please help me with this problem

Answer:

[tex]\displaystyle \nabla f(\sqrt{2}, 1) = 2\sqrt{2} \hat{\i} - 2 \hat{\j}[/tex]

General Formulas and Concepts:
Calculus

Differentiation

Derivative Rule [Basic Power Rule]:

Multivariable Calculus

Differentiation

Gradient:                                                                                                               [tex]\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}[/tex]

Gradient Property [Addition/Subtraction]:                                                           [tex]\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)[/tex]

Step-by-step explanation:

Step 1: Define

Identify.

[tex]\displaystyle f(x, y) = x^2 - y^2[/tex]

[tex]\displaystyle P(\sqrt{2}, 1)[/tex]

Step 2: Find Gradient

∴ the gradient of the given f(x, y) function is equal to <2√2, -2>.

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Topic: Multivariable Calculus

Unit: Directional Derivatives

Answer:

  D)  infinitely many solutions

Step-by-step explanation:

The equation reduces to ...

  -2x +6 = -2x +6

This is true for all possible values of x, so there are infinitely many solutions.

Answer: [tex]n^{2}[/tex]

Step-by-step explanation:

(n + n) represents the definition of the sum of n and itself.

(2n) means two times n, wich is the same as adding n + n.

[tex]n^{2}[/tex] actually represents the multiplication of n two times: [tex]n*n[/tex]

For example, if n=3:

[tex]n+n=3+3=6[/tex]

[tex]2n=2(3)=6[/tex]

[tex]n^{2}=3^{2} =9[/tex]

The roc of the volume of a cube with respect to the length of its side s is 3s².

What is rate of change?

The rate of change of a function is the rate at which the output values change over the input value.

The rate of change of a function is one degree less than the actual function.

The volume of a cube is :

V = s³

The rate of change of volume V with respect to length a will be :

dV/ds = d(s³ )/ds

          = 3s²

This means that, a cubic function will have a quadratic rate of change

Hence, The roc of the volume of a cube with respect to the length of its side s is 3s².

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Answer:

1. (3, 2)

2. Option C. 12.2

3. Option A. 165 units

Step-by-step explanation:

1. The midpoint of two coordinates (x₁, x₂) and (y₁, y₂) is calculate by,

[tex](x, y) = (\frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2})[/tex]

⇒[tex](x, y) = (\frac{-1 + 7}{2},\frac{3 + 1}{2})[/tex]

Thus (x, y) = (3, 2)

Hence, none of given options are true.

2. The distance between two coordinates is calculate by,

[tex]Distance=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2[/tex]

⇒ Distance = 12.16 ≈ 12.2 unit

Hence, option (C) is correct.

3. The distance between T(80, 20) and V(110, 85) is comparatively smaller than T(80, 20) and U(20, 60).

Using the Distance formula,

[tex]Distance=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2[/tex]

Distance between T(80, 20) and V(110, 85) is 71.59 unit

and Distance between U(20, 60) and V(110, 85) is 93.41 unit

So, Airplane firstly go to point V from point T and then point U.

Total shortest distance = 71.60 + 93.40 = 165 unit.

Hence, option (A) is correct.

Answer:

y=3x

Step-by-step explanation:

For a line to be proportional it must have the form y=mx where the y-intercept is (0,0) through the origin. To change y=3x+7 to be proportional, write it as y=3x.

When x is 4, y is approximately -3.43.

To find y when x is 4, we can use the given information that y varies directly as x. This means that the ratio of y to x remains constant. We can set up a proportion using the initial values of y and x and solve for the unknown value:

6 / -7 = y / 4

By cross-multiplying, we get y = -24 / 7. Therefore, when x is 4, y is approximately -3.43.

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Answer:

The constant rate of change of the account is $50 or Increasing by $50 per month.

Step-by-step explanation:

Consider the provided information.

Joanne is depositing money into a bank account. After 3 months there is $150 in the account. After 6 months there is $300 in the account.

Rate of change is known as how one quantity change in relation to other.

The rate of change can be calculated as:

[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Now use the above formula to calculated the rate of change.

[tex]\frac{300-150}{6-3}[/tex]

[tex]\frac{150}{3}[/tex]

[tex]50[/tex]

Hence, the constant rate of change of the account is $50 or Increasing by $50 per month.

To calculate Mary's car's gas mileage, divide the number of miles driven by the number of gallons of gas used. In this scenario, her car's gas mileage is approximately 19.06 miles per gallon.

To calculate Mary's car's gas mileage, we need to divide the number of miles driven by the number of gallons of gas used. In this scenario, Mary traveled 324 miles and used 17 gallons of gas. Therefore, her car's gas mileage can be calculated as:

Gas mileage = Miles driven / Gallons of gas used

Substituting the values:

Gas mileage = 324 miles / 17 gallons

Simplifying the equation:

Gas mileage = 19.06 miles per gallon (rounded to two decimal places)

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Trigonometric Identities are equalities that utilize trigonometry functions and hold true for all variables in the equation. The correct option is C, -7/25.

Trigonometric Identities are equalities that utilize trigonometry functions and hold true for all variables in the equation. There are several trigonometric identities relating to the side length and angle of a triangle.

Given that the value of tan(x) = -4/3 and x is in quadrant 2.

In order to find the value of cos(2x), we can use the trigonometric identity of cos(2x), therefore, for cos(2x) we can write,

[tex]\cos(2x) = \dfrac{1-\tan^2(x)}{1+\tan^2(x)}[/tex]

Since the value of tan(x) is known, substitute the value in the identity,

[tex]\cos(2x) = \dfrac{1-(-\frac{4}{3})^2}{1+(-\frac{4}{3})^2}\\\\\cos(2x) = \dfrac{1-(\frac{16}{9})}{1+(\frac{16}{9})}\\\\[/tex]

cos(2x) = (-7/9) × (9/25)

Cancelling 9 from the numerator and the denominator,

cos(2x) = -7/25

Hence, if tanx=-4/3 and x are in quadrant 2 then cos2x=-7/25.

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The correct answer is c. [tex]$\cos(2x) = -\frac{7}{25}$[/tex].

First, we find [tex]$\sin(x)$[/tex] and [tex]$\cos(x)$[/tex] using the given value of [tex]$\tan(x)$[/tex]:

[tex]$\tan(x) = \frac{\sin(x)}{\cos(x)} = -\frac{4}{3}$.[/tex]

We can use the Pythagorean identity [tex]$\sin^2(x) + \cos^2(x) = 1$[/tex] to find [tex]$\sin(x)$[/tex] and [tex]$\cos(x)$[/tex]. Let's assume [tex]$\sin(x) = \frac{4}{5}$[/tex] (since [tex]$\tan(x) = -\frac{4}{3}$[/tex], and we are looking for a positive value of [tex]$\sin(x)$[/tex] in quadrant 2). Using the Pythagorean identity:

[tex]$\left(\frac{4}{5}\right)^2 + \cos^2(x) = 1$ \\ $\frac{16}{25} + \cos^2(x) = 1$ \\ $\cos^2(x) = 1 - \frac{16}{25}$ \\ $\cos^2(x) = \frac{25}{25} - \frac{16}{25}$ \\ $\cos^2(x) = \frac{9}{25}$. \\[/tex]

Since [tex]$\cos(x)$[/tex] is negative in quadrant 2, we have [tex]$\cos(x) = -\frac{3}{5}$[/tex].

 Now, we can use the double-angle formula for cosine:

[tex]$\cos(2x) = 2\cos^2(x) - 1$ \\ $\cos(2x) = 2\left(-\frac{3}{5}\right)^2 - 1$ \\ $\cos(2x) = 2\left(\frac{9}{25}\right) - 1$ \\ $\cos(2x) = \frac{18}{25} - 1$ \\ $\cos(2x) = \frac{18}{25} - \frac{25}{25}$ \\ $\cos(2x) = -\frac{7}{25}$.[/tex]

Therefore, the value of [tex]$\cos(2x)$[/tex] = [tex]$ -\frac{7}{25}$[/tex].

Answer: The solution is,

[tex]n = 0\text{ or }n = \frac{1}{10}[/tex]

Step-by-step explanation:

Since, Zero product property states that if the product of two numbers or expression is equal to zero then either of the numbers or expressions must be equal to zero.

That is, If a.b = 0 ⇒ a = 0 or b = 0

Here, the given expression is,

[tex]-8n(10n-1)=0[/tex]

[tex]\implies (-8n)(10n-1)=0[/tex]

Thus, by the above property,

[tex]-8n = 0\text{ or }(10n-1)=0[/tex]

[tex]\implies n = \frac{0}{-8}\text{ or }10n= 1[/tex]

[tex]\implies n = 0\text{ or }n = \frac{1}{10}[/tex]

Answer:

f(x) = don't spend more than 200 dollars on staplers

Step-by-step explanation:

The prime factorization of 37 is 37

We have,

The prime factorization of a number involves expressing it as a product of prime numbers.

However, in the case of prime numbers themselves, their prime factorization is simply the number itself.

In this case,

The number 37 is a prime number because it is only divisible by 1 and itself.

Since it has no other factors, its prime factorization is simply 37.

Thus,

The prime factorization of 37 is 37.

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