fail if the hypothesis that f1, f2, ... are nonnegative functions is dropped. 3.11 Monotone Convergence Theorem Suppose (X, S, u) is a measure space and 0 < fi < f2 <... is an increasing sequence of S-measurable functions. Define f: X → [0,00] by f(x) = lim fx(x). koo Then lim k+00 | fx du = / f du.
The Monotone Convergence Theorem can be demonstrated by considering the decreasing sequence {a_n} = 1/n, which is bounded below by zero and converges to zero.
Consider the sequence of real numbers {a_n} defined as a_n = 1/n. We want to show that the sequence converges to zero.
First, notice that the sequence is decreasing since a_n+1 = 1/(n+1) < 1/n = a_n for all n ≥ 1. Moreover, the sequence is bounded below by zero since a_n > 0 for all n. Thus, the sequence {a_n} is a decreasing bounded sequence and by the Monotone Convergence Theorem, it must converge to some limit L.
Let's now calculate the limit L. Since the sequence is decreasing and bounded below by zero, its limit L must be greater than or equal to zero. Furthermore, for any ε > 0, there exists an N such that 1/n < ε for all n > N, since the sequence converges to zero. Therefore, we have
|a_n - 0| = |1/n - 0| = 1/n < ε for all n > N.
This shows that the limit of the sequence is zero, i.e., lim (n → ∞) 1/n = 0.
Thus, we have demonstrated that the Monotone Convergence Theorem applies to the sequence {a_n}, which is decreasing and converges to zero.
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I have solved the question in general, as the given question is incomplete.
The complete question is:
Give an example to show that the Monotone Convergence Theorem?
At a certain instant, the base of a triangle is 5 inches and is increasing at the rate of 1 inch per minute. At the same instant, the height is 10 inches and is decreasing at the rate of 2.5 inches per minute. Is the area of the triangle increasing or decreasing? Justify your answer.
Using differentiation, the area of the triangle is decreasing at the given time.
Is the area of the triangle increasing or decreasing?The formula for the area of a triangle is:
A = (1/2)bh
where b is the base and h is the height.
Differentiating both sides of the equation with respect to time t, we get:
[tex]\frac{dA}{dt} = (1/2)[(\frac{db}{dt}) h + b(\frac{dh}{dt}) ][/tex]
Substituting the given values, we get:
[tex]\frac{dA}{dt} = (1/2)[(1)(10) + (5)(-2.5)] = (1/2)(10 - 12.5) = -1.25[/tex]
Since the derivative of the area with respect to time is negative (-1.25), the area of the triangle is decreasing at the given instant.
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HELP Whats the Answer to this Stand Deviation Question?
Answer: he would be 2 standard deviations above the
Step-by-step explanation:
Values of the Born exponents for Rb+ and l-are 10 and 12, respectively. The Born exponent for Rbl is therefore: O A. 2 O B.22 C. 1/11 OD. 11
The Born exponent or interatomic potential energy for Rbl is 11 ( approximately). The correct option is D).
The Born exponent for RbI can be calculated using the relationship between the Born exponent and the interionic distance. The Born exponent is defined as the ratio of the repulsive to attractive contributions to the interatomic potential energy, and it depends on the charges and sizes of the ions.
For Rb+ and I-, the Born exponents are 10 and 12, respectively. This means that the repulsive interaction between Rb+ and I- is weaker than the attractive interaction, as the repulsion is proportional to Rb+^10 and the attraction is proportional to I^-12. Therefore, the attractive interaction dominates.
For RbI, we can use the relationship between the Born exponent and the interionic distance to calculate the Born exponent. This relationship is given by:
B = (1/d) * ln[(l1 + l2)/|l1 - l2|]
where B is the Born exponent, d is the interionic distance, and l1 and l2 are the ionic radii of the cation and anion, respectively.
Assuming the ionic radii of Rb+ and I- are additive, we have:
l1 + l2 = l(RbI) = l(Rb+) + l(I-) = 1.52 + 1.81 = 3.33 Å
|l1 - l2| = |l(Rb+) - l(I-)| = |1.52 - 1.81| = 0.29 Å
Substituting these values into the equation for B, we get:
B = (1/d) * ln[(l1 + l2)/|l1 - l2|] = (1/d) * ln[3.33/0.29] ≈ 11.02
Therefore, the Born exponent for RbI is approximately 11.02.
The correct answer is D).
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Hi, any one can solve it ?
For given function f(x)= x³ + 2x , the complete table is mentioned below. f⁻¹(3)= 1, f⁻¹(-12) = -2.
Describe function ?In mathematics, a function is a rule that assigns a unique output value for every input value in a specified set. It is a fundamental concept in algebra, calculus, and other areas of mathematics.
A function is typically denoted by a symbol, such as f(x), where x is the input variable, and f(x) is the output variable. The set of all input values for which the function is defined is called the domain, and the set of all output values is called the range.
To complete the table of values, we simply plug in the given values of x into the expression for f(x) and evaluate:
x f(x)
0 0
1 3
2 14
To find f⁻¹(3), we need to solve for x in the equation f(x) = 3:
x³ + 2x = 3
x³ + 2x - 3 = 0
We can use trial and error to find that x = 1 is a solution to this equation:
1³ + 2(1) - 3 = 0
Therefore, f⁻¹(3) = 1.
To find f⁻¹(-12), we need to solve for x in the equation f(x) = -12:
x³ + 2x = -12
x³ + 2x + 12 = 0
We can use trial and error to find that x = -2 is a solution to this equation:
(-2)³ + 2(-2) + 12 = 0
Therefore, f⁻¹(-12) = -2.
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Solve using the correct order of
operations.
P
E
MD
AS
15-(4-3) 2= [?]
Enter
Help
Using the correct order of operations, the value is 13
What is PEDMAS?PEDMAS is simply described as a mathematical acronym that represents the different arithmetic operations in order from least to greatest of application.
The alphabets represents;
P represents parentheses.E represents exponents.D represents division.M represents multiplication.A represents addition.S represents subtraction.From the information given, we have;
15-(4-3)2
solve the parentheses first
15 - (1)2
Multiply the values
15 - 2
Subtract the values
13
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The complete question:
Solve using the correct order of
operations of PEMDAS
15 - (4-3)2
Please help me with this question
The slope-intercept version of the equation for the tangent line to f(x) at the position (-5, -1) is y = (-1/5)x -2. Thus,
m = -1/5
y = (-1/5)x -2
What can you infer from a tangent line?A tangent line is a straight line that οnly has οne cοntact with a functiοn. (See earlier.) The instantaneοus rate οf change οf the functiοn at that exact place is shοwn by the tangent line. At each given pοint οn the functiοn, the slοpe οf the tangent line is equal tο the derivative οf the functiοn at that same lοcatiοn.
We must determine the derivative οf the functiοn and evaluate it at x = -5 in οrder tο determine the slοpe οf f(x) = 5/x at the pοint (-5, -1).
f(x) = 5/x
f'(x) = [-5/x²]
When we enter x = -5, we obtain:
f'(-5) = [-5/(-5)²] = -1/5
As a result, the tangent line to f(x) at the point (-5, -1) has a slope of -1/5.
y - y1 = m(x - x1)
y - (-1) = (-1/5)(x - (-5))
y + 1 = (-1/5)(x + 5)
y = (-1/5)x -10/5
y = (-1/5)x -2
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For each of the following propositions, either i. use a case-based proof to demonstrate that the proposition holds true or ii. Use a counterexample to demonstrate the proposition does not hold.
(a) Assume x is an integer that is not divisible by 3, and y is an integer that is not divisible by 3. Then the sum of x and y cannot be divisible by 3.
(b) Assume x is an integer that is not divisible by 3, and y is an integer that is divisible by 3. Then the sum of x and y cannot be divisible by 3.
In both cases, the sum of x and y is not divisible by 3, we have demonstrated that the proposition is true. and the proposition is false, and we have shown a counterexample where the sum of two integers, one of which is not divisible by 3 and the other is divisible by 3, can be divisible by 3.
(a) To prove that the sum of two integers, x and y, neither of which is divisible by 3, cannot be divisible by 3, we can use a case-based proof.
Case 1: x and y leave a remainder of 1 when divided by 3.
Let x = 3m + 1 and y = 3n + 1, where m and n are integers. Then, the sum of x and y is 3m + 3n + 2, which leaves a remainder of 2 when divided by 3. Therefore, x + y is not divisible by 3.
Case 2: x and y leave a remainder of 2 when divided by 3.
Let x = 3m + 2 and y = 3n + 2, where m and n are integers. Then, the sum of x and y is 3m + 3n + 4, which leaves a remainder of 1 when divided by 3. Therefore, x + y is not divisible by 3.
Since in both cases, the sum of x and y is not divisible by 3, we have demonstrated that the proposition is true.
(b) To prove that the sum of two integers, x and y, where x is not divisible by 3 and y is divisible by 3, cannot be divisible by 3, we can use a counterexample.
Let x = 2 and y = 6. Then, x is not divisible by 3 and y is divisible by 3. However, x + y = 8, which is not divisible by 3.
Therefore, the proposition is false, and we have shown a counterexample where the sum of two integers, one of which is not divisible by 3 and the other is divisible by 3, can be divisible by 3.
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Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading between 0.59°C and 0.88°C.
The probability of obtaining a reading between 0.59°C and 0.88°C is 0.7224 and 0.8106.
What is mean?The sum of all possible values, weighted by the chance of each value, is equal to the mean of a discrete probability distribution of the random variable X. Each possible number of X must be multiplied by its probability P(x) before being added as a whole to determine the mean. In statistics, the mean is one measure of central trend in addition to the mode and median. The mean is simply the average of the numbers in the specified collection. It suggests that values in a specific data gathering are evenly distributed. In order to find the mean, the total values given in a datasheet must be added, and the result must be divided by the total number of values.
In this question, using the formula,
z-score = (x – μ) / σ
where:
x: individual data value
μ: population mean
σ: population standard deviation
for x=0.59
μ= 0
σ= 1
z-score= 0.59
Probability=0.7224
for x=0.88
z-score= 0.88
Probability=0.8106
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what is x?
what is m?
what is b?
x=?
m=?
b=?
There is a vertical asymptote at x = 2 and the slope and intercept of the oblique asymptote are 2 and - 1, respectively.
How to determine the vertical asymptote and the oblique asymptote
In this problem we find the definition of a rational function:
f(x) = (2 · x² - 5 · x + 3) / (x - 2)
The vertical asympote correspond to the vertical line at the x-value where the function is undefined. And the oblique asymptote is defined by a equation of the form:
y = m · x + b
Where:
m - Slopeb - InterceptAnd the slope and the intercept of the asymptote can be found by means of the following equation:
Slope
[tex]m = \lim_{x \to \pm \infty} \left[\frac{f(x)}{x}\right][/tex]
Intercept
[tex]b = \lim_{x \to \pm \infty} [f(x) - m \cdot x][/tex]
First, factor and simplify the rational equation to determine whether any zero is evitable:
f(x) = (2 · x² - 5 · x + 3) / (x - 2)
f(x) = (2 · x - 3) · (x - 1) / (x - 2)
The discontinuity at x = 2 is not evitable. Then, the equation for the vertical asymptote is x = 2.
Second, determine the slope and the intercept of the oblique asymptote:
[tex]m = \lim_{x \to \pm \infty} \left[\frac{2\cdot x^{2}-5\cdot x + 3}{x^{2} - 2\cdot x} \right][/tex]
m = 2
[tex]b = \lim_{x \to \pm \infty} \left[\frac{2\cdot x^{2}-5\cdot x + 3}{x - 2} - 2 \cdot x\right][/tex]
[tex]b = \lim_{x \to \pm \infty} \left[\frac{2\cdot x^{2}-5\cdot x + 3-2 \cdot x^{2}+4\cdot x}{x-2}\right][/tex]
[tex]b = \lim_{x \to \pm \infty} \left[\frac{3 - x}{x-2} \right][/tex]
b = - 1
The slope and the intercept of the oblique asymptote are 2 and - 1, respectively.
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PLS HELP FAST 50 POINTS + BRAINLIEST
Answer:
Anna had 23 sweets in her bag at the start of the day.
Step-by-step explanation:
Let's use working backwards to find out how many sweets were in the bag at the start of the day.
At the end of lesson 4, Anna had 1 sweet left in her bag. So, before she gave a sweet to her teacher in lesson 4, she had 2 sweets left in her bag.
In lesson 3, she gave out half of the sweets left in her bag and then gave one to the teacher. So, before she gave a sweet to her teacher in lesson 3, she had 2 x 2 + 1 = 5 sweets in her bag.
In lesson 2, she gave out half of the sweets left in her bag and then gave one to the teacher. So, before she gave a sweet to her teacher in lesson 2, she had 5 x 2 + 1 = 11 sweets in her bag.
In lesson 1, she gave out half of the sweets in her bag and then gave one to the teacher. So, before she gave a sweet to her teacher in lesson 1, she had 11 x 2 + 1 = 23 sweets in her bag.
Therefore, Anna had 23 sweets in her bag at the start of the day.