Find the slope m of the tangent to the curve y = 8 + 4x2 − 2x3 at the point where x =
a.

Final answer:

A postulate becomes a theorem through a process of logical deduction and proof. Starting from basic assumptions or postulates, mathematicians derive theorems using step-by-step logical inferences, forming the basis for further deductions, such as with Newton's second law or the Schrödinger equation in physics.

Explanation:

A postulate is a statement assumed to be true without proof and serves as a starting point for deducing and inferring other truths. The transformation from a postulate into a theorem involves a process of logical deduction and rigorous proof. When we establish a theorem, what we're doing is showing that the statement necessarily follows from the previously accepted postulates and theorems by a sequence of logical deductions. For example, in geometry, one might postulate that a straight line can be drawn between any two points, and from there, using a series of logical steps, derive the Pythagorean theorem. The Pythagorean theorem's validity is then supported by the fact that it can be derived from these fundamental postulates through a sequence of unassailable logical steps within a proof.

To illustrate, a textbook might first introduce a straightforward theorem like: "If we start out with four lines that form a rectangle, we can draw a new, fifth line that divides the rectangle into two identical triangles." From this, using the fifth line as a postulate, more complicated theorems can be constructed. Similarly, in physics, Newton's second law (f = ma) and the Schrödinger equation in quantum mechanics begin as postulates that explain a wide range of observations. Once propositions like these are established, they can be used to deduce further principles that build upon them.

Answer:

what's the answer

Step-by-step explanation:

Answer:

d. 96

Step-by-step explanation:

on usa test prep

Answer:

[tex]-2\sqrt{1+\cos(x)}+C[/tex]

Step-by-step explanation:

[tex]\boxed{\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))}[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{\sin(x)}{\sqrt{1+\cos(x)}}\;\text{d}x[/tex]

To evaluate the given integral, we can use the method of substitution.

Integration by substitution is a way of integrating a function of a function by simplifying the integral. To integrate by substitution, write part of the function in terms of u, where u is some function of x.

Use the substitution u = 1 + cos(x).

Start by differentiating u with respect to x:

[tex]\begin{aligned}u&=1+\cos(x)\\\\\dfrac{\text{d}u}{\text{d}x}&=0-\sin(x)\\\\\dfrac{\text{d}u}{\text{d}x}&=-\sin(x)\end{aligned}[/tex]

Now rearrange the equation for du/dx to get dx on its own:

[tex]\text{d}u=-\sin(x)\:\text{d}x[/tex]

[tex]\text{d}x=-\dfrac{1}{\sin(x)}\:\text{d}u[/tex]

Substitute everything into the original expression:

[tex]\begin{aligned}\displaystyle \int \dfrac{\sin(x)}{\sqrt{1+\cos(x)}}\;\text{d}x&=\int \dfrac{\sin(x)}{\sqrt{u}}\cdot -\dfrac{1}{\sin(x)}\:\text{d}u\\\\ &=\int -\dfrac{1}{\sqrt{u}}\:\text{d}u\end{aligned}[/tex]

Simplify the integral and evaluate:

                              [tex]\begin{aligned}&=\displaystyle \int -u^{-\frac{1}{2}}\:\text{d}u\\\\&=\dfrac{-u^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+C\\\\&=\dfrac{-u^{\frac{1}{2}}}{\frac{1}{2}}+C\\\\&=-2\sqrt{u}+C\end{aligned}[/tex]

[tex]\textsf{Finally, substitute $u=1+\cos(x)$ back in:}[/tex]

                              [tex]=-2\sqrt{1+\cos(x)}+C[/tex]

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For this case we have the following expression:

[tex](-4) (- 4)[/tex]

Using law of exponents we have:

[tex](a ^ m) (a ^ n) = a ^ {m + n}[/tex]

We use the definition to rewrite the given expression

Rewriting we have:

[tex](-4) (- 4) = (- 4)^{ 1 + 1}\\(-4) (- 4) = (- 4) ^ 2[/tex]

Answer:

The rewritten expression is given by:

[tex](-4) (- 4) = (- 4) ^ 2[/tex]

Answer:

b: 2x + 1 = 73

Step-by-step explanation:

Let x be the smaller integer,

Then the larger integer consecutive to x is x + 1,

Sum of these integers = x + x + 1 = 2x + 1

According to the question,

The sum is 73

⇒ 2x + 1 = 73

Which is the required equation that could be used to solve the problem.

Option 'b' is correct.

The number that can be added to √7 to get a rational number is -√7.

A number of the form a/b where a and b are integers and b ≠ 0 is called a rational number.

The given number is √7.

√7 is an irrational number.

Note that the square root of prime numbers is irrational, therefore, to get a rational number by addition, the only way is to add the negative reciprocal of the given irrational number.

The negative reciprocal of √7 is -√7, therefore, add -√7 to √7:

√7 + (-√7) = 0 (a rational number).

Hence, the number that can be added to √7 to get a rational number is -√7.

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Points A, B and Q are coplanar.

Points or lines are said to be coplanar if they lie in the same plane. Example 1: The points P , Q , and R lie in the same plane A . They are coplanar.

A line which is in the same plane as another line. Any two intersecting lines must lie in the same plane, and therefore be coplanar.

Three or more points that lie on the same line are collinear points . Example : The points A , B and C lie on the line m.

From the figure we can observe

Points A, B and Q are not in the same line.

Three points are collinear only if they are on the same line

3 points are collinear if we can draw a line that connects the points.

In the image A, Q are in a same line but B is not in the line of A and Q.

Points A, B and Q are on the base, So the three points are coplanar but not collinear.

So, Points A, B and Q are coplanar.

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Answer:

D) Amplitude;2 ; midline: y= 2

Step-by-step explanation:

The total height of the curve = 4

Amplitude = 4/2 = 2

The middle line is y = 2

Therefore, answer is D) Amplitude;2 ; midline: y= 2

Hope this will helpful.

Thank you.

Answer:

x = 43.25

Step-by-step explanation:

In the given figure :

In ΔDEF ,

DF = FE

⇒ ∠DEF = ∠FDE = x ( Angles opposite to equal sides are equal)

Now, using angles sum property of a triangle in ΔDEF

∠DEF + ∠FDE + ∠DFE = 180

⇒ ∠FDE + ∠FDE + 93.5 = 180

⇒ x + x = 180 - 93.5

⇒ 2x = 86.5

⇒ x = 43.25

Answer: D. 43.25

Step-by-step explanation:

I have just taken the test

Answer:

10 or answer B.

Step-by-step explanation:

Commission is calculated by multiplying sales revenue by the commission rate; this incentivizes sales, aiming to balance salary, commission, and overall business objectives like profitability and customer satisfaction.

The formula for calculating commission is usually based on the sales revenue multiplied by the commission rate. If, for example, a sales employee sells $10,000 worth of products and the commission rate is 5%, the calculation would be $10,000 x 0.05 = $500.

Commissions serve as a motivational tool in sales. They provide sales employees with an incentive to increase sales, as their earnings are directly linked to the success of their efforts. Optimal sales volume, profitability, and customer satisfaction are central to a well-balanced compensation plan that incorporates both a base salary and a commission structure.

Answer:

[tex]10x^2-25x-35[/tex]

Step-by-step explanation:

The FOIL method is used to multiply two binomials.

FOIL stand for:

First : Multiply the first term in each set of parenthesis.

Outer : Multiply the outer term in each set of parenthesis.

Inner : Multiply the inner term in each set of parenthesis.

Last: Multiply the last term in each set of parenthesis.

Given that:

[tex](2x-7)(5x+5)[/tex]

Using the foil method:

[tex]2x(5x+5)-7(5x+5)[/tex]     [Using distributive property]

⇒[tex]10x^2+10x -(35x+35)[/tex]

⇒[tex]10x^2+10x -35x-35[/tex]

Combine like terms:

⇒[tex]10x^2-25x-35[/tex]

Therefore, the product of (2x – 7)(5x+5) using FOIL method is, [tex]10x^2-25x-35[/tex]

Answer:

Option 2 - log base 2 of 6 over log base 2 of 3

Step-by-step explanation:

Given : Expression [tex]\log_3 6[/tex]

To find : Write the expression as a  logarithm of base 2?

Solution :

Expression [tex]\log_3 6[/tex]

Applying the change-of-base formula,

[tex]\log_a(x)= \frac{\log_b(x)}{\log_b(a)}[/tex]

On comparing, a = 3, x = 6, b = 2

Substitute in the formula,

[tex]\log_3 (6)= \frac{\log_2(6)}{\log_2(3)}[/tex]

Therefore, Option 2 is correct.

Option 2 - log base 2 of 6 over log base 2 of 3

Answer: The correct option is C. The domain of  [tex](b\circ a)(x)[/tex]  is  [tex][0,\infty)[/tex].

Explanation:

It is given that,

[tex]a(x)=3x+1[/tex]

[tex]b(x)=\sqrt{x-4}[/tex]

First we have to find the composite function [tex](b\circ a)(x)[/tex].

[tex](b\circ a)(x)=b(a(x))[/tex]

[tex](b\circ a)(x)=b(3x+1)[/tex]

[tex](b\circ a)(x)=\sqrt{3x+1-4}[/tex]

[tex](b\circ a)(x)=\sqrt{3x-3}[/tex]

We know that a root function is defined for only positive values therefore the composite function is defined if,

[tex]3x-3\geq 0[/tex]

[tex]3x\geq 3[/tex]

[tex]x\geq 1[/tex]

The function is defined for all the values of x which are greater than or equal to 1. Therefore the domain of  [tex](b\circ a)(x)[/tex]  is  [tex][0,\infty)[/tex] and option C is correct.

The numbers in order from least to greatest are:

To order the numbers from least to greatest, we need to convert them all to decimals.

25/7 = 3.57142857...

√12 = 3.4641016151377544

49 V4 = 49/4 = 12.25

Therefore, the numbers in order from least to greatest are:

√12,

3.6,

25/7,

49 √4

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To write the height h of the rectangle as a function of x, you need to understand the relationship between the height and the other variables involved.

To write the height h of the rectangle as a function of x, you need to understand the relationship between the height and the other variables involved. In a rectangle, the height is typically perpendicular to the base.

So, if we assume that x represents the base length, then the height h can be written as a function of x using an equation.

For example, if the rectangle has a fixed width, the height can be calculated using the following equation: h = f(x), where f is the function that describes the relationship between the base length x and the height h.

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The equation (a) y = 3/4x, equation (b) y = 0.6x, equation (d) y = 2/3x are correct option because they have greater rate than A

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

We have;

Relationship B has a greater rate than Relationship A.

From the graph, the rate of A or slope will be:

A(m) = (4-2)/(8-4) = 2/4

A(m) =1/2 = 0.5

From the given option, the equation  y = 3/4x, y = 0.6x, and y = 2/3x has a greater rate than A which is 3/4, 0.6, and 2/3 respectively.

Thus, the equation (a) y = 3/4x, equation (b) y = 0.6x, equation (d) y = 2/3x are correct option because they have greater rate than A

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Using the distance formula, we can calculate the distances between the vertices of quadrilateral JKLM. After performing these calculations, we can determine that not all sides have the same length, thus quadrilateral JKLM is not a rhombus.

To determine if quadrilateral JKLM is a rhombus, we need to check if all four sides are of equal length because a defining property of a rhombus is that it has all sides of equal length. The length between two points (x1, y1) and (x2, y2) can be found using the distance formula which is √[(x2-x1)² + (y2-y1)²].

After calculating, we find:
J-K distance = √[(5-2)² + (-3-3)²] = √42
K-L distance = √[(-1-5)² + (9-3)²] = √52
L-M distance = √[(-3- -1)² + (2-9)²] = √52
M-J distance = √[(2- -3)² + (-3-2)²] = √42

Therefore, the distance between vertices J-K and L-M are the same, and the distance between vertices K-L and M-J are the same, but the two pairs are not in themselves equal to each other. As such, quadrilateral JKLM is not a rhombus because all sides do not have the same length.

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