Prove that if function, f is differentiable at x,, then f is continuous at Xo.

\(\large\huge\green{\sf{Answer:-}}\)

\( \red {\mathbb{ \underline { \tt By \: using \: Pythagoras \: theorm}}}\)

\(h {}^{2} = p {}^{2} + b {}^{2} \\ x {}^{2} = (7 {}^{2} + 6 {}^{2} )ft {}^{2} \\ x {}^{2} = (49 +3 6)ft {}^{2} \\ x {}^{2} = 85 ft\\ x = \sqrt{85 } ft \\ \)

Answer:

\(\huge\boxed{\sf H = 9.2\ ft}\)

Step-by-step explanation:

Since the triangle is a right-angled triangle, we can apply Pythagoras Theorem on this triangle.

Pythagoras Theorem:

\((Hypotenuse)^2= (Base)^2+(Perpendicular)^2\)

Given:

Base = 6 ft

Perpendicular = 7 ft

Required:

Hypotenuse = x ft

Solution:

Put the givens in the above formula of the Pythagoras Theorem:

(x)² = (6)² + (7)²

(x)² = 36 + 49

(x)² = 85

Taking sqrt on both sides

x = 9.2 ft

\(\rule[225]{225}{2}\)

Hope this helped!

\(24^{1/3}=\sqrt[3]{24}=\sqrt[3]{8 \cdot 3}=\sqrt[3]{8}\sqrt[3]{3}=\boxed{2\sqrt[3]{3}}\)

Answer:

The radius of the circle is 6.

Explanation:

Take the square root of 36.

The correct answer is OD. 8x + 1.

To divide 8x³ + x² - 32x - 4 by x² - 4, we can use polynomial long division.

The dividend is 8x³ + x² - 32x - 4, and the divisor is x² - 4.

We start by dividing the highest degree term, which is 8x³, by x². This gives us 8x.

Next, we multiply the divisor x² - 4 by the quotient 8x. The result is 8x³ - 32x.

Subtracting 8x³ - 32x from the dividend, we get x² - 32x.

Now, we divide x² - 32x by x² - 4. This gives us 1.

Multiplying the divisor x² - 4 by the quotient 1, we get x² - 4.

Subtracting x² - 4 from the remaining dividend, which is -32x, we get -32x + 4.

Since we can no longer divide, the final result is the quotient we obtained: 8x + 1.

Therefore, the correct answer is:

OD. 8x + 1.

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Answer:

-4 I think

Step-by-step explanation:

In school suspension

The transformation of the function Y = X^4 / (6X + 6) can be described as follows:

The function has a domain of all real numbers except X = -1.

The range of the function is all non-negative real numbers.

The function is not symmetric about the y-axis.

It has a Y-intercept at (0, 0).

It does not have any X-intercepts.

The function approaches 0 as X approaches positive or negative infinity.

To describe the transformation of the function Y = X^4 / (6X + 6), we can analyze its characteristics, such as the domain, range, symmetry, intercepts, and behavior as X approaches positive and negative infinity.

1. Domain:

The function is defined for all real values of X except when the denominator 6X + 6 equals zero. Solving for X gives X = -1. Therefore, the domain of the function is all real numbers except X = -1.

2. Range:

Since the numerator is X^4, which is always non-negative, and the denominator 6X + 6 is always positive (except at X = -1), the range of the function is all non-negative real numbers (including zero).

3. Symmetry:

To determine if the function has any symmetry, we check if replacing X with -X results in an equivalent function. Substituting -X for X in the original function gives Y = (-X)^4 / (6(-X) + 6), which simplifies to Y = X^4 / (-6X + 6). Since the function does not remain unchanged under this substitution, it is not symmetric about the y-axis.

4. Intercepts:

a. Y-intercept: To find the Y-intercept, we set X = 0. Plugging in X = 0 into the function gives Y = 0^4 / (6(0) + 6) = 0 / 6 = 0. Therefore, the Y-intercept is at the point (0, 0).

b. X-intercept: To find the X-intercept, we set Y = 0 and solve for X. However, since the numerator X^4 is always non-negative, the function does not have any X-intercepts.

5. Behavior as X approaches infinity:

As X becomes large positive or negative, the 6X term dominates the 6X + 6 denominator. Therefore, the function approaches 0 as X approaches positive or negative infinity.

In summary, the transformation of the function \(Y = X^4 / (6X + 6)\)can be described as follows:

- The function has a domain of all real numbers except X = -1.

- The range of the function is all non-negative real numbers.

- The function is not symmetric about the y-axis.

- It has a Y-intercept at (0, 0).

- It does not have any X-intercepts.

- The function approaches 0 as X approaches positive or negative infinity.

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Answer: 18.4 inches

Step-by-step explanation:

Using operations with functions, we have that:

a) w(t) = -3t + 5.

b) The pool will leak all the water in 1.67 hours.

c) 1.67 hours.

d) Functions f(t) and d(t) intersect on the graph when all the water of the pool has been leaked.

e) The domain of the function is 0 ≤ t ≤ 1.67.

The amount of water in the pool is determined by the subtraction of the amount flowing into the pool by the amount being drained out the pool.

For this problem, these following functions are given:

Hence the amount of water in the pool is given by:

w(t) = f(t) - d(t)

w(t) = t² + 8t + 9 - t² - 11t - 4

w(t) = -3t + 5.

Hence:

The function is: w(t) = -3t + 5.

The pool will have leaked all the water when:

w(t) = 0.

Hence:

-3t + 5 = 0.

3t = 5

t = 1.67.

The pool will leak all the water in 1.67 hours.

Equaling f(t) and d(t), we have that:

t² + 8t + 9 = t² + 11t + 4.

3t = 5.

t = 1.67.

Functions f(t) and d(t) intersect on the graph when all the water of the pool has been leaked.

The domain of a function is the set that contains all input values of the function. For this problem, the input is the time, hence:

Hence:

The domain of the function is 0 ≤ t ≤ 1.67.

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Answer:

111 meters, break each area into a rectangle, find the areas, then add.

Answer:

"Rotation" means turning around a center: The distance from the center to any point on the shape stays the same.Every point makes a circle around the centre.

The Figure 2 was constructed using figure 1.Among all the given options those which apply for the transformation to be defined as a rotation are :

A)The segment connecting the center of rotation, C, to a point on the pre-image (figure 1) is equal in length to the segment that connects the center of rotation to its corresponding point on the image (figure 2)

(B)The transformation is rigid.

(C)Every point on figure 1 moves through the same angle of rotation about the center of rotation, C, to create figure 2.

(E)If figure 1 is rotated 360° about point C, it will be mapped onto itself.

Options A,B,C,E are the correct ones.

Answer:

Step-by-step explanation:

did edge test said to pick 3

There are 392 acres of old-growth trees.

What is the total area?

The area is the region bounded by the shape of an object. The space covered by the figure or any two-dimensional geometric shape. The surface area of a solid object is a measure of the total area that the surface of the object occupies.

Here, we have

The total area of the forest is 49,000 acres.

0.8% of 49,000 is (0.008)(49,000) = 392 acres.

Therefore, there are 392 acres of old-growth trees.

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Answer:

ytyt

Step-by-step explanation:

tyt5667

Answer:

let the position vector of A,B,C,D and E be OA,OB,OC,OD andOE respectively.

IN triangle OBD , OE = OD+OB/2

OE=OA+AB+OD/2. (in triangle,OB=OA+AB)

OE=OA+OD+DC/2

OE=OA+OC/2. (in triangle ODC,OD+DC=OC)

proved....

The measure of all the angles is given below:

angle 1= 30

angle 2=  150

angle 3= 30

angle 4= 150

angle 5= 30

angle 6= 150

angle 7= 30

angle 8= 73.5

An angle is formed when two straight lines or rays meet at a common endpoint. The common point of contact is called the vertex of an angle.

As,

angle 1= 30 (Vertically opposite angle)

angle 7= 30 (corresponding angle)

angle 4+30= 180 (linear pair)

 angle 4= 150

angle 4= angle 2= 150 (Vertically opposite angle)

angle 2= angle 6= 150  (corresponding angle)

angle 1= angle 5= 30  (corresponding angle)

angle (2x+3)= angle 4  (corresponding angle)

     2x+3= 150

      x= 73.5

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Answer:

x = 4

Step-by-step explanation:

4x+2=18

subtract 2 to both sides

2 cancels out

4x=16

divide both by 4

4 cancels out

x= 4

Answer:

\(x\) = 4

Stage One: Subtract two from both sides.

4x + 2 - 2 = 18 - 2.

Stage Two: Divide both by 4.

\(\frac{4x}{4}\) = \(\frac{16}{4}\)

x = 4

the vector → u with magnitude 3 in the opposite direction as → v = ⟨ 4 , − 4 ⟩ is ⟨ -3/8 , 3/8 ⟩.

The magnitude of a vector is the length or size of the vector. In this case, we want to find a vector with magnitude 3, so we need to scale the vector → v to have a length of 3. Additionally, we want the resulting vector to be in the opposite direction as → v.

To achieve this, we can calculate the unit vector in the direction of → v by dividing → v by its magnitude:

→ u = → v / |→ v |

→ u = ⟨ 4/√(4^2+(-4)^2) , -4/√(4^2+(-4)^2) ⟩

→ u = ⟨ 4/√32 , -4/√32 ⟩

Next, we can scale → u to have a magnitude of 3 by multiplying it by -3/|→ v |:

→ u = -3/|→ v | * → u

→ u = -3/√32 * ⟨ 4/√32 , -4/√32 ⟩

→ u = ⟨ -34/32 , -3(-4)/32 ⟩

→ u = ⟨ -3/8 , 3/8 ⟩

Therefore, the vector → u with magnitude 3 in the opposite direction as → v = ⟨ 4 , − 4 ⟩ is ⟨ -3/8 , 3/8 ⟩.

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The surface integral ∬F · dS over the tetrahedron surface S is 128/3.

To evaluate the surface integral of the vector field F = yi - (z - y)j + xk over the tetrahedron surface S, we can use the surface integral formula:

∬F · dS = ∭div(F) dV,

where ∬ represents the surface integral, ∭ represents the volume integral, div(F) is the divergence of F, dS is the differential surface area vector, and dV is the differential volume.

To apply this formula, we need to find the divergence of F. The divergence of a vector field F = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is given by:

div(F) = ∂P/∂x + ∂Q/∂y + ∂R/∂z.

In our case, P(x, y, z) = 0, Q(x, y, z) = y - (z - y), and R(x, y, z) = x. Let's calculate the divergence:

∂P/∂x = 0,

∂Q/∂y = 1 - (-1) = 2,

∂R/∂z = 0.

Therefore, div(F) = 0 + 2 + 0 = 2.

Since the divergence of F is constant, we can simplify the surface integral formula:

∬F · dS = ∭div(F) dV = 2 ∭dV.

Now, we need to set up the triple integral over the volume of the tetrahedron bounded by the given vertices. The tetrahedron has three sides lying on the coordinate planes, so we can use the limits of integration:

0 ≤ x ≤ 4,

0 ≤ y ≤ 4 - x,

0 ≤ z ≤ 4 - x.

Let's set up the triple integral and evaluate it:

∬F · dS = 2 ∭dV

= 2 ∫₀⁴ ∫₀⁴(4-x) ∫₀⁴(4-x) dz dy dx.

Integrating the innermost integral:

∫₀⁴(4-x) ∫₀⁴(4-x) dz dy = ∫₀⁴(4-x) (4-x) dy = (4-x)(4-x) = (4-x)².

Now integrating the next integral:

∫₀⁴ (4-x)² dx

= ∫₀⁴ (16 - 8x + x²) dx

= 16x - 4x² + (1/3)x³ ∣₀⁴

= (64 - 64 + 64/3)

= 64/3.

Therefore, the surface integral ∬F · dS over the tetrahedron surface S is equal to 2(64/3) = 128/3.

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To solve this problem, we multiply and divide the given expression by 2-i:

Recall that:

Therefore:

Finally, multiplying the factors in the numerator, we get:

Finally, simplifying the above result, we get:

Answer:

The zero is 2.

Because b > 1, the correct response in this situation would be: "1.022 is the growth factor for the GDP since 1950, and the GDP increases by a factor of 1.022 every year."

In this instance, we're assuming that we can use the following function to simulate the US GDP, or gross domestic product, in dollars:

\(GDP = 11(1.0220)^{t}\)

We can also see that the exponential model formula given by: governs this formula.

\(GDP = 11(1.0220)^{t\)

Where a denotes the starting sum, b the model's growth/decay rate, and t is the number of years since 1950.

The value of b in this instance is provided by:

b = 1.022

And when we calculated r as the growth rate, we obtained:

1.022 = 1 + r

r = 1.022 - 1

r = 0.022

Because b > 1, the correct response in this situation would be: "1.022 is the growth factor for the GDP since 1950, and the GDP increases by a factor of 1.022 every year."

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The answer to this question is c.The use of a correlation coefficient.

The use of a correlation coefficient would tell you that an association claim is being made in a research study. A correlation coefficient is a statistical measure that shows the strength and direction of a relationship between two variables. It ranges from -1 to 1, with 0 indicating no relationship and -1 or 1 indicating a perfect negative or positive relationship, respectively.

A scatterplot or bar graph may be used to visually display the relationship between two variables, but they do not necessarily indicate that an association claim is being made. The measurement of two variables is necessary for any type of research study, but it does not inherently imply that an association claim is being made.

Interrogation of internal validity is a process used to ensure that the study's results accurately reflect the relationship between the variables being studied. It is important for any research study but does not specifically indicate that an association claim is being made.

Therefore,the answer to this question is c.The use of a correlation coefficient.

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Answer:

It is equal to each other

Step-by-step explanation:

\(\sqrt{\frac{1}{x^2}} = \frac{1}{x}\\\sqrt[3]{\frac{1}{x^3}} = \frac{1}{x}\)

Answer:

A translation of 3 units to the right, followed by a vertical stretch by a factor of 2, followed by a translation of 4 units up.

Step-by-step explanation:

Transformations

\(f(x-a) \implies f(x) \: \textsf{translated $a$ units right}.\)

\(\begin{aligned} y =a\:f(x) \implies & \textsf{$f(x)$ stretched/compressed vertically by a factor of $a$}.\\& \textsf{If $a > 1$ it is stretched by a factor of $a$}.\\& \textsf{If $0 < a < 1$ it is compressed by a factor of $a$}.\end{aligned}\)

\(f(x)+a \implies f(x) \: \textsf{translated $a$ units up}\)

Therefore, the series of transformations of:

\(y=x^2 \quad \textsf{to} \quad y=2(x-3)^2+4\quad \textsf{is}:\)

Translated 3 units to the right:

\(f(x-3)\implies y=(x-3)^2\)

Stretched vertically by a factor of 2:

\(2f(x-3)\implies y=2(x-3)^2\)

Translated 4 units up:

\(2f(x-3)+4\implies y=2(x-3)^2+4\)

Therefore, the series of transformations is:

A translation of 3 units to the right, followed by a vertical stretch by a factor of 2, followed by a translation of 4 units up.

Answer:

68

Step-by-step explanation:

4 + (11 - 3)^2

4 + 8^2

4 + 64

= 68

The total present value of these payments at the time the first payment is made is 1,735.85 (747.26 + 988.59).

To calculate the present value of these payments, we need to use the formula for the present value of an annuity:

\(PV = (P/i) x [1 - (1+i)^-n]\)

Where:
P = payment amount
i = annual effective rate
n = number of payments

Using this formula, we can calculate the present value of the first 10 payments:

\(PV = (100/0.07) x [1 - (1+0.07)^-10] = 747.26\)

To calculate the present value of the remaining 10 payments, we need to first calculate the payment amounts. To do

this, we can use the following formula:

\(Pn = P1 x (1 + g)^n\)

Where:
Pn = payment in year n
P1 = first payment amount
g = growth rate
n = number of years since first payment

For the 11th payment:

\(P11 = 105 x (1 + 0.05)^1 = 110.25\)

For the 12th payment:

\(P12 = 110.25 x (1 + 0.05)^1 = 115.76\)

And so on, until the 20th payment:

\(P20 = 163.32 x (1 - 0.05)^8 = 79.24\)

Now we can calculate the present value of these payments:

PV = \((110.25/0.07) x [1 - (1+0.07)^-10] + (115.76/0.07) x [1 - (1+0.07)^-9] + ... + (79.24/0.07) x [1 - (1+0.07)^-1]\)

PV = 988.59

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The velocity of flow in a 300-mm-diameter pipe carrying a flow of 90 L/s can be calculated. The velocity is approximately 0.238 m/s.

To find the velocity of flow in the pipe, we need to use the equation Q = A * V, where Q is the flow rate, A is the cross-sectional area of the pipe, and V is the velocity of flow.

The diameter of the pipe is 300 mm, which is equal to 0.3 meters. From the diameter, we can calculate the radius by dividing it by 2, giving us a radius of 0.15 meters.

Using the formula for the area of a circle, A = π * r^2, we can substitute the values and find the area: A = π * (0.15)^2.

The flow rate Q is given as 90 L/s, which is equivalent to 0.09 m^3/s.

Now we can rearrange the equation Q = A * V to solve for V. Plugging in the known values, we have 0.09 = π * (0.15)^2 * V.

Simplifying the equation, we find V ≈ 0.238 m/s.

Therefore, the velocity of flow in the 300-mm-diameter pipe carrying a flow of 90 L/s is approximately 0.238 m/s.

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Imagine a list of all 5-digit numbers that have distinct digits. For example, 70364 and 93145 are two such numbers on the list, while 80628 is not on the list since it repeats the digit 8. 1. There are 27,216 numbers are listed. 2. There are 15,120 numbers on the list that use the digit 0 at least once.

1. For the first digit, we have 9 choices (1-9), as 0 cannot be the leading digit.

For the second digit, we have 9 choices (0-9, excluding the digit used in the first position).

For the third digit, we have 8 choices (0-9, excluding the digits used in the first and second positions).

For the fourth digit, we have 7 choices (0-9, excluding the digits used in the first, second, and third positions).

For the fifth digit, we have 6 choices (0-9, excluding the digits used in the first, second, third, and fourth positions).

Using the counting principle, the total number of 5-digit numbers with distinct digits is:

9 × 9 × 8 × 7 × 6 = 27,216

Therefore, there are 27,216 numbers listed.

2. To find the number of numbers on the list that use the digit 0 at least once, we can analyse the cases where the digit 0 is present.

Case 1: 0 is in the first position

In this case, we have 1 choice for the first position (0) and then proceed as before, so we have:

1 × 9 × 8 × 7 × 6 = 3,024 possibilities.

Case 2: 0 is in one of the other four positions

In this case, we have 4 choices for the position of 0 (second, third, fourth, or fifth), and the remaining digits can be filled in with the available choices. Therefore, we have:

4 × 9 × 8 × 7 × 6 = 12,096 possibilities.

Adding the possibilities from both cases, we have a total of:

3,024 + 12,096 = 15,120 numbers on the list that use the digit 0 at least once.

Therefore, there are 15,120 numbers on the list that use the digit 0 at least once.

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Answer:

x¹=-6 x²=0

Step-by-step explanation:

Substitute y =0

Swap sides

Factor expression

Split into possible cases

Solve equation