The positive real number that satisfies the given condition is 30.
Let's assume the positive real number as x. The given condition states that its square is equal to 4 times the number, increased by 780. Mathematically, we can express this as:
x² = 4x + 780
To solve this equation, we can rearrange it to obtain a quadratic equation:
x² - 4x - 780 = 0
Now we can factorize or use the quadratic formula to find the roots of the equation. Factoring may not be straightforward in this case, so we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For our equation, the coefficients are a = 1, b = -4, and c = -780. Substituting these values into the quadratic formula, we have:
x = (-(-4) ± √((-4)² - 4(1)(-780))) / (2(1))
x = (4 ± √(16 + 3120)) / 2
x = (4 ± √(3136)) / 2
x = (4 ± 56) / 2
Simplifying further, we have two solutions:
x₁ = (4 + 56) / 2 = 60 / 2 = 30
x₂ = (4 - 56) / 2 = -52 / 2 = -26
Since we are looking for a positive real number, the solution is x = 30.
To learn more about real number click here : brainly.com/question/551408
#SPJ11
Consider the matrix A (a) rank of A. (b) nullity of 4. 1 1 -1 1 1 -1 1 1 -1 -1 1 -1-1, then find [5] (5)
To determine the rank and nullity of matrix A, we need to perform row reduction to its reduced row echelon form (RREF).
The given matrix A is:
A = [1 1 -1; 1 1 -1; 1 -1 1; -1 1 -1]
Performing row reduction on matrix A:
R2 = R2 - R1
R3 = R3 - R1
R4 = R4 + R1
[1 1 -1; 0 0 0; 0 -2 2; 0 2 0]
R3 = R3 - 2R2
R4 = R4 - 2R2
[1 1 -1; 0 0 0; 0 -2 2; 0 0 -2]
R4 = -1/2 R4
[1 1 -1; 0 0 0; 0 -2 2; 0 0 1]
R3 = R3 + 2R4
R1 = R1 - R4
[1 1 0; 0 0 0; 0 -2 0; 0 0 1]
R2 = -2 R3
[1 1 0; 0 0 0; 0 1 0; 0 0 1]
Now, we have the matrix in its RREF. We can see that there are three pivot columns (leading 1's) in the matrix. Therefore, the rank of matrix A is 3.
To find the nullity, we count the number of non-pivot columns, which is equal to the number of columns (in this case, 3) minus the rank. So the nullity of matrix A is 3 - 3 = 0.
Now, to find [5] (5), we need more information or clarification about what you mean by [5] (5). Please provide more details or rephrase your question so that I can assist you further.
To know more about matrix visit:
brainly.com/question/29132693
#SPJ11
Haruki lemmui Given tuo non intersecting chards Авај ср a circle а variable point P the are renote from points Card P. Let E AJ be the intersection of chonds PC, AB of PA, AB respectivals. the value of BF Joes not Jepan on AE position of P. u U = = corstart. x = -
In Haruki Lemmui's scenario, there are two non-intersecting chords, a circle, and a variable point P. The intersection points between the chords and the line segment PA are labeled E and F. The value of BF does not depend on the position of point P on AE.
In this scenario, we have a circle and two non-intersecting chords, PC and AB. The variable point P is located on chord AB. We also have two intersection points labeled E and F. Point E is the intersection of chords PC and AB, while point F is the intersection of line segment PA and chord AB.
The key observation is that the value of BF, the distance between point B and point F, does not depend on the position of point P along line segment AE. This means that regardless of where point P is located on AE, the length of BF remains constant.
The reason behind this is that the length of chord AB and the angles at points A and B determine the position of point F. As long as these parameters remain constant, the position of point P along AE does not affect the length of BF.
Therefore, in Haruki Lemmui's scenario, the value of BF does not change based on the position of point P on AE.
Learn more about circle here: https://brainly.com/question/30106097
#SPJ11
A certain triangle has a perimeter of 3076 mi. The shortest side measures 77 mi less than the middle side, and the longest side measures 372 mi more than the middle side. Find the lengths of the three sides.
The shortest side is _________ mi long.
The shortest side of the triangle is 876 miles long. we have the following relationships x = y - 77 , z = y + 372
Let's denote the lengths of the sides of the triangle as follows: Shortest side: x Middle side: y Longest side: z According to the given information, we have the following relationships x = y - 77 (the shortest side is 77 miles less than the middle side z = y + 372 (the longest side is 372 miles more than the middle side)
The perimeter of a triangle is the sum of the lengths of its sides: Perimeter = x + y + z Substituting the given relationships, we get: 3076 = (y - 77) + y + (y + 372) Simplifying the equation: 3076 = 3y + 295 Rearranging and solving for y: 3y = 3076 - 295 3y = 2781 y = 927
Substituting the value of y into the relationships, we can find the lengths of the other sides: x = y - 77 = 927 - 77 = 850, z = y + 372 = 927 + 372 = 1299
Learn more about triangle here:
https://brainly.com/question/29083884
#SPJ11
List all inclusions that hold among the following sets: (a) A = all integers from 1 to 9 satisfying x² — 5x = - 14, (b) B = {2,7}, (c) C = {-2, 7}, (d) D = {7}.
The inclusions among the given sets are as follows: B ⊆ A, C ⊆ A, and D ⊆ A.
The set A consists of integers from 1 to 9 that satisfy the equation x² - 5x = -14. To find the elements of A, we can solve the equation. By rearranging the equation, we have x² - 5x + 14 = 0. However, this equation does not have any real solutions. Therefore, the set A is empty.
Moving on to the other sets, we have B = {2, 7}, C = {-2, 7}, and D = {7}. We can see that B is a subset of A because both 2 and 7 are included in A. Similarly, C is also a subset of A since 7 is present in both sets. Finally, D is a subset of A because 7 is an element of both sets.
In summary, the inclusions among the given sets are B ⊆ A, C ⊆ A, and D ⊆ A. It means that B, C, and D are subsets of A, or in other words, all elements of B, C, and D are also elements of A. However, since set A is empty, this means that all the other sets (B, C, and D) are also empty.
Learn more about subset here:
https://brainly.com/question/28705656
#SPJ11
Solve The Initial Value Problemfind An Explicit Formula For Y(T)Ty' = 1 + Y Y(1) = 3
solve the initial value problem
find an explicit formula for y(t)
ty' = 1 + y y(1) = 3
We have the explicit formula for y(t): y(t) = (t^3/6 + (A - K)t) + K
where A is a constant determined by the initial condition, and K is the constant of integration.
To solve the initial value problem and find an explicit formula for y(t), we can use the method of separating variables and integrating.
Given: ty' = 1 + y, y(1) = 3
Step 1: Separate the variables
ty' - y = 1
Step 2: Integrate both sides with respect to t
∫(ty' - y) dt = ∫1 dt
Integrating the left side:
∫ty' dt - ∫y dt = t²/2 - ∫y dt
Integrating the right side:
t²/2 - ∫y dt = t²/2 + C
Step 3: Solve for y
Now we need to solve for y. To do that, we need to find the integral of y.
∫y dt = ∫(t²/2 + C) dt
Integrating the right side:
∫y dt = (t³/6 + Ct) + K
Where K is the constant of integration.
Step 4: Substitute the initial condition to find the value of the constant
Using the initial condition y(1) = 3, we can substitute t = 1 and y = 3 into the equation:
∫y dt = (t³/6 + Ct) + K
∫3 dt = (1³/6 + C(1)) + K
3t = 1/6 + C + K
Step 5: Simplify and solve for C
3 = 1/6 + C + K
Simplifying:
C + K = 3 - 1/6
C + K = 17/6
Since C + K is a constant, we can let C + K = A, where A is a new constant.
So we have:
C = A - K
Step 6: Substitute back into the equation and simplify
∫y dt = (t³/6 + Ct) + K
∫y dt = (t³/6 + (A - K)t) + K
Finally, we have the explicit formula for y(t):
y(t) = (t³/6 + (A - K)t) + K
where A is a constant determined by the initial condition, and K is the constant of integration.
To learn more about explicit formula visit:
brainly.com/question/25094536
#SPJ11
Determine p'(x) when p(x) = 0.08 √z Select the correct answer below: OP(x) = 0.08 2√/2 O p'(x) = 0.08 (*))(√²)(1²) Op'(x)=0.08(- (ze²-¹)(√²)(¹)(27)) (√√z)² Op'(x) = 0.08 (¹)-(*))).
The value of p'(x) is Op'(x) = 0.04 z^(-1/2).The answer is option (D). Op'(x) = 0.08 (¹)-(*))).
A function is a mathematical relationship that maps each input value to a unique output value. It is a rule or procedure that takes one or more inputs and produces a corresponding output. In other words, a function assigns a value to each input and defines the relationship between the input and output.
Given function is, p(x) = 0.08 √z
To find p'(x), we can differentiate the given function with respect to z.
So, we have, dp(x)/dz = d/dz (0.08 z^(1/2)) = 0.08 d/dz (z^(1/2))= 0.08 * (1/2) * z^(-1/2)= 0.04 z^(-1/2)
Therefore, the value of p'(x) is Op'(x) = 0.04 z^(-1/2).The answer is option (D). Op'(x) = 0.08 (¹)-(*))).
Learn more about function
https://brainly.com/question/31062578\
#SPJ11
Let f be the function defined x^3 for x< or =0 or x for x>o. Which of the following statements about f is true?
(A) f is an odd function
(B) f is discontinuous at x=0
(C) f has a relative maximum
(D) f ‘(x)>0 for x not equal 0
(E) none of the above
"f ‘(x)>0 for x not equal 0 " is true statement about function f.
This is option D.
The function `f` defined by `f(x) = x^3` for `x≤0` or `f(x) = x` for `x>0`.
Statement (A) - False: If `f` is odd, then `f(-x) = -f(x)` for every `x` in the domain of `f`.
However, `f(-(-1)) = f(1) = 1` and `f(-1) = -1`, so `f` is not odd.
Statement (B) - False:There are no limits of `f(x)` as `x` approaches `0` because `f` has a "sharp point" at `x = 0`, which means `f(x)` will be continuous at `x = 0`.Therefore, `f` is not discontinuous at `x = 0`.
Statement (C) - False:There is no maximum value in the function `f`.The function `f` is defined as `f(x) = x^3` for `x≤0`.
There is no maximum value in this domain.The function `f(x) = x` is strictly increasing on the interval `(0,∞)`, and there is no maximum value.
Therefore, `f` does not have a relative maximum.
Statement (D) - True:
For all `x ≠ 0`, `f'(x) = 3x^2` if `x < 0` and `f'(x) = 1` if `x > 0`.Both `3x^2` and `1` are positive numbers, which means that `f'(x) > 0` for all `x ≠ 0`.Therefore, statement (D) is true.
Statement (E) - False: Since statement (D) is true, statement (E) must be false.
Therefore, the correct answer is (D) `f ‘(x)>0 for x not equal 0`.
Learn more about function at
https://brainly.com/question/31780291
#SPJ11
Let P =(,0). For each of the following pairs of points Q₁ and Q2, say which hyperbolic distance dr(P,Q₁) or dn (P, Q2) is greater than the other: 1. Q₁ = (0,0), Q₂ = (,0). 2. Q₁-(,0), Q₂ = (). 3. Q₁=(-1,0), Q₂ = (-)
We can conclude that the function f(x) = ln(1 + x) on the interval (-1, +[infinity]0) has no absolute maximum or minimum.
In order to prove that the function f(x) = ln(1+x) on the interval (-1, +[infinity]0) has no absolute maximum or absolute minimum, we must examine the behavior of this function on the boundary points and its behavior at the endpoints of the interval.
To analyze the behavior of this function at the boundary points of the interval, we must analyze the limits of this function. Since ln(1+x) is a continuous function, its limit as x approaches -1 from the right side is equal to its value at x = -1, which is ln(0) = -∞. Similarly, the limit of this function as x approaches +[infinity]0 is equal to +∞. Thus, since both limits exist and are unbounded, the function does not have an absolute maximum or minimum at the boundary points of the interval.
Next, we must analyze the endpoint behavior of the function. For the endpoint at x = -1, the function is ln(0) = -∞, so it clearly has no absolute maximum or minimum here. For the endpoint +[infinity]0, the function is +∞ and therefore has no absolute maximum or minimum here either. Therefore, the function has no absolute maximum or minimum at either endpoint of the interval.
Therefore, we can conclude that the function f(x) = ln(1 + x) on the interval (-1, +[infinity]0) has no absolute maximum or minimum.
To know more about function click-
http://brainly.com/question/25841119
#SPJ11
Give all your answers of Question 6 in 4 decimal places. The weight of IVE students is normally distributed, with a mean of 80 kg and standard deviation of 10 kg. (a) Find the probability that a randomly selected student weighs: (i) less than 56 kg. (3 marks) (3 marks) (ii) more than 104 kg. (b) Find how many students would be expected to weigh between 74 kg and 80 kg if the samples are randomly taken from 400 number of students. (4 marks)
a) i) Probability that a randomly selected student weighs less than 56 kg is 0.0082. ; ii) probability that a randomly selected student weighs more than 104 kg is 0.0082 ; b) approximately 90 students would be expected to weigh between 74 kg and 80 kg if the samples are randomly taken from 400 number of students.
Given : The weight of IVE students is normally distributed, with a mean of 80 kg and standard deviation of 10 kg.
(a) Probability that a randomly selected student weighs:
i) less than 56 kg.
We need to find P(x < 56)Now, calculating z-score,
[tex]\[z=\frac{x-\mu }{\sigma }[/tex]
[tex]=\frac{56-80}{10}[/tex]
=-2.4
From the z-score table, the corresponding probability is 0.0082
Therefore, the probability that a randomly selected student weighs less than 56 kg is 0.0082.
ii) more than 104 kg.
We need to find P(x > 104)
Now, calculating[tex][tex]z-score,[/tex]
[tex]z=\frac{x-\mu }{\sigma }[/tex]
[tex]=\frac{104-80}{10}[/tex]
=2.4
From the z-score table, the corresponding probability is 0.0082
Therefore, the probability that a randomly selected student weighs more than 104 kg is 0.0082.
(b) Now, calculating z-score,
[tex][tex]\[z=\frac{74-80}{10}[/tex]
[tex]=-0.6\][/tex]and,[tex][/tex]
[tex]\[z=\frac{80-80}{10}[/tex]
=0
From the z-score table,[tex]\[P( -0.6 < z < 0)[/tex]
=[tex]P(z < 0) - P(z < -0.6)\]\[[/tex]
= 0.5 - 0.2743
= 0.2257
Therefore, the probability that a student weighs between 74 kg and 80 kg is 0.2257.Then, the expected number of students who weigh between 74 kg and 80 kg if the samples are randomly taken from 400 number of students is,
[tex][tex]\[n=pN[/tex]
=0.2257×400
=90.28
Therefore, approximately 90 students would be expected to weigh between 74 kg and 80 kg if the samples are randomly taken from 400 number of students.
To know more about probability, refer
https://brainly.com/question/24756209
#SPJ11
 1. Emily and Joe design a fenced backyard play space for their children Max and Caroline. They start by considering two designs for a triangular play space. They have made measurements in their yard and determined that either design would fit into the available space.
A) Explain how Emily and Joe can use trigonometry to calculate the area and perimeter of the possible play spaces
B) Calculate the area of the play space for each design.
C) Calculate the perimeter of the play space for each design.
D) Which design do you think Emily and Joe should choose? Explain your reasoning.
Show work, calculation, and step-by-step.
a.) The area and perimeter of the both spaces can be calculated using the Pythagorean formula to determine the length of the missing side
b.) The area of each play space design would be=44ft²
c.) The perimeter of play space=31.6ft
D.) The design Emily and Joe should choose would be= The design that they should use would be the first design.
How to calculate the length of the missing side of the triangular play space?To calculate the missing length of the triangular play space, the formula for Pythagorean theorem should be used and it's given as follows:
C² = a²+b²
where;
a= 11ft
b= 8ft
c²= 11²+8²
= 121+64
= 185
c=√185
= 12.6
The area of the triangular play space can be calculated using the formula such as;
= 1/2base ×height
For the first triangular space:
= 1/2 × 11×8
= 44ft²
The perimeter= a+b+c
= 11+8+12.6
= 31.6ft.
Learn more about area here:
https://brainly.com/question/28470545
#SPJ1
The given point is on the curve. Find the lines that are (a) tangent and (b) normal to the curve at the given point 7x² + 3xy + 2y² +17y-7=0.(-1,0) (a) Give the equation of the line that is tangent to the curve at the given point (b) Give the equation of the line that is normal to the curve at the given point y=0
a) The equation of the line tangent to the curve at the point (-1,0) is y = -3x - 7, and b) the equation of the line normal to the curve at the same point is y = 1/3x + 1/3.
To find the equation of the tangent line, we first need to find the derivative of the curve at the given point (-1,0). Taking the derivative of the given equation, we get dy/dx = (-6x - 3y) / (3x + 4y + 17). Substituting x = -1 and y = 0, we find the slope of the tangent line to be m = -3.
Using the point-slope form of a line, we can write the equation of the tangent line as y - y1 = m(x - x1), where (x1, y1) is the given point (-1,0). Plugging in the values, we get y - 0 = -3(x + 1), which simplifies to y = -3x - 3.
To find the equation of the normal line, we know that the slope of the normal line is the negative reciprocal of the slope of the tangent line. Therefore, the slope of the normal line is m' = -1/(-3) = 1/3. Using the point-slope form again, we can write the equation of the normal line as y - y1 = m'(x - x1), where (x1, y1) is (-1,0). Plugging in the values, we get y - 0 = 1/3(x + 1), which simplifies to y = 1/3x + 1/3.
Learn more about derivative here: https://brainly.com/question/32963989
#SPJ11
Find the open intervals where the function is concave upward or concave downward. Find any inflection points. f(x) = -2x+3x²+168x-1 Where is the function concave upward and where is it concave downward? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice OA. The function is concave upward on the interval(s) and concave downward on the interval(s) (Type your answers in interval notation. Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as needed.) OB. The function is concave upward on the interval(s) The function is never concave downward. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) The function is never concave upward. OC. The function is concave downward on the interval(s) (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) OD. The function is never concave upward or downward.
The function is concave upward on the interval(s). The function is never concave downward.
To determine where the function f(x) = -2x + 3x^2 + 168x - 1 is concave upward or concave downward, we need to find the intervals where its second derivative is positive (concave upward) or negative (concave downward). First, let's find the first derivative f'(x) of the function: f(x) = -2x + 3x^2 + 168x - 1, f'(x) = -2 + 6x + 168
Now, let's find the second derivative f''(x) by differentiating f'(x): f''(x) = 6. The second derivative f''(x) is a constant, which means it is always positive. Therefore, the function f(x) = -2x + 3x^2 + 168x - 1 is concave upward on its entire domain, and it is never concave downward. So, the correct choice is: OB. The function is concave upward on the interval(s). The function is never concave downward.
To learn more about derivative, click here: brainly.com/question/2159625
#SPJ11
The marginal revenue (in thousands of dollars) from the sale of x gadgets is given by the following function. 2 3 R'(x) = )= 4x(x² +26,000) (a) Find the total revenue function if the revenue from 120 gadgets is $15,879. (b) How many gadgets must be sold for a revenue of at least $45,000?
To find the total revenue function, we need to integrate the marginal revenue function R'(x) with respect to x.
(a) Total Revenue Function:
We integrate R'(x) = 4x(x² + 26,000) with respect to x:
R(x) = ∫[4x(x² + 26,000)] dx
Expanding and integrating, we get:
R(x) = ∫[4x³ + 104,000x] dx
= x⁴ + 52,000x² + C
Now we can use the given information to find the value of the constant C. We are told that the revenue from 120 gadgets is $15,879, so we can set up the equation:
R(120) = 15,879
Substituting x = 120 into the total revenue function:
120⁴ + 52,000(120)² + C = 15,879
Solving for C:
207,360,000 + 748,800,000 + C = 15,879
C = -955,227,879
Therefore, the total revenue function is:
R(x) = x⁴ + 52,000x² - 955,227,879
(b) Revenue of at least $45,000:
To find the number of gadgets that must be sold for a revenue of at least $45,000, we can set up the inequality:
R(x) ≥ 45,000
Using the total revenue function R(x) = x⁴ + 52,000x² - 955,227,879, we have:
x⁴ + 52,000x² - 955,227,879 ≥ 45,000
We can solve this inequality numerically to find the values of x that satisfy it. Using a graphing calculator or software, we can determine that the solutions are approximately x ≥ 103.5 or x ≤ -103.5. However, since the number of gadgets cannot be negative, the number of gadgets that must be sold for a revenue of at least $45,000 is x ≥ 103.5.
Therefore, at least 104 gadgets must be sold for a revenue of at least $45,000.
Learn more about inequality here -: brainly.com/question/25944814
#SPJ11
Haruki commui Given tuo non intersecting chords Авај ср a circle CA variable point p On the are renate from points. Can D. Let F ve the intersection of chonds PC, AB and of PA, AB respectively. the value of BF Joes not Jepa EF on the position of P. F 5 1/0 W 0 *=constart.
In a circle with non-intersecting chords AB and CD, let P be a variable point on the arc between A and B. The intersection points of chords PC and AB are denoted as F and E respectively. The value of BF does not depend on the position of P, given that F = 5 and E = 1/0 * constant.
Let's consider the given situation in more detail. We have a circle with two non-intersecting chords, AB and CD. The variable point P lies on the arc between points A and B. We are interested in the relationship between the lengths of chords and their intersections.
We are given that the intersection of chords PC and AB is denoted as point F, and the intersection of chords PA and AB is denoted as point E. The value of F is specified as 5, and E is given as 1/0 * constant, where the constant remains constant throughout the problem.
Now, to understand why the value of BF does not depend on the position of point P, we can observe that points F and E are defined solely in terms of the lengths of chords and their intersections. The position of P on the arc does not affect the lengths of the chords or their intersections, as long as it remains on the same arc between points A and B.
Since the position of P does not influence the lengths of chords AB, CD, or their intersections, the value of BF remains constant regardless of the specific location of P. This conclusion is supported by the given information, where F is defined as 5 and E is a constant multiplied by 1/0. Thus, the value of BF remains unchanged throughout the problem, independent of the position of P.
Learn more about chords here:
https://brainly.com/question/30845529
#SPJ11
write the sequence of natural numbers which leaves the remainder 3 on dividing by 5
Answer: 3, 8, 13, 18, 23, etc...
Step-by-step explanation:
To get a remainder of 3 upon dividing by 5, we must get multiples of 5 then add 3 to each. So, we start with 0+3=3, then 5+3=8, 10+3=13, etc... So, we end up with the sequence 3, 8, 13, 18, 23... notice how each term is 5 more than the previous.
Solve the following questions about functions: a) (6 pts) Are the following two functions from R to R one-to-one correspondences, respectively? 1) f(x) = x² +1 2) f(x) = (x + 4)/(x + 2) b) (4 pts) Let g: A → B and f: B C where A = {a,b,c,d}, B = {1,2,3}, C = {2,3,6,8), and g and f and defined by g = {(a, 2), (b, 1), (c, 3), (d, 2)} and ƒ = {(1,8), (2,3), (3,2)}. 1) Find fog. 2) Find f-¹.
The elements of each ordered pair, we get:
f⁻¹ = {(8, 1), (3, 2), (2, 3)}.
a) To determine if the given functions are one-to-one correspondences, we need to check if they are injective (one-to-one) and surjective (onto).
Function f(x) = x² + 1:
To check injectivity, we need to show that if f(x₁) = f(x₂), then x₁ = x₂.
Assume f(x₁) = f(x₂):
x₁² + 1 = x₂² + 1
x₁² = x₂²
Taking the square root of both sides:
|x₁| = |x₂|
Since the square root of a number is always non-negative, we can conclude that x₁ = x₂.
Thus, the function f(x) = x² + 1 is injective.
To check surjectivity, we need to show that for every y in the range of f(x), there exists an x in the domain such that f(x) = y.
Since the function f(x) = x² + 1 is a quadratic function, its range is all real numbers greater than or equal to 1 (i.e., [1, ∞)).
Therefore, for every y in the range, we can find an x such that f(x) = y.
Thus, the function f(x) = x² + 1 is surjective.
Based on the above analysis, the function f(x) = x² + 1 is a one-to-one correspondence.
Function f(x) = (x + 4)/(x + 2):
To check injectivity, we need to show that if f(x₁) = f(x₂), then x₁ = x₂.
Assume f(x₁) = f(x₂):
(x₁ + 4)/(x₁ + 2) = (x₂ + 4)/(x₂ + 2)
Cross-multiplying and simplifying:
(x₁ + 4)(x₂ + 2) = (x₂ + 4)(x₁ + 2)
Expanding and rearranging terms:
x₁x₂ + 2x₁ + 4x₂ + 8 = x₂x₁ + 2x₂ + 4x₁ + 8
Canceling like terms:
2x₁ = 2x₂
Dividing both sides by 2:
x₁ = x₂
Thus, the function f(x) = (x + 4)/(x + 2) is injective.
To check surjectivity, we need to show that for every y in the range of f(x), there exists an x in the domain such that f(x) = y.
The range of f(x) is all real numbers except -2 (i.e., (-∞, -2) ∪ (-2, ∞)).
There is no x in the domain for which f(x) = -2 since it would result in division by zero.
Therefore, the function f(x) = (x + 4)/(x + 2) is not surjective.
Based on the above analysis, the function f(x) = (x + 4)/(x + 2) is not a one-to-one correspondence.
b) Given:
g: A → B
f: B → C
A = {a, b, c, d}
B = {1, 2, 3}
C = {2, 3, 6, 8}
g = {(a, 2), (b, 1), (c, 3), (d, 2)}
f = {(1, 8), (2, 3), (3, 2)}
Finding fog (composition of functions):
fog represents the composition of functions f and g.
To find fog, we need to apply g first and then f.
g(a) = 2
g(b) = 1
g(c) = 3
g(d) = 2
Applying f to the values obtained from g:
f(g(a)) = f(2) = 3 (from f = {(2, 3)})
f(g(b)) = f(1) = 8 (from f = {(1, 8)})
f(g(c)) = f(3) = 2 (from f = {(3, 2)})
f(g(d)) = f(2) = 3 (from f = {(2, 3)})
Therefore, fog = {(a, 3), (b, 8), (c, 2), (d, 3)}.
Finding f⁻¹ (inverse of f):
To find the inverse of f, we need to switch the roles of the domain and the range.
The original function f = {(1, 8), (2, 3), (3, 2)}.
Swapping the elements of each ordered pair, we get:
f⁻¹ = {(8, 1), (3, 2), (2, 3)}.
Learn more about injective function here:
https://brainly.com/question/13656067
#SPJ11
If a person obliquely leans on the wall with a protruding part at the bottom as follows and is measured at 180cm, how can I calculate the difference with the height in case there was no protruding part but the person stood straight with the wall to be measured?
If a person obliquely leans on the wall with a protruding part at the bottom as follows and is measured at 180cm, to calculate the difference with the height .
When a person leans on a wall that has a protruding part at the bottom, the measurement is taken as 180cm. If we need to find out the person's actual height without leaning against the wall, we can use the Pythagoras theorem. To apply Pythagoras theorem, we can consider the person to be the hypotenuse of a right-angled triangle, and the length of the person when leaning on the wall as one of the sides. The distance between the protruding part and the wall can be considered as the other side of the triangle. Now, we can apply the Pythagoras theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.To find the difference in height, we can use the formula:
Height = √(Length of person when leaning on wall)² - (Distance between protruding part and wall)²
Suppose the length of the person when leaning on the wall is 180cm, and the distance between the protruding part and the wall is 20cm.
Then, the calculation for the person's actual height would be
:Height = √(180cm)² - (20cm)²
Height = √(32400cm² - 400cm²)
Height = √32000cm²
Height = 178.9cm
Therefore, the person's actual height is 178.9cm.
We can use the Pythagoras theorem to calculate the difference in height when a person leans on a wall with a protruding part. The length of the person when leaning on the wall can be considered as one of the sides of the triangle, and the distance between the protruding part and the wall can be considered as the other side of the triangle. By using the formula:
Height = √(Length of person when leaning on wall)² - (Distance between protruding part and wall)²
we can find out the actual height of the person.
To know more about Pythagoras theorem visit:
brainly.com/question/21926466
#SPJ11
use inverse interpolation to find x such that f(x) = 3.6
x= -2 3 5
y= 5.6 2.5 1.8
Therefore, using inverse interpolation, we have found that x = 3.2 when f(x) = 3.6.
Given function f(x) = 3.6 and x values i.e., -2, 3, and 5 and y values i.e., 5.6, 2.5, and 1.8.
Inverse interpolation: The inverse interpolation technique is used to calculate the value of the independent variable x corresponding to a particular value of the dependent variable y.
If we know the value of y and the equation of the curve, then we can use this technique to find the value of x that corresponds to that value of y.
Inverse interpolation formula:
When f(x) is known and we need to calculate x0 for the given y0, then we can use the formula:
f(x0) = y0.
x0 = (y0 - y1) / ((f(x1) - f(x0)) / (x1 - x0))
where y0 = 3.6.
Now we will calculate the values of x0 using the given formula.
x1 = 3, y1 = 2.5
x0 = (y0 - y1) / ((f(x1) - f(x0)) / (x1 - x0))
x0 = (3.6 - 2.5) / ((f(3) - f(5)) / (3 - 5))
x0 = 1.1 / ((2.5 - 1.8) / (-2))
x0 = 3.2
Therefore, using inverse interpolation,
we have found that x = 3.2 when f(x) = 3.6.
To know more about inverse interpolation visit:
https://brainly.com/question/31494775
#SPJ11
There is a one-to-one correspondence between the set of topologies on a set and the set of all nearness relations on that set. Proof: Let X be a set. Suppose 3 is a topology on X. For A CX, y EX define y 8 A iff y E A. It is easy to check that this gives a nearness relation on X. Conversely suppose a nearness relation on X is given. For A CX we let 0(A) = (y EX: y8 A). The conditions (i) to (iv) then easily imply that is a closure operator and thus determines a unique topology 3 on X. The proof that these two correspondences are inverses of each other is left to the reader. Thun m 3.4 Theorem Let (X, 3), (Y, U) be spaces and f: X→ Y a function. Then the following statements are equivalent: 1. fis continuous (i.e. 3-U continuous). 2. For all VE U,f(V) € 3. 3. There exists a sub-base S for U such that f(V) E 3 for all VES. 4. For any closed subset A of Y, f(A) is closed in X. 5. For all A CX, f(A) c f(A). Proof (
The given statement asserts a one-to-one correspondence between the set of topologies on a set and the set of all nearness relations on that set.
The proof consists of two parts. Firstly, it shows that for any given topology on a set, a corresponding nearness relation can be defined. Secondly, it demonstrates that for any given nearness relation, a corresponding topology can be determined.
To establish the first part of the proof, let X be a set and consider a topology Ƭ on X. For any A ⊆ X and y ∈ X, define y ∼ A if and only if y ∈ A. It can be easily verified that this defines a nearness relation on X.
Conversely, suppose a nearness relation is given on X. For any A ⊆ X, define the closure of A, denoted as Ƭ(A), as the set of all y ∈ X such that y ∼ A. The conditions (i) to (iv) of closure operators can be shown to hold, implying that Ƭ determines a unique topology on X.
The proof that these two correspondences are inverses of each other is left to the reader.
To know more about topologies click here: brainly.com/question/31913704
#SPJ11
. Consider the boundary value problem y"'(x) + x²y(x) = x, y(0) = -1, y(2) = 2, where yd and y" =. Use central differences with the grid spacing h=1 to set = dz up a system of linear equations for the computation of the values for y(1) and y(2). Give the answer in the form [3] = B, where A is a 2 x 2 matrix and B is a 2x 1 column vector. DO NOT attempt to solve the system. (10 marks)
The system of linear equations, obtained using central differences, for the values of y(1) and y(2) in the given boundary value problem is represented as [2, -2; 5, -2] [y₁; y₂] = [1; 2].
To set up a system of linear equations using central differences, we can approximate the derivatives using finite differences.
Let's define y(0) = y₀, y(1) = y₁, and y(2) = y₂. The grid spacing is h = 1.
Using central differences, we can approximate the second derivative as:
y"(x) ≈ (y(x+h) - 2y(x) + y(x-h))/h²
Substituting this approximation into the given boundary value problem, we have:
(y(x+h) - 2y(x) + y(x-h))/h² + x²y(x) = x
Replacing x with the corresponding grid points, we obtain the following equations:
For x = 1:
(y₂ - 2y₁ + y₀)/1² + 1²y₁ = 1
For x = 2:
(y₃ - 2y₂ + y₁)/1² + 2²y₂ = 2
Since we are interested in finding the values for y(1) and y(2), we can rewrite the equations as a system of linear equations in the form A [y₁, y₂]ᵀ = B:
[1² + 1², -2] [y₁] [1]
[1² + 2², -2] [y₂] = [2]
Simplifying the matrix equation, we get:
[2, -2] [y₁] [1]
[5, -2] [y₂] = [2]
Therefore, the system of linear equations is represented as:
[2, -2] [y₁] [1]
[5, -2] [y₂] = [2]
In the form [A] [y] = B, we have:
A = [2, -2; 5, -2]
B = [1; 2]
To know more about linear equations,
https://brainly.com/question/32205318
#SPJ11
F(x,y,z) = (xy²z4,2x²y+z, y³z² ) a) Find curlF. b) Find divF.
a) To find the curl of the vector field F, denoted as curl F or ∇ × F, we need to calculate the determinant of the curl matrix. The curl F is given by the vector (∂F3/∂y - ∂F2/∂z, ∂F1/∂z - ∂F3/∂x, ∂F2/∂x - ∂F1/∂y).
b) To find the divergence of the vector field F, denoted as div F or ∇ · F, we need to calculate the sum of the partial derivatives of the components of F with respect to x, y, and z. The divergence of F is given by (∂F1/∂x + ∂F2/∂y + ∂F3/∂z).
a) The vector field F is given as F(x, y, z) = (xy²z⁴, 2x²y + z, y³z²). We need to find the curl of F, which is the vector (∂F₃/∂y - ∂F₂/∂z, ∂F₁/∂z - ∂F₃/∂x, ∂F₂/∂x - ∂F₁/∂y).
Calculating the partial derivatives:
∂F₁/∂x = y²z⁴, ∂F₁/∂y = 0, ∂F₁/∂z = 4xy²z³
∂F₂/∂x = 4xy, ∂F₂/∂y = 2x² + z, ∂F₂/∂z = 0
∂F₃/∂x = 0, ∂F₃/∂y = 3y²z², ∂F₃/∂z = 2y³z
Now, calculating the curl components:
∂F₃/∂y - ∂F₂/∂z = 3y²z² - 0 = 3y²z²
∂F₁/∂z - ∂F₃/∂x = 4xy²z³ - 0 = 4xy²z³
∂F₂/∂x - ∂F₁/∂y = 4xy - 0 = 4xy
Therefore, the curl of F is (∂F₃/∂y - ∂F₂/∂z, ∂F₁/∂z - ∂F₃/∂x, ∂F₂/∂x - ∂F₁/∂y) = (3y²z², 4xy²z³, 4xy).
b) The divergence of F, denoted as div F or ∇ · F, is the sum of the partial derivatives of the components of F with respect to x, y, and z.
Calculating the partial derivatives:
∂F₁/∂x = y²z⁴, ∂F₁/∂y = 0, ∂F₁/∂z = 4xy²z³
∂F₂/∂x = 4xy, ∂F₂/∂y = 2x² + z, ∂F₂/∂z = 0
∂F₃/∂x = 0, ∂F₃/∂y =
3y²z², ∂F₃/∂z = 2y³z
Now, calculating the divergence:
∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z = y²z⁴ + (2x² + z) + 2y³z
Therefore, the divergence of F is ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z = y²z⁴ + 2x² + z + 2y³z.
To learn more about partial derivatives Click Here: brainly.com/question/32387059
#SPJ11
Ice Cream The function C(g) represents the cost to produce g gallons of chocolate chunk ice cream. Interpret the meaning of the statement: C(686) = 140. It costs $ to produce gallons of chocolate chunk ice cream.
The statement "C(686) = 140" means that the cost to produce 686 gallons of chocolate chunk ice cream, as represented by the function C(g), is $140. In other words, if you want to make 686 gallons of chocolate chunk ice cream, it will cost you $140.
This statement provides insight into the relationship between the quantity of ice cream produced and the corresponding cost. The function C(g) represents a mathematical model that describes how the cost varies with the amount of ice cream produced.
By evaluating C(686), we obtain the specific cost associated with producing 686 gallons of chocolate chunk ice cream, which is $140. This information allows us to understand the financial implications of scaling up production or estimating the production cost for a given quantity of ice cream.
Learn more about function here: brainly.com/question/31062578
#SPJ11
Differentiate the function. g(t) = 9t + 8t² g'(t) =
In summary, the derivative of the function g(t) = 9t + 8t² is g'(t) = 9 + 16t. The derivative represents the rate at which the function is changing with respect to the variable t.
To find the derivative of the function g(t) = 9t + 8t², we can use the power rule for derivatives. According to the power rule, the derivative of t raised to the power n is n times t raised to the power (n-1).
Taking the derivative of each term separately, we have:
The derivative of 9t with respect to t is 9.
The derivative of 8t² with respect to t is 2 times 8t, which simplifies to 16t.
Therefore, the derivative of g(t) is g'(t) = 9 + 16t. This derivative represents the rate at which the function is changing with respect to t.
To learn more about power rule click here : brainly.com/question/23418174
#SPJ11
Suppose g(x) = f(3 + 6(x − 5)) and ƒ' (3) = 4. Find g' (5). g' (5)= 3 (1 point) Suppose u(t) = w(t² + 5) and w'(6) = 8. Find u' (1). u'(1) = (1 point) Suppose h(x) = √√√f(x) and the equation of the tangent line to f(x) at x = 1 is y = 4 +5(x - 1). Find h'(1). h'(1) = 2
To find g'(5), we need to differentiate g(x) with respect to x and evaluate it at x = 5. Let's start by finding g'(x) using the chain rule:
g'(x) = f'(3 + 6(x - 5)) * (d/dx)(3 + 6(x - 5))
The derivative of the inner function is simply 6, and we know that f'(3) = 4. Substituting these values, we get:
g'(x) = 4 * 6
g'(x) = 24
Now we can evaluate g'(5):
g'(5) = 24
Therefore, g'(5) = 24.
To find u'(1), we need to differentiate u(t) with respect to t and evaluate it at t = 1.
Using the chain rule, we have:
u'(t) = w'(t² + 5) * (d/dt)(t² + 5)
The derivative of the inner function is 2t, and we know that w'(6) = 8. Substituting these values, we get:
u'(t) = 8 * (2t)
u'(t) = 16t
Now we can evaluate u'(1):
u'(1) = 16(1)
u'(1) = 16
Therefore, u'(1) = 16.
To find h'(1), we need to differentiate h(x) with respect to x and evaluate it at x = 1.
Using the chain rule, we have:
h'(x) = (1/2)(1/2)(1/2)(1/√f(x)) * f'(x)
Since the equation of the tangent line to f(x) at x = 1 is y = 4 + 5(x - 1), we can deduce that f'(1) = 5.
Substituting these values, we get:
h'(x) = (1/2)(1/2)(1/2)(1/√(4 + 5(x - 1))) * 5
Simplifying further:
h'(x) = (1/8)(1/√(4 + 5(x - 1)))
Now we can evaluate h'(1):
h'(1) = (1/8)(1/√(4 + 5(1 - 1)))
h'(1) = (1/8)(1/√(4))
h'(1) = (1/8)(1/2)
h'(1) = 1/16
Therefore, h'(1) = 1/16.
Learn more about differentiation here -: brainly.com/question/954654
#SPJ11
Linear Programming: Nonstandard Simplex P Use the simplex method to find x₁ 20 and x2 20 such that X₁ + 2x₂ ≥ 23 X₁ + X₂ ≤50 and such that z = 12x₁ + 10x2 is maximized. The maximum is z = when x₁ = and x₂ =
Using simplex method the maximum value of z is 24 when x₁ = 11 and x₂ = 6.
To solve the given linear programming problem using the simplex method, we need to convert the inequalities into equations and set up the initial simplex tableau. Let's start by introducing slack variables and converting the inequalities into equations:
Let s₁ and s₂ be slack variables for the first and second inequalities, respectively. The problem can be rewritten as follows:
Maximize z = 12x₁ + 10x₂
Subject to:
x₁ + 2x₂ + s₁ = 23
x₁ + x₂ + s₂ = 50
x₁, x₂, s₁, s₂ ≥ 0
Now, we set up the initial simplex tableau:
┌───┬───┬───┬───┬───┬───┬───┐
│ │ x₁ │ x₂ │ s₁ │ s₂ │ RHS │
├───┼───┼───┼───┼───┼───┼───┤
│ s₁│ 1 │ 2 │ 1 │ 0 │ 23 │
├───┼───┼───┼───┼───┼───┼───┤
│ s₂│ 1 │ 1 │ 0 │ 1 │ 50 │
├───┼───┼───┼───┼───┼───┼───┤
│ z │ -12 │ -10 │ 0 │ 0 │ 0 │
└───┴───┴───┴───┴───┴───┴───┘
Now, we will apply the simplex method to find the optimal solution. The steps involved are as follows:
Select the most negative coefficient in the bottom row (z-row). In this case, it is -12.
Determine the pivot column by selecting the variable corresponding to the smallest positive ratio in the pivot column. The ratio is calculated by dividing the right-hand side (RHS) value by the value in the pivot column.
For the first pivot column, the ratio for s₁ is 23/2 = 11.5, and for s₂ is 50/1 = 50. We choose s₁ as the pivot column since it has the smallest positive ratio.
Determine the pivot row by selecting the variable corresponding to the smallest nonnegative ratio in the pivot column. The ratio is calculated by dividing the RHS value by the value in the pivot column.
For s₁, the ratio is 23/1 = 23, and for s₂, the ratio is 50/1 = 50. We choose s₁ as the pivot row since it has the smallest nonnegative ratio.
Perform row operations to make the pivot element (intersection of the pivot row and pivot column) equal to 1 and clear the other elements in the pivot column.
Divide the pivot row by the pivot element (1/1).
Replace the other rows by subtracting appropriate multiples of the pivot row to make their elements in the pivot column equal to 0.
Repeat steps 1-4 until there are no negative values in the z-row or all the ratios in the pivot column are negative.
Using these steps, we will perform the simplex iterations:
Iteration 1:
Pivot column: s₁
Pivot row: s₁
Divide the pivot row by the pivot element:
s₁: 1, x₁: 2, x₂: 1, s₁: 0, s₂: 23
Perform row operations:
x₁: -1, x₂: -1, s₁: 1, s₂: 23
┌───┬───┬───┬───┬───┬───┬───┐
│ │ x₁ │ x₂ │ s₁ │ s₂ │ RHS │
├───┼───┼───┼───┼───┼───┼───┤
│ s₁│ 0 │ 1 │ 0 │ 2 │ 11 │
├───┼───┼───┼───┼───┼───┼───┤
│ s₂│ 0 │ 2 │ -1 │ -1 │ 12 │
├───┼───┼───┼───┼───┼───┼───┤
│ z │ 0 │ 2 │ 12 │ -10 │ 24 │
└───┴───┴───┴───┴───┴───┴───┘
Iteration 2:
Pivot column: x₂
Pivot row: s₁
Divide the pivot row by the pivot element:
x₂: 1, x₁: 0, x₂: 1, s₁: 2, s₂: 11
Perform row operations:
s₂: -2, x₁: 1, s₁: -2, x₂: 0
┌───┬───┬───┬───┬───┬───┬───┐
│ │ x₁ │ x₂ │ s₁ │ s₂ │ RHS │
├───┼───┼───┼───┼───┼───┼───┤
│ s₁│ 1 │ 0 │ 1 │ 2 │ 11 │
├───┼───┼───┼───┼───┼───┼───┤
│ x₂│ 0 │ 1 │ -1 │ -1 │ 6 │
├───┼───┼───┼───┼───┼───┼───┤
│ z │ 0 │ 2 │ 12 │ -10 │ 24 │
└───┴───┴───┴───┴───┴───┴───┘
Iteration 3:
No negative values in the z-row. The current tableau is the final tableau.
From the final tableau, we can read the optimal solution and the maximum value of z:
x₁ = 11
x₂ = 6
z = 24
Therefore, the maximum value of z is 24 when x₁ = 11 and x₂ = 6.
Learn more about Pivot here:
https://brainly.com/question/31261482
#SPJ11
Solve the following problem using proper numerical methods and time steps (Gerald, Applied numerical analysis). Report your results along with analytical solutions, comparisons, plots and MATLAB scripts. Check the table for your parameters. APP7. A vibrating string, with a damping force-opposing its motion that is proportional to the velocity, fol- lows the equation where B is the magnitude of the damping force. Solve the problem if the length of the string is 5 ft with 7-24 lb, w=0.1 lb/ft, and B= 2.0. Initial conditions are XXX--05x<3, XXX 3≤x≤5, = x(x - 5). Compute a few points of the solution by difference equations. 1/s B TP parameters Newton kg/m T W 140 2.4 14
To solve the problem of a vibrating string with damping using numerical methods and time steps, follow these steps:
1. Discretize the string into a set of points along its length.
2. Use a finite difference method, such as the central difference method, to approximate the derivatives in the equation.
3. Apply the difference equation to each point on the string, considering the damping force and given parameters.
4. Set up the initial conditions for the string's displacement and velocity at each point.
5. Iterate over time steps to update the displacements and velocities at each point using the finite difference equation.
6. Compute and store the values of the solution at selected points for analysis.
7. Compare the numerical solution with the analytical solution, if available, to assess accuracy.
8. Plot the results to visualize the behavior of the vibrating string over time.
9. If using MATLAB, write a script to implement the numerical method and generate plots.
Note: The specific equation and initial conditions are missing from the given question, so adapt the steps accordingly.
To know more about Function visit-
brainly.com/question/31062578
#SPJ11
ind f(x,y) for f(x,y) = -5x²y² +8x² + 2y² + 5x OA. f(x,y)= - 10xy + 16x+4y+5 OB. f(x,y) = -10xy² + 16x+5 OC. fx(x,y) = -20xy + 16x + 4y +5 O D. f(x,y)= - 10x + 16x + 5 OE. f(x,y)=-20x²y² + 16x + 4y² +5
The given function is
`f(x, y) = -5x²y² + 8x² + 2y² + 5x`.
The partial derivative with respect to x,
`fx(x, y)` is obtained by considering y as a constant and differentiating the expression with respect to x.
`fx(x, y) = d/dx[-5x²y² + 8x² + 2y² + 5x]`
Now, differentiate each term of the expression with respect to x.
`fx(x, y) = -d/dx[5x²y²] + d/dx[8x²] + d/dx[2y²] + d/dx[5x]`
Simplifying this expression by applying derivative rules,
`fx(x, y) = -10xy² + 16x + 0 + 5`
Therefore, the correct option is O C.
`fx(x, y) = -20xy + 16x + 4y + 5`is incorrect.
To know more about partial derivative visit:
brainly.com/question/15342361
#SPJ11
Determine if the following piecewise defined function is differentiable at x = 0. 2x-5, x20 f(x) = x² + 5x -5, x < 0
The following piecewise function f(x)= 3x-5, x≥0 and f(x) = x² + 5x -5, x < 0 is not differentiable at x = 0 .
To determine if the piecewise defined function is differentiable at x = 0, we need to check if the left-hand limit and the right-hand limit of the function are equal at x = 0, and if the derivative exists at x = 0.
First, let's find the left-hand limit:
lim (x→0⁻) f(x) = lim (x→0⁻) (x² + 5x - 5)
= (0² + 5(0) - 5)
= -5
Next, let's find the right-hand limit:
lim (x→0⁺) f(x) = lim (x→0⁺) (3x - 5)
= (3(0) - 5)
= -5
Since the left-hand limit (-5) and the right-hand limit (-5) are equal, we can proceed to find the derivative of the function at x = 0.
For x ≥ 0, f(x) = 3x - 5. Taking the derivative of this function:
f'(x) = 3
For x < 0, f(x) = x² + 5x - 5. Taking the derivative of this function:
f'(x) = 2x + 5
Now, let's evaluate the derivative at x = 0 from both sides:
lim (x→0⁻) f'(x) = lim (x→0⁻) (2x + 5) = 2(0) + 5 = 5
lim (x→0⁺) f'(x) = lim (x→0⁺) 3 = 3
The left-hand derivative (5) and the right-hand derivative (3) are not equal.
Since the left-hand and right-hand derivatives are not equal, the derivative of the function does not exist at x = 0. Therefore, the piecewise defined function is not differentiable at x = 0.
To know more about piecewise function here
https://brainly.com/question/28225662
#SPJ4
Find de and for the parametric equation dy d²y dx² dx x=t-et, y=t+e=t
The values of t for which the curve is concave upward are: t > 3.
Given the parametric equations are:
x = eᵗ
y = t e⁻ᵗ
Differentiating with respect to 't' we get,
dx/dt = eᵗ
dy/dt = e⁻ᵗ [1 - t]
So,
dy/dx = (dy/dt)/(dx/dt) = (e⁻ᵗ [1 - t])/eᵗ = e⁻²ᵗ [1 - t]
differentiating the above term with respect to 'x' we get,
d²y/dx² = d/dx [e⁻²ᵗ [1 - t]] = e⁻²ᵗ [(-1) - 2(1 - t)] = e⁻²ᵗ [t - 3]
Since the curve is concave upward so,
d²y/dx² > 0
e⁻²ᵗ [t - 3] > 0
either, t - 3 > 0
t > 3
Hence the values of t for which the curve is concave upward are: t > 3.
Know more about concave upward,
brainly.com/question/31990866
#SPJ4
Correct question:
Find dy/dx and d²y/dx².
x = [tex]e^{t}[/tex], y = t[tex]e^{-t}[/tex]
d/dx (y) = |(- [tex]e^{-t}[/tex] * (t - 1))/([tex]e^{t}[/tex])|
(d² * y)/(d * x²) = |([tex]e^{-2t}[/tex] * (2t - 3))/([tex]e^{t}[/tex])|
For which values of t is the curve concave upward? (Enter your answer using interval notation.)
a) Suppose that A and B are 4×4 matrices, det (4) = 2 and det((3ATB)-¹)= Calculate det (B). b) Let A, B. and C be nxn matrices and suppose that ABC is invertible. Which of A, B, and C are necessarily invertible? Justify your answer.
Since ABC is invertible, each of A, B, and C must be invertible since we cannot have an invertible product of matrices with a non-invertible matrix in it.
a) For a matrix A of order n, the determinant of A transpose is equal to the determinant of the original matrix A, i.e., det(A transpose) = det(A).
So, we have:
det(3ATB) = 3⁴ × det(A) × det(B)
Now,
det(3ATB)⁻¹ = (1/det(3ATB))
= (1/3⁴) × (1/det(A)) × (1/det(B))
Given that det(4) = 2,
we have det(A) = 2
So, (1/3⁴) × (1/2) × (1/det(B))
= (1/24) × (1/det(B))
= det((3ATB)⁻¹)
Now, equating the two values of det((3ATB)⁻¹),
we have:
(1/24) × (1/det(B)) = 2/3
Solving for det(B),
we get:
det(B) = 9
b) We know that the product of invertible matrices is also invertible. Hence, since ABC is invertible, each of A, B, and C must be invertible since we cannot have an invertible product of matrices with a non-invertible matrix in it.
To know more about determinant visit:
https://brainly.com/question/14405737
#SPJ11
