For which values of t is the parametric curve
x=6t^3,y=t+t^2,−[infinity]≤t≤[infinity]
concave up? (Enter your answer using interval notation i.e., (a,b),[a,b),(a,b] or [a,b])
Answers
The parametric curve x = 6t³ and y = t + t² is concave up for all values of t within the given interval (-∞, ∞). This means that the curve is always curving upwards, regardless of the value of t.
To determine when the parametric curve given by x = 6t³ and y = t + t² is concave up, we need to analyze the concavity of the curve. Concavity is determined by the second derivative of the curve. Let's find the second derivative of y with respect to x and determine the values of t for which the second derivative is positive.
Find dx/dt and dy/dt:
Differentiating x = 6t³ with respect to t gives dx/dt = 18t².
Differentiating y = t + t² with respect to t gives dy/dt = 1 + 2t.
Find dy/dx:
Dividing dy/dt by dx/dt gives dy/dx = (1 + 2t)/(18t²).
Find d²y/dx²:
Differentiating dy/dx with respect to t gives d²y/dx² = d/dt((1 + 2t)/(18t²)).
Simplifying, we have d²y/dx² = (36t - 36)/(18t²) = (2t - 2)/t² = 2(1 - 1/t²).
Analyze the sign of d²y/dx²:
To determine the concavity, we need to find when d²y/dx² is positive. Setting (2 - 2/t²) > 0, we have:
2 - 2/t² > 0,
2 > 2/t²,
1 > 1/t².
As 1/t² is always positive for all t ≠ 0, the inequality holds true for all t.
To analyze the concavity of the parametric curve, we first found the second derivative of y with respect to x by taking the derivatives of x and y with respect to t and then dividing them. The resulting second derivative was (2 - 2/t²).
To determine when the curve is concave up, we examined the sign of the second derivative. We simplified the expression and found that (2 - 2/t²) is always positive for all t ≠ 0. Therefore, the curve is concave up for all values of t within the interval (-∞, ∞).
This means that regardless of the value of t, the curve defined by the parametric equations x = 6t³ and y = t + t² always curves upward, indicating a concave upward shape throughout the entire interval.
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Related Questions
What is the length of AB?
Answers
Answer:
We are given to find the length of side AB. We know that in a triangle, if two angles have equal measures, then the sides opposite to them are equal in length. Thus, the length of side AB is 6 units.
Hope this helps!!
Answer:
12
Step-by-step explanation:
4x=2×+6
them you use the inverse operation and them the result is and you multiply by 2 the answer is 12
50 points! Which expression is equivalent to
Answers
We have,
= -6(3x - 2/3)
= -6(3x × 3 - 2)/3
= -6(9x - 2)/3
= -2(9x - 2)
= (-18x + 4)
Your answer will be option number 4.
Answer:
- 18x + 4
Step-by-step explanation:
\(-6(3x-\frac{2}{3})\)
- 18x + 4
if f(2x+3)=7x-10, g(x)=x+1/2 and ff(x)=g-1(x) find x
Answers
Answer: x=2.6
Step-by-step explanation:
You only need to solve the first equation to find x
2x + 3 = 7x - 10
Subtract 2x from both sides
3 = 5x - 10
Add 10 to both sides
13 = 5x
Divide 13 and 5x by 5 (13/5=2.6 5x/5=x)
2.6 = x
HELPPP W TRIANGLE ABC graph!!??
Answers
-3,4. i adding this cause this answer needs atleast 20 characters
Given that 3=a+r/a-r,make r the subject of the formula?
Answers
i- dont know...sorry
The graph of f(x) is shown.
On a coordinate plane, a parabola opens up. It goes through (negative 1.75, 4), has a vertex of (negative 0.25, negative 3), and goes through (1.4, 4). Solid circles appear on the parabola at (negative 1.6, 3.1), (negative 1.2, 0), (negative 0.25, negative 3), (0.8, 0), (1.2, 3.1).
Over which interval on the x-axis is there a negative rate of change in the function?
–2 to –1
–1.5 to 0.5
0 to 1
0.5 to 1.5
Answers
Answer:
The answer to this question is A, or -2 to -1.
Hope this helps.
Answer: A
Step-by-step explanation:
What is the slope of the line that passes through the points (-2, 3) and (-14, -5) ? ( Write your answer in the simplest form.
Answers
Answer:
⅔
Step-by-step explanation:
slope = (-5-3)/(-14+2) = -8/-12= ⅔
Which function represents g(x), a reflection of f(x) = 6(one-third) Superscript x across the y-axis?
g(x) = −6(one-third) Superscript x
g(x) = −6(one-third) Superscript negative x
g(x) = 6(3)x
g(x) = 6(3)−x
Answers
The function g(x) is g(x) = 6(1/3)^-x
How to determine the function g(x)From the question, we have the following parameters that can be used in our computation:
f(x) = 6(one-third) Superscript x
Rewrite as
f(x) = 6(1/3)^x
The transformation rule is given as
Reflection across the y-axis
This is represented as
g(x) = f(-x)
So, we have
g(x) = 6(1/3)^-x
Hence, the function is g(x) = 6(1/3)^-x
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13 Here are the first three terms of a sequence. 26 20 32 Find the first two terms in the sequence that are less than zero.
Answers
The first two terms in the sequence that are less than zero are -6 and -18.
To find the first two terms in the sequence that are less than zero, let's analyze the given sequence: 26, 20, 32. Since none of these terms are less than zero, we need to generate additional terms to identify the first two terms that satisfy this condition.
Let's assume the next term in the sequence follows a pattern where we add or subtract a constant value. We can observe that the first term (26) decreases by 6 to reach the second term (20), and then increases by 12 to reach the third term (32). Based on this pattern, we can continue generating terms in the sequence.
The fourth term would be obtained by subtracting 6 from the third term: 32 - 6 = 26.
The fifth term would be obtained by adding 12 to the fourth term: 26 + 12 = 38.
The sixth term would be obtained by subtracting 6 from the fifth term: 38 - 6 = 32.
Continuing this pattern, we can see that the sequence alternates between subtracting 6 and adding 12.
Now, let's check which terms in the sequence are less than zero:
The first term less than zero is -6, which is obtained by subtracting 6 from the fourth term.
The second term less than zero is -18, which is obtained by subtracting 6 from the fifth term.
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Compare the strengths and weaknesses of the horizontal and vertical methods for adding and subtracting polynomials. Include common errors to watch out for when using each of these methods.
Answers
Answer:
Overall vertical is visually better, if done correctly
it forces you to "line up" all the common exponents.
The disadvantage is that it usually requires re-writing the problem, and it takes up space.
most problems are presented horizontally, that becomes the issue to locate the common exponents.
in both cases the biggest issue is people forget
that when subtracting "subtracting a negative is like adding a positive"
-5x - (-8x) = 3x [that is a positive 3x]
or:
-7x
- - 10x
-------------
3x
everyone misses those eventually so you have to watch out for that in both methods
Step-by-step explanation:
A committee of four students is being randomly selected from a group of twenty students. How many combinations of four students are possible?
Answers
Using the combination formula, it is found that 4845 combinations of four students are possible.
What is the combination formula?\(C_{n,x}\) is the number of different combinations of x objects from a set of n elements, given by:
\(C_{n,x} = \frac{n!}{x!(n-x)!}\)
In this problem, four students are taken from a set of 20, hence the number of combinations is given by:
\(C_{20,4} = \frac{20!}{4!16!} = 4845\)
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A group of 20 students and 4 adults are going on a trip to the museum. Show the ratio.
Thank you for helping!♡
Do not send links because they don't work.
Answers
Answer:
5:1
Step-by-step explanation:
I think
PLS HELP IM BEGGING U
Answers
Answer:
answer is 57.
Step-by-step explanation:
Given
m<6= x - 8
m<7 = 2x - 7
We know
<6 + <7 = 180° [ being linear pair ]
x - 8 + 2x - 7 = 180°
3x - 15 = 180°
3x = 180° + 15°
3x = 195°
x = 195° / 3
x = 65°
Now
m<6= x - 8 = 65° - 8° = 57°
hope it helps :)
A survey was given to a random sample of 1350 residents of a town to determine whether they support a new plan to raise taxes in order to increase education spending. Of those surveyed, 64% of the people said they were in favor of the plan. At the 95% confidence level, what is the margin of error for this survey expressed as a percentage to the nearest tenth?
Answers
The margin of error for the survey, rounded to the nearest tenth, is approximately 4.0% when expressed as a percentage.
To determine the margin of error for a survey at the 95% confidence level, we need to calculate the standard error. The margin of error represents the range within which the true population proportion is likely to fall.
The formula for calculating the standard error is:
Standard Error = sqrt((p * (1 - p)) / n)
where p is the sample proportion and n is the sample size.
In this case, the sample proportion is 64% (or 0.64) since 64% of the 1350 surveyed residents support the plan.
Plugging in the values:
Standard Error = \(\sqrt{(0.64 * (1 - 0.64)) / 1350)}\)
\(= \sqrt{(0.2304 / 1350)} \\= \sqrt{(0.0001707)}\)
≈ 0.0131
Now, to find the margin of error, we multiply the standard error by the appropriate critical value for a 95% confidence level. The critical value corresponds to the z-score, which is approximately 1.96 for a 95% confidence level.
Margin of Error = z * Standard Error
= 1.96 * 0.0131
≈ 0.0257
Finally, to express the margin of error as a percentage, we divide it by the sample proportion and multiply by 100:
Margin of Error as Percentage = (Margin of Error / Sample Proportion) * 100
= (0.0257 / 0.64) * 100
≈ 4.0%
Therefore, the margin of error for this survey, expressed as a percentage to the nearest tenth, is approximately 4.0%.
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Find the SURFACE AREA of the rectangular prism. LABEL your answer.
5 cm
6 cm
6 cm
What formula will you use to find the area of each side? A=
How many sides does the prism have?
What is the TOTAL surface area? (label your answer)
Answers
Answer:
surface area of prism= 2(lb+bh+lb)
surface area of prism= 2(6×6+6×5+5×6)
surface area of prism= 168cm²
formula=2(lb+bh+hl)
sides=12 sides
total surface area =168cm²
volume of prism=l×b×h
volume of prism=6×6×5
volume of prism=180cm³
A coin jar contains 10 quarters, 6 dimes, and 8 nickels. Find the probability that a dime is chosen at random
from the jar and then a nickel is chosen at random from the jar.
Answers
Answer:
6/24 - 1/4
Step-by-step explanation:
IM ON MY LAST TOPIC AND MY LAST QUESTION!!! Amanda bought 5 CDs that were each the same price. Including sales tax, she paid a total of 67.50. Each CD had a tax of 0.60. What was the price of each CD before tax?
Answers
Answer:
$ 12.90
Step-by-step explanation:
(0.60)*(5)=$3
$67.50 - 3 = $ 64.50
$64.50/5 = $12.90
7. The perimeter of a circle of diameter 10 cm is-
(a) 30.4 cm. (b) 30.14 cm
(c) 31.4 cm. (d) 30.01 cm.
(Take π = 3.14)
Answers
\(\qquad\qquad\qquad\boxed{\bf\:\:{(c) \: 31.4 \: cm \: ans \: }}\)
Explanation -:
The diameter is = 10 cm
( Take π = 3.14 )
\( \\ \: \tt :\implies \: 10 \: \times 3.14 \\ \\ 31.4 \: ans\)
More Information -:
Area of circle= πr² Circumference= 2 πr π = 22/7 or 3.14 ( approx )Annulas = π(R² - r ²)2. Triangle LMN is isosceles with LM = LN; X and Y are points on LM, LN respectively such that LX = LY. Prove that triangles LMY and LNX are congruent.
Answers
Through the SAS rule of congruence, we prove that triangles LMY and LNX are congruent.
Congruence is the condition when both of the figures have the same size, and shape and are mirror images of each other. SAS rule of congruence states when two sides and one angle is equal to the corresponding sides of the other triangle, then the two triangles are congruent.
Now, we have triangles LMY and LNX
In these two triangles, we have :
LX = LY { equal sides of isosceles triangle }
LM = LN { given}
∠LMY = ∠LNX {corresponding angles of the equal sides of the isosceles triangle }
Hence, triangles LMY and LNX are congruent by the SAS rule of congruence.
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what will the the measures of the interior be?
Answers
The sum of the measures all the interior angles of a polygon having 14 sides is 2160°.
How to calculate the sumThe formula we use for this is
Sum of interior angles of a polygon
= ( n-2)*180
Where “n” is the number of sides of the polygon.
Here n=14
So, sum = (14–2)*180 = 2160 degrees
Therefore, the sum of the measures all the interior angles of a polygon having 14 sides is 2160°.
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What will be the sum of the measures all the interior angles of a polygon having 14 sides?
The arc length x = True O False 4(3 + y)² on the interval [1, 4] is approximately 131 units.
Answers
The statement is false. The arc length of the curve defined by x = 4(3 + y)² on the interval [1, 4] is not approximately 131 units.
To find the arc length of a curve, we use the formula ∫ √(1 + (dx/dy)²) dy, where the integral is taken over the given interval.
In this case, the equation x = 4(3 + y)² represents a parabolic curve. By differentiating x with respect to y and substituting it into the arc length formula, we can calculate the exact arc length over the interval [1, 4].
However, it is clear that the arc length of the curve defined by x = 4(3 + y)² cannot be approximately 131 units, as this would require a specific calculation using the precise integral formula mentioned above.
Therefore, the statement is false, and without further calculations, we cannot determine the exact arc length of the given curve on the interval [1, 4].
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(9 – x² , x < 2 2. Given f(x) = { vx + 7, 2< x < 10, [x – 4] ,x 2 10 give the values of the following:
(a) f(2)
(b) f(15)
(c) f(-3)
(d) f(5)
(e) f(1.5)
(f) f(12.5)
please
Answers
Answer:
kailangan lang po
Step-by-step explanation:
plese pa brainlest po ty po plzzzzz
Jonah has read 265 pages. He has 147 more to read. How many pages did he read in all.
Answers
Answer:
the pages read =265 more pages=147hedidnotread so pages he read =265
does the data suggest that true average total body bone mineral content during post weaning exceeds that during lactation by more than 25 g? state and test the appropriate hypotheses using a significance level of 0.05. (note: the appropriate normal probability plot shows some curvature but not enough to cast substantial doubt on a normality assumption.)
Answers
The appropriate hypotheses using a significance level of 0.05 reject H0.
The hypothesis :
H0 : μL - μP ≥ 25
H0 : μL - μP < 25
The sample mean difference; Xd
d = L - P
Xd = Σd/n
d = -198,-336,-70,-18,-122,-9,-50,-5,-163,-86
Xd = - 1057 / 10
Xd = - 105.7
Using calculator ;
Standard deviation of difference, Sd = 103.845
The test statistic :
T = Xd ÷ (Sd/√n)
T = -105.7 ÷ (103.845/√10)
T = - 3.219
Decision region :
Reject H0 ; If P-value < α
The P-value : df = n - 1 ; 10 - 1 = 9
P-value(-3.219, 9) ; two-tailed = 0.00525
Hence the answer is, the appropriate hypotheses using a significance level of 0.05 reject H0.
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Out of 1,000 bees in a colony 500 are drones. Out of these 500 drones, 100 are outside the hive. Out of the 500 bees that are not drones 300 are outside the hive. What is the probability that a randomly selected bee outside the hive is a drone?
Answers
The probability that a randomly selected bee outside the hive is a drone is 0.25 or 25%.
To find the probability that a randomly selected bee outside the hive is a drone, we need to consider the total number of bees outside the hive and the number of drones outside the hive.
Given information:
Total number of bees in the colony: 1,000
Number of drones in the colony: 500
Number of drones outside the hive: 100
Number of non-drone bees in the colony: 1,000 - 500 = 500
Number of non-drone bees outside the hive: 300
To calculate the probability, we divide the number of favorable outcomes (drones outside the hive) by the total number of possible outcomes (bees outside the hive).
Probability of selecting a drone outside the hive = Number of drones outside the hive / Number of bees outside the hive
Probability = 100 / (100 + 300) = 100 / 400 = 0.25
Therefore, the probability that a randomly selected bee outside the hive is a drone is 0.25 or 25%.
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Find the missing segment in the image below
Answers
Answer:
15/5=?/8
Step-by-step explanation:
3=?/8 ?=24 it can help you
Factor the expression 4x + 32. Explain each step you take in the process. 100 points goes to brainliest
Answers
Answer:
4(x+8)
Step-by-step explanation:
4x+32
x+8 in parentheses
and put the 4 on the outside of the parentheses
like this 4(x+8)
Answer:
4(x+8)
Step-by-step explanation:
4x + 32
Rewriting
4*x + 4*8
Factor out 4
4(x+8)
Identify the quadratic function that is in standard form and has zeros −2 and 8.
f(x) = x2 − 6x − 16
f(x) = x2 + 10x + 16
f(x) = x2 − 6x + 16
f(x) = x2 + 10x − 16
Answers
The quadratic function is f(x) = x² -6x + 16.
What is a polynomial?An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial.
Let the standard form of quadratic function polynomial is ax² + bx + c = 0
And the zeros of the equation is (x +2) and (x-8)
Now to find the polynomial,
we multiply both zeros
(x +2)(x -8)
= x² + 2x -8x + 16
= x² -6x + 16
And the function in the standard form.
Therefore, the quadratic function is f(x) = x² -6x + 16.
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in an effort to remember the quadratic formula, justice writes the equation 50 times. she is using a technique known as
Answers
Justice is using a technique known as chunking.
Short-term memory can function more effectively thanks to chunking, however the quantity of chunks that can be stored efficiently declines as chunk size grows.
Large pieces of information may be divided into smaller bits and then grouped together using relevant information or attributes before being stored in order to improve retention of that information in the short term memory. To establish relational information or similarity between the information to be stored in order to improve retention and recall in the short term memory, information must be broken down before being regrouped. However, as chunk size grows, so does the efficiency of information chunking.
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Which of the following fractions is the largest? a. 2/3 b. 1/2 c. 4/9 d. 3/7 Mark to review later... 2. Which of the following fractions is the largest? a. 3/5 b. 9/8 c. 6/11 d. 7/19 Mark to review later... 3. The sum of 191/4,21,4,26,9, and 21 is closest to which of the following? a. 91 b. 97 c. 100 d. 102 Mark to review later... 4. $327.45 subtracted from $572.98 is closest to which of the following? a. $240.99 b. $245.57 c. $247.18 d. $248.99
Answers
1. The largest fraction is 2/3. option a
2. The largest fraction is 3/5. Option a
3. The sun=m of the given fractions is closest to 102. Option d
4. The difference of the two given number is closes to $245.57. Option b
Determination of largest fractionTo determine the largest among these fractions 2/3, 1/2, 4/9, and 3/7, convert each of them to a common denominator.
The smallest common denominator is 126, which is the product of 2, 3, 7, and 9.
a. 2/3 = 84/126
b. 1/2 = 63/126
c. 4/9 = 56/126
d. 3/7 = 54/126
Since 84/126 > 63/126, 56/126, and 54/126, the largest fraction is 2/3.
Similarl
For these fractions 3/5, 9/8, 6/11, and 7/19,
The smallest common denominator is 2095, which is the product of 5, 8, 11, and 19.
a. 3/5 = 3348/2095
b. 9/8 = 2315/2095
c. 6/11 = 1140/2095
d. 7/19 = 735/2095
Since 3348/2095 > 2315/2095, 1140/2095, and 735/2095, the largest fraction is 3/5.
Sum of 191/4, 21/4, 26/9, and 21,
The smallest common denominator is 36, which is the product of 4 and 9.
a. 191/4 = 4775/36
b. 21/4 = 1575/36
c. 26/9 = 104/36
d. 21 = 756/36
Adding these fractions
4775/36 + 1575/36 + 104/36 + 756/36 = 8210/36 = 227.78
Approximately 228
228 is closest to 102 than any other among the given numbers.
The difference between $572.98 and $327.45
$572.98 - $327.45 = $245.53
The closest is $245.53
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