A. The function, f(x) is differentiable at x = 0 but is not continuously differentiable at x = 0.
B. f(x) = sin[tex]\frac{1}{x}[/tex] is not differentiable at x = [tex]\frac{1}{n\pi}[/tex] for all integers n and It is defined on the interval [0, 1]. However, it is not continuous at x = 0 and at all points of the form x = 1/nπ for all integers n. This is because the function oscillates wildly as x approaches these points.
C. If f is differentiable on (a,b), with f'(x) ≠ 1 for all x in (a,b), then f can have at most one fixed point in (a, b).
Let say f = (x₁, x₂) and x₁ < x₂
Which means f(x₁) = x₁ and f(x₂) = x₂
According to the Mean Value Theorem therefore f'(c) = [tex]\frac{f(x_2) - f(x_1)}{ (x_2 - x_1).}[/tex] =1
But f(x₁) = x₁ and f(x₂) = x₂,
so f'(c) = 1, a contradiction.
Therefore, f can have at most one fixed point in (a, b)
How do we show that the function is differentiable at x = 0, but not continuously differentiable?(A) To show that the function f(x) = x² sin[tex]\frac{1}{x}[/tex] is differentiable at x = 0, we find the derivative of f(x) to know if it exists at x = 0.
For f(x) = x²sin[tex]\frac{1}{x}[/tex]
⇒ f'(x) = 2xsin[tex]\frac{1}{x}[/tex] - cos[tex]\frac{1}{x}[/tex] become the derivative, using the product and chain rule.
To find f'(0), we use the limit definition of the derivative:
lim_(x→0) [f(x) - f(0)] / (x - 0) = lim_(x→0) [x × sin(1/x)] = 0.
∴This limit exists, so f(x) is differentiable at x = 0.
However, derivative f'(x) = 2xsin[tex]\frac{1}{x}[/tex] - cos[tex]\frac{1}{x}[/tex] does not have a limit as x approaches 0 (it oscillates indefinitely),
∴ f(x) is not continuously differentiable at x = 0.
(C) The Mean Value Theorem states that for any differentiable function f and any interval [a,b], there exists a point c in (a,b) such that
[tex]f'(c) =\frac{ f(b) - f(a) }{(b - a)}[/tex]
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PLEASE HELP!!!! 69 points my guy!!!
Answer:
D. f(q) = 2q + 3Step-by-step explanation:
Given equation:
6q = 3s - 9q is independent variable, we need to solve it for s:
6q = 3s - 93s = 6q + 9s = 2q + 3Correct choice is
D. f(q) = 2q + 3solve the equation 3a - 6 = -12.
Answer:
a=-2
Step-by-step explanation:
3a−6=−12
Step 1: Add 6 to both sides.
3a−6+6=−12+6
3a=−6
Step 2: Divide both sides by 3.
3a/ 3 = −6/ 3
a=−2
What is the probability of getting a number greater than or equal to 5 when rolling a number cube numbered 1 to 6?
Answer:
There is a 1/3 (or 0.33%) probability of rolling a number greater than or equal to 5.
Step-by-step explanation:
First, find what numbers are greater than or equal to 5:
5 and 6
Find what options you can get on a number cube:
1, 2, 3, 4, 5, and 6
Out of the 6 possible outcomes, there are only 2 that will get a number greater than or equal to 5. Write this as a fraction:
2/6
Simplify:
1/3
A coastline recedes at a rate of 3cm per year. How much of the coastline dissappears after 4 years?
Answer:
12 cm of coastline
Step-by-step explanation:
Multiply the years (factor) by the receding amount, 3cm, (constant) BAM answer.
PLEASE HELP! What does x equal?
Step-by-step explanation:
(FE+BC)/2=<FDE
<FDE=<BDC
it is basically the average of x and 70.
(x+70)/2=104
x=138
Hope that helps :)
Answer:
The answer is probly 70 degrees
i might be wrong
i just hope this works
good luck
Your grandmother always has a jar of cookies on her counter. One day while you are visiting, you eat 5 cookies from the jar. In the equation below, c is the number of cookies remaining in the jar and b is the number of cookies in the jar before your visit.
Answer:
b - 5 = c
Step-by-step explanation:
if you take the number of cookies before the visit (B) and then you eat 5, that would be (B - 5). after you eat the cookies, C is the amount left. so it’s a subtraction problem without 2 numbers. so the equation would be
b - 5 = c
or
before - 5 = after
A certain statistic bˆ is being used to estimate a population parameter B. The expected value of bˆ is equal to B. What property does bˆ exhibit?
Answer:
Unbiased
Step-by-step explanation:
If b^ is equal to B this means that it is an unbiased estimator. When there is an absence of bias, we have an unbiased estimator. As an unbiased estimator it gives accurate information most of the time. The result it gives is not over estimated and also it is not underestimated.
Expected value = true value
Parameter estimates are correct on average
Thank you
Is this statement true or false? You calculate a finance charge by subtracting the cost of the purchase from the total payment,
Answer:
True
Step-by-step explanation:
Brainliest?
Answer:
I Believe the answer is True
Step-by-step explanation:
Find the general solution to the differential equation dy xoay + 3y = x2 dx b) Find the particular solution to the differential equation dy dx = (y + 1)(3x2 – 1) E subject to the condition that y = 0 at x = 0 c) Find the particular solution to the differential equation dy dx = y X- subject to the condition that y = 2 at x = 1
a) The differential equation is y = [tex]e^{(4x)/x}[/tex] + 3 b) The particular solution is y = [tex]e^{x^3 - x}[/tex] - 1 c) The particular solution to the differential equation is given by the equations is y = 2x or y = -2x.
a) To find the general solution to the differential equation:
x(dy/dx) + 3y = [tex](e^{4x})/{x^2}[/tex]
We can start by rearranging the equation:
dy/dx = [[tex](e^{4x})/{x^2}[/tex] - 3y]/x
This equation is linear, so we can use an integrating factor to solve it. The integrating factor is given by:
μ(x) = e^(∫(1/x) dx) = [tex]e^{ln|x|}[/tex] = |x|
Multiplying both sides of the equation by the integrating factor:
|x| * dy/dx - 3|xy| = [tex]e^{(4x)/x}[/tex]
Now, let's integrate both sides with respect to x:
∫(|x| * dy/dx - 3|xy|) dx = ∫([tex]e^{(4x)/x}[/tex]) dx
Using the properties of absolute values and integrating term by term:
∫(|x| * dy) - 3∫(|xy|) dx = ∫([tex]e^{(4x)/x}[/tex]) dx
Integrating each term separately:
∫(|x| * dy) = ∫([tex]e^{(4x)/x}[/tex]) dx + 3∫(|xy|) dx
To integrate ∫(|x| * dy), we need to know the form of y. Let's assume y = y(x). Integrating ∫[tex](e^{4x)/x}[/tex] dx gives us a natural logarithm term.
Integrating 3∫(|xy|) dx can be done using different cases for the absolute value of x.
By solving these integrals and rearranging the equation, you can find the general solution for y(x).
b) To find the particular solution to the differential equation:
dy/dx = (y + 1)(3x² - 1)
subject to the condition that y = 0 at x = 0.
We can solve this equation using separation of variables. Rearranging the equation:
dy/(y + 1) = (3x² - 1) dx
Now, let's integrate both sides:
∫(dy/(y + 1)) = ∫((3x² - 1) dx)
The left-hand side can be integrated using the natural logarithm function:
ln|y + 1| = x³ - x + C1
Solving for y, we have:
[tex]y + 1 = e^{x^3 - x + C1}\\y = e^{x^3 - x + C1} - 1[/tex]
Using the initial condition y = 0 at x = 0, we can find the particular solution. Substituting these values into the equation:
0 = [tex]e^{0 - 0 + C1}[/tex] - 1
1 = [tex]e^{C1}[/tex]
C1 = ln(1) = 0
Therefore, the particular solution is:
y = [tex]e^{x^3 - x}[/tex] - 1
c) To find the particular solution to the differential equation:
x(dy/dx) - y = y
subject to the condition that y = 2 at x = 1.
We can simplify the equation:
x(dy/dx) = 2y
Now, let's separate variables and integrate:
(1/y) dy = (1/x) dx
Integrating both sides:
ln|y| = ln|x| + C2
Simplifying further:
ln|y| = ln|x| + C2
ln|y| - ln|x| = C2
ln(|y/x|) = C2
|y/x| = [tex]e^{C2}[/tex]
Since we are given the initial condition y = 2 at x = 1, we can substitute these values into the equation:
|2/1| = [tex]e^{C2}[/tex]
2 = [tex]e^{C2}[/tex]
C2 = ln(2)
Therefore, the particular solution is:
|y/x| = [tex]e^{ln(2)}[/tex]
|y/x| = 2
Solving for y, we have two cases:
y/x = 2
y = 2x
y/x = -2
y = -2x
So, the particular solution to the differential equation is given by the equations:
y = 2x or y = -2x.
The complete question is:
a) Find the general solution to the differential equation
x dy/dx + 3y = (e⁴ˣ)/(x²)
b) Find the particular solution to the differential equation dy/dx = (y + 1)(3x² - 1)
subject to the condition that v = 0 at x = 0
c) Find the particular solution to the differential equation
x dy/dx (y) = y
subject to the condition that y = 2 at x = 1
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The radius of a circle is 3 kilometers. What is the circle's area
Answer:
28.27
Step-by-step explanation:
A=πr2=π·32≈28.27433
Answer:
If you're using 3.14 for pi, it's 28.26
Step-by-step explanation:
What is the approximate surface area of this right prism with triangular bases?
Answer:
394.2 ft
Step-by-step explanation:
7.8(9)=70.2
This is the measurement for the two triangular sides
9(12)(3)=324 This is the measurement for the three triangular sides
70.2 +324 = 394.2 APPROX
Can somebody help me!!??
Answer:
D) 8
Step-by-step explanation:
We're looking for [tex]x[/tex], which means the 7x and 5x will have the SAME x.
7(8) = 56
For the C corner, it is cornered as a 90 degree angle, so that means the (7x) + 34 degrees NEEDS to equal 90.
7(8) = 56 + 34 = 90!!
Now we have to see if 5(8) is correct
5(8) = 40 + 50 [The degree mark] = WHICH EQUALS 90 TOO..
Therefore, the answer is D) 8
Luis's car used 4/5
of a gallon to travel 29 miles. At what rate does the car use gas, in miles per gallon?
Answer:
36.25 miles per gallon
Step-by-step explanation:
4/5 = .8
29/.8 = x/1
cross-multiply:
.8x = 29
x = 36.25
what angle is opposite the longest side?
I will mark as brainliest
Answer:
75
Step-by-step explanation:
Answer:
the angle that is 75 degrees
Step-by-step explanation:
Because the longest side is always opposite of the largest angle and the smallest side is always opposite of the smallest angle.
Question 8 of 9
Carlita has a swimming pool in her backyard that is rectangular with a length of 26 feet and a width of 16
feet. She wants to install a concrete walkway of width c around the pool. Surrounding the walkway, she
wants to have a wood deck that extends w feet on all sides. Find an expression for the perimeter of the wood
deck.
Then, the width of the pool covered by the concrete walkway is 16 + 2c feet (2c feet on each side), and the length of the pool covered by the concrete walkway is 26 + 2c feet (2c feet on each end).
We have given a rectangular swimming pool with a length of 26 feet and a width of 16 feet. We need to find the perimeter of the wood deck that surrounds the concrete walkway of width c around the pool and extends w feet on all sides.
Let's solve the given problem as follows:Firstly, let's calculate the dimensions of the concrete walkway. Let the width of the concrete walkway be 'c' feet.
Then, the width of the pool covered by the concrete walkway is 16 + 2c feet (2c feet on each side), and the length of the pool covered by the concrete walkway is 26 + 2c feet (2c feet on each end).
So, the dimensions of the pool and concrete walkway are (26 + 2c) ft. x (16 + 2c) ft.The dimensions of the wood deck that surrounds the concrete walkway by w feet on all sides will be (26 + 2c + 2w) ft. x (16 + 2c + 2w) ft.Now, let's write the expression for the perimeter of the wood deck.P = 2(Length + Width)P = 2[(26 + 2c + 2w) + (16 + 2c + 2w)]P = 2[42 + 4c + 4w]P = 84 + 8c + 8wThe expression for the perimeter of the wood deck is 84 + 8c + 8w. Hence, the answer is 84 + 8c + 8w.
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determine the inteerval on which the following function is convave upor concave down. identify any inflection points. f(x)= x^4-4x^3 3
To determine the intervals on which the function [tex]f(x) = x^4 - 4x^3 + 3[/tex] is concave up or concave down, we need to find the second derivative of the function and analyze its sign.
First, let's find the first derivative of f(x):
[tex]f'(x) = 4x^3 - 12x^2[/tex]
Next, let's find the second derivative of f(x) by taking the derivative of f'(x):
[tex]f''(x) = 12x^2 - 24x[/tex]
To determine the intervals of concavity, we need to find where f''(x) is positive (concave up) or negative (concave down). We can do this by analyzing the sign of f''(x).
Now, let's find the values of x for which f''(x) = 0, as these will give us potential inflection points:
[tex]12x^2 - 24x = 0[/tex]
12x(x - 2) = 0
x = 0 or x = 2
The potential inflection points are x = 0 and x = 2.
To analyze the concavity of the function, we'll use test points within the intervals between the potential inflection points and beyond.
For x < 0, let's choose x = -1 as a test point:
[tex]f''(-1) = 12(-1)^2 - 24(-1) = 12 + 24 = 36[/tex]
Since f''(-1) = 36 > 0, the function is concave up for x < 0.
For 0 < x < 2, let's choose x = 1 as a test point:
[tex]f''(1) = 12(1)^2 - 24(1) = 12 - 24 = -12[/tex]
Since f''(1) = -12 < 0, the function is concave down for 0 < x < 2.
For x > 2, let's choose x = 3 as a test point:
[tex]f''(3) = 12(3)^2 - 24(3) = 108 - 72 = 36[/tex]
Since f''(3) = 36 > 0, the function is concave up for x > 2.
In summary:
The function is concave up for x < 0 and x > 2.
The function is concave down for 0 < x < 2.
The inflection points are x = 0 and x = 2.
I hope this clarifies the concavity and inflection points of the given function. Let me know if you have any further questions!
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[tex](2x5y6)(4x - 3y - 3) [/tex]
hey guys could yall solve this problem for me? thanks
Answer:
Given ABCD ~ EFGH
FG = BC(EF/AB)
FG = 7(9/6)
FG = 63/6
FG = 10.5
GH = CD(EF/AB)
GH = 11(9/6)
GH = 99/6
GH = 16.5
EH = AD(EF/AB)
EH = 12(9/6)
EH = 108/6
EH = 18
CAN SOMEONE HELP PLS :D will mark brainliest ;)
Answer: the second one/ \/36/6
Step-by-step explanation:
An experiment consists of tossing 3 fair (not weighted) coins, except one of the 3 coins has a head on both sides. Compute the probability of obtaining exactly 3 heads The probability of obtaining exactly 3 heads is
The probability of obtaining exactly 3 heads is 5/16.
We can find the probability of obtaining exactly 3 heads by considering the different ways in which this can happen.
First, suppose we toss the two normal coins and the biased coin with the two heads.
There is a probability of getting heads on each toss of the biased coin: 1/2
A probability of getting heads on each toss of the normal coins: 1/2
Therefore, the probability of getting exactly 3 heads in this case is:
(1/2) * (1/2) * (1/2) = 1/8
Now suppose we toss the two normal coins and the biased coin with the two heads, but we choose to use the biased coin twice. In this case, we need to get two heads in a row with the biased coin, and then a head with one of the normal coins.
The probability of getting two heads in a row with the biased coin is: 1/2, and the probability of getting a head with one of the normal coins is: 1/2. Therefore, the probability of getting exactly 3 heads in this case is:
(1/2) * (1/2) * (1/2) = 1/8
Finally, suppose we use the biased coin and one of the normal coins twice each. In this case, we need to get two heads in a row with the biased coin, and then two tails in a row with the normal coin. The probability of getting two heads in a row with the biased coin is: 1/2,
and the probability of getting two tails in a row with the normal coin is :
(1/2) * (1/2) = 1/4.
Therefore, the probability of getting exactly 3 heads in this case is:
(1/2) * (1/2) * (1/4) = 1/16
Adding up the probabilities from each case, we get:
1/8 + 1/8 + 1/16 = 5/16
Therefore, the probability of obtaining exactly 3 heads is 5/16.
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calculate the exact distance between the points (8, -3) and (-2, 4). sophia calculus
The exact distance between the points (8, -3) and (-2, 4) can be calculated using the distance formula in mathematics.
The formula for finding the distance between two points (x1, y1) and (x2, y2) in a two-dimensional Cartesian coordinate system is given by: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). Using the coordinates (8, -3) and (-2, 4), we can substitute the values into the distance formula: Distance = sqrt((-2 - 8)^2 + (4 - (-3))^2) = sqrt((-10)^2 + (7)^2) = sqrt(100 + 49) = sqrt(149) ≈ 12.207
Therefore, the exact distance between the points (8, -3) and (-2, 4) is approximately 12.207 units.
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A whale is at the surface of the ocean to breathe. What is the whale's elevation?
Which dot plot represents the data in this frequency table?
Number 3 4 5 7 8
Frequency 3 2 4 2 3
Answer:
Im so sorry im late! The answer is A i just took the quiz!
Step-by-step explanation:
The correct dot plot is given in option 1.
What is a dot plot?Any data that may be shown as dots or tiny circles is called a dot plot. Given that the height of the bar created by the dots indicates the numerical value of each variable, it is comparable to a bar graph or a simple histogram. Little amounts of data are shown using dot plots.
As per the given data:
There are 4 options, with each option represented by a diagram also the number and frequency table is given
Number: 3 4 5 7 8
Frequency: 3 2 4 2 3
We can find the correct diagram of the dot plot by observing the number of cross against each value of the number on the line and then matching the obtained value with the given number and frequency table.
Hence, the correct dot plot is given in option 1.
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PARK is a parallelogram. Find the value of x.
Answer: 40
Step-by-step explanation:
complete the table... plz help
Answer:
3=20
4=15
5=12
Step-by-step explanation:
Find the inverse of the following matrix. Write entries as integers or fractions in lowest terms. If the matrix is not invertible, type "N" for all entries. -5-1021 A = -2-5 9 1 2 -4
The inverse of matrix A is given by;
A^-1 = |5/139 -189/139 29/139 |
|-10/139 129/139 -27/139 |
|-5/139 19/139 -3/139 |
The given matrix is A =
| -5 -10 21 |
| -2 -5 9 |
| 1 2 -4 |
To find the inverse of a matrix, first find the determinant of that matrix. The determinant of matrix A is given as;
|A| = -5(-5(-4) - 2(9)) - (-10)(-2(-4) - 1(21)) + (21)(-2(2) - 1(-5))
|A| = -5(10) + 100 - 21(9)
|A| = -50 + 100 - 189
|A| = -139
Thus, the determinant of matrix A is -139. Now, we can use the formula of inverse of a 3x3 matrix;
A^-1 = 1/|A| * |(b22b33 - b23b32) (b13b32 - b12b33) (b12b23 - b13b22)|
| (b23b31 - b21b33) (b11b33 - b13b31) (b13b21 - b11b23)|
| (b21b32 - b22b31) (b12b31 - b11b32) (b11b22 - b12b21)|
where b is the cofactor of each element of matrix A.
The cofactor of element aij is denoted as Aij and given as Aij = (-1)i+j|Mij|.
Thus, the cofactors of matrix A are;
|-5 -10 21|
| -2 -5 9 |
| 1 2 -4 |
M11 = | -5 9 |
| 2 -4 |
M12 = | -2 9 |
| 2 -5 |
M13 = | -2 -5 |
M21 = | -10 21 |
| 2 -9 |
M22 = | -5 -21 |
| -2 5 |
M23 = | -2 -2 |
M31 = | -10 -5 |
| 2 9 |
M32 = | -5 -9 |
| 2 2 |
M33 = | -2 -2 |
Now we can find the inverse of matrix A as follows;
A^-1 = 1/-139 * |(5 189 -29)|
|(-10 -129 27)|
|(-5 19 -3) |
Hence, the inverse of matrix A is given by;
A^-1 = |5/139 -189/139 29/139 |
|-10/139 129/139 -27/139 |
|-5/139 19/139 -3/139 |
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Can someone answer all of them? Tysm!
In a river bank
you can put the answers from the end
1.) 1/12
2.) 2/13
3.) 8/41
4.) 1/4
5.) 11/74
6.) 2/35
7.) 15/58
8.) 5/18
9.) 1/7
10.) 1/13
2. Translate the sentence to function notation:
The cost of shipping depends on the distance shipped.
Answer:
see the explanation below
Step-by-step explanation:
The expression for the cost can be given as
y=mx+c
where x= the distance
m= the cost to move x distance
c= the fixed cost
Therefore the total cost to ship the cost will be y
What does the best fit line estimate for the y value when x is 100
The best fit line estimate for the y value when x is 100 in this case is 205.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
To determine what the best fit line estimate for the y value is when x is 100.
Assuming that you have a linear regression model with the equation y = mx + b, where "m" is the slope and "b" is the y-intercept, you would need to know the values of "m" and "b" to estimate the y value for a given x value.
If you have these values, you can substitute x = 100 into the equation and solve for y.
The resulting value will be the estimated y value for the given x value.
If the equation of the best fit line is y = 2x + 5, then the estimated y value when x is 100 would be:
y = 2(100) + 5
y = 200 + 5
y = 205
Hence, the best fit line estimate for the y value when x is 100 in this case is 205.
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what is 1/12 in simplest form
It cant be written any way else its already in its simplest form