Suppose that ƒ is a function given as f(x) = 4x² + 5x + 3.
Simplify the expression f(x + h).
f(x + h)
Simplify the difference quotient,
ƒ(x + h) − ƒ(x)
h
=
Submit Question
The derivative of the function at x is the limit of the difference quotient as h approaches zero.
f(x+h)-f(x)
f'(x) =lim
h→0
h
ƒ(x + h) − f(x)
h
=

Answer:

64 and 65

Step-by-step explanation:

Let the two lockers be x and (x+1) since they are consecutive

The sum is 129 so,

x + (x+1) = 129

2x + 1 = 129

2x = 128

x = 64

x + 1 = 65

The locker numbers are 64 and 65

Answer:

W = V/(LH)

Step-by-step explanation:

All we are doing is isolating W. Since V=LWH, then dividing both sides by LH will put W by itself on the right-hand side, you have V/(LH) = W as your equation

con un conjunto de deshielo determinante igual a 4, es x = 2.

Para determinar el valor de x en la matriz x 2

5 7

dado que el conjunto de deshielo determinante es igual a 4, necesitamos utilizar la propiedad de que el determinante de una matriz 2x2 se puede calcular utilizando la siguiente formula:

determinante = (a × d) - (b × c)

Donde a, b, c, y d son los elementos de la matriz.

En este caso, tenemos la matriz:

x 2

5 7

Aplicando la formula del determinante, podemos establecer la siguiente ecuacion:

( x × 7 ) - ( 2 × 5 ) = 4

Simplificando la ecuacion, obtenemos:

7x - 10 = 4

A continuacion, vamos a resolver la ecuacion para encontrar el valor de x:

7x = 4 + 10

7x = 14

Dividiendo ambos lados de la ecuacion por 7, obtenemos:

x = 2

Por lo tanto, el valor de x en la matriz x 2

5 7

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The equations of the asymptotes of the hyperbola are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.

To find the equations of the asymptotes of the hyperbola defined by the equation:

[tex]-25x^2 + 81y^2 + 100x + 1134y + 1844 = 0[/tex]

We can rewrite the equation in the standard form by isolating the x and y terms:

[tex]-25x^2 + 100x + 81y^2 + 1134y + 1844 = 0[/tex]

Rearranging the terms:

[tex]-25x^2 + 100x + 81y^2 + 1134y = -1844[/tex]

Next, let's complete the square for both the x and y terms:

[tex]-25(x^2 - 4x) + 81(y^2 + 14y) = -1844\\-25(x^2 - 4x + 4 - 4) + 81(y^2 + 14y + 49 - 49) = -1844\\-25((x - 2)^2 - 4) + 81((y + 7)^2 - 49) = -1844[/tex]

Expanding and simplify

[tex]-25(x - 2)^2 + 100 - 81(y + 7)^2 + 3969 = -1844\\-25(x - 2)^2 - 81(y + 7)^2 = -1844 - 100 - 3969\\-25(x - 2)^2 - 81(y + 7)^2 = -4913[/tex]

Dividing both sides by -4913:

[tex](x - 2)^2/(-4913/25) - (y + 7)^2/(-4913/81) = 1[/tex]

Comparing this equation to the standard form of a hyperbola:

[tex](x - h)^2/a^2 - (y - k)^2/b^2 = 1[/tex]

We can determine that the center of the hyperbola is (h, k) = (2, -7). The value of [tex]a^2[/tex] is (-4913/25), and the value of [tex]b^2[/tex] is (-4913/81).

The equations of the asymptotes can be found using the formula:

y - k = ±(b/a)(x - h)

Substituting the values we found:

y + 7 = ±(√(-4913/81) / √(-4913/25))(x - 2)

Simplifying:

y + 7 = ±(√(4913) / √(81)) × √(25/4913) × (x - 2)

y + 7 = ±(√(4913) / 9) × √(25/4913) × (x - 2)

Rationalizing the denominators and simplifying:

y + 7 = ±(5/9) ×(x - 2)

Finally, rearranging the equation to isolate y:

y = ±(5/9)x - 10/9 - 7

Simplifying further:

y = ±(5/9)x - 79/9

In light of this, the equations for the hyperbola's asymptotes are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.

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Answer:

[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]

Step-by-step explanation:

First, rewrite the given equation in the standard form of a hyperbola by completing the square.

Given equation:

[tex]-25x^2+81y^2+100x+1134y+1844=0[/tex]

Arrange the equation so all the terms with variables are on the left side and the constant is on the right side:

[tex]-25x^2+100x+81y^2+1134y=-1844[/tex]

Factor out the coefficient of the x² term and the coefficient of the y² term:

[tex]-25(x^2-4x)+81(y^2+14y)=-1844[/tex]

Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:

[tex]-25(x^2-4x+4)+81(y^2+14y+49)=-1844-25(4)+81(49)[/tex]

Factor the two perfect trinomials on the left side and simplify the right side:

[tex]-25(x-2)^2+81(y+7)^2=2025[/tex]

Divide both sides by the number of the right side so the right side equals 1:

[tex]\dfrac{-25(x-2)^2}{2025}+\dfrac{81(y+7)^2}{2025}=\dfrac{2025}{2025}[/tex]

      [tex]\dfrac{-(x-2)^2}{81}+\dfrac{(y+7)^2}{25}=1[/tex]

        [tex]\dfrac{(y+7)^2}{25}-\dfrac{(x-2)^2}{81}=1[/tex]

As the y²-term is positive, the hyperbola is vertical (opening up and down).

The standard equation of a vertical hyperbola is:

[tex]\boxed{\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1}[/tex]

Therefore, comparing this with the rewritten equation:

The formula for the equations of the asymptotes of a vertical hyperbola is:

[tex]\boxed{y=\pm \dfrac{a}{b}(x-h)+k}[/tex]

Substitute the values of h, k, a and b into the formula:

[tex]y=\pm \dfrac{5}{9}(x-2)-7[/tex]

Therefore, the equations for the asymptotes are:

[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]

Answer:

Step-by-step explanation:

step 1:

determining the values for standard form for the equation of a line,

y = mx + c

Step 2:

calculation of m, where m is the gradient or slope which determines how steep the line is.

step 3:

calculation of c, where c is the height at which the line crosses the y - axis also known as y - intercept

The cafeteria can conclude that a majority of the senior high school students like the newly served snack.

To determine if more than 50% of the students like the newly served snack, we need to analyze the responses of the 60 randomly selected students.

Analyzing the responses:

Out of the 60 students surveyed, we have:

- Number of students who responded with "1" (liking the snack): 32 students.

- Number of students who responded with "0" (not liking the snack): 28 students.

To determine the percentage of students who liked the snack, we divide the number of students who liked it by the total number of students surveyed and multiply by 100: (32/60) * 100 = 53.33%.

Since the percentage of students who liked the newly served snack is 53.33%, which is greater than 50%, we can conclude that more than 50% of the students like the snack based on the given survey results.

Therefore, the cafeteria can conclude that a majority of the senior high school students like the newly served snack.

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the center of the given ellipse is (-1, 1).Hence, the required answer is (-1, 1).

The given equation of the ellipse is

100pts9x²+4y²+18x - 8y-23=0.

To find the center of the ellipse, Rearrange the given equation of the ellipse to standard form by completing the square. To complete the square for x terms, we need to add

(18/2)²=9²=81

to both sides of the equation. To complete the square for y terms, we need to add

(-8/2)²=4²=16

to both sides of the equation.

100pts9x²+18x+4y²-8y=23+81+16-100pts100pts(9x²+18x+81) + 100pts(4y²-8y+16) = 120100pts(3x+3)² + 100pts(2y-2)² = 120 + 100pts100pts3(x+1)² + 100pts2(y-1)² = 180

The standard form of the given equation of the ellipse is:

100pts(3(x+1)²)/180 + (2(y-1)²)/180 = 1

Divide throughout by

180:100pts(3(x+1)²)/180 + (2(y-1)²)/180 = 1 Simplify:100pts(3(x+1)²)/36 + (2(y-1)²)/90 = 1

The center of the ellipse is (-1, 1) (h, k), where h is the x-coordinate of the center and k is the y-coordinate of the center.

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Answer:

Center = (-1, 1)

Step-by-step explanation:

To find the center of the ellipse, we first need to write the equation in its standard form by completing the square.

Given equation:

[tex]9x^2+4y^2+18x-8y-23=0[/tex]

Arrange the equation so that all the terms with variables are on the left side and the constant is on the right side.

[tex]9x^2+18x+4y^2-8y=23[/tex]

Factor out the coefficient of the x² term and the coefficient of the y² term:

[tex]9(x^2+2x)+4(y^2-2y)=23[/tex]

Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:

[tex]9(x^2+2x+1)+4(y^2-2y+1)=23+9(1)+4(1)[/tex]

Factor the two perfect trinomials on the left side and simplify the right side:

[tex]9(x+1)^2+4(y-1)^2=36[/tex]

Divide both sides by 36 so the right side equals 1:

[tex]\dfrac{9(x+1)^2}{36}+\dfrac{4(y-1)^2}{36}=\dfrac{36}{36}[/tex]

[tex]\dfrac{(x+1)^2}{4}+\dfrac{(y-1)^2}{9}=1[/tex]

The standard form of the equation of an ellipse with center (h, k) is:

[tex]\boxed{\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1}[/tex]

Comparing the rewritten equation with the standard equation, we can determine that h = -1 and k = 1.

Therefore, the center (h, k) of the ellipse is (-1, 1).

a. There are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.

b. There are 14,950 ways to form a 4-person committee from a class of 26 members.

a. To select a president, a vice president, and a secretary from a class of 26 members, we can use the concept of permutations.

For the president position, we have 26 choices. After selecting the president, we have 25 choices remaining for the vice president position. Finally, for the secretary position, we have 24 choices left.

The total number of ways to select a president, a vice president, and a secretary is obtained by multiplying the number of choices for each position:

Number of ways = 26 * 25 * 24 = 15,600

Therefore, there are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.

b. To form a 4-person committee from a class of 26 members, we can use the concept of combinations.

The number of ways to choose a committee of 4 people can be calculated using the formula for combinations:

Number of ways = C(n, r) = n! / (r!(n-r)!)

where n is the total number of members (26 in this case) and r is the number of people in the committee (4 in this case).

Plugging in the values, we have:

Number of ways = C(26, 4) = 26! / (4!(26-4)!)

Calculating this expression, we get:

Number of ways = 26! / (4! * 22!)

Using factorials, we simplify further:

Number of ways = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1) = 14,950

Therefore, there are 14,950 ways to form a 4-person committee from a class of 26 members.

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Answer:

Step-by-step explanation:

To calculate the future value of Francine's 401k account in 10 years, considering an annual contribution of $8,000 and an average annual return of 4%, we can use the formula for the future value of a series of regular payments, also known as an annuity.

The formula for the future value of an annuity is:

FV = P * [(1 + r)^n - 1] / r

Where:

FV is the future value

P is the payment amount

r is the interest rate per period

n is the number of periods

In this case:

P = $8,000 (annual contribution)

r = 4% or 0.04 (annual interest rate)

n = 10 (number of years)

Calculating the future value:

FV = $8,000 * [(1 + 0.04)^10 - 1] / 0.04

FV = $8,000 * (1.04^10 - 1) / 0.04

FV ≈ $8,000 * (1.480244 - 1) / 0.04

FV ≈ $8,000 * 0.480244 / 0.04

FV ≈ $8,000 * 12.0061

FV ≈ $96,048.80

Therefore, Francine will have approximately $96,048.80 in her 401k account in 10 years if the account averages a 4% annual return and she contributes $8,000 each year.

Answer:

53)

[tex]f(x) = \frac{7(x - 4)(x + 6)}{(x + 4)(x + 5)} [/tex]

The minor arc UB in the circle measures 46 degrees,

The external angles theorem states that "the measure of an angle formed by two secant lines, two tangent lines, or a secant line and a tangent line from a point outside the circle is half the difference of the measures of the intercepted arcs.

It is expressed as;

External angle = 1/2 × ( major arc - minor arc )

From the diagram:

External angle I = 48 degrees

Major arc ZC = 142 degrees

Minor arc UB = ?

Plug these values into the above formula and find the minor arc UB:

External angle = 1/2 × ( major arc - minor arc )

48 = 1/2 × ( 142 - minor arc )

Multiply both sides by 2:

2 × 48 = 2 × 1/2 × ( 142 - minor arc )

2 × 48 = ( 142 - minor arc )

96 = ( 142 - minor arc )

96 = 142 - Minor arc

Minor arc = 142 - 96

Minor arc = 46°

Therefore, the minor arc measures 46 degrees.

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Answer:

Step-by-step explanation:

Answer:

A) [tex]\displaystyle y=\left \{ {{x^3-4,\,x\leq 1} \atop {x^2-3,\,x > 1}} \right.[/tex]

Step-by-step explanation:

The first "piece" of the piecewise function, [tex]y=x^3-4[/tex], contains [tex]x=1[/tex] because of the closed dot there.

The second "piece" of the piecewise function, [tex]y=x^2-3[/tex], doesn't contain [tex]x=1[/tex] because of the open dot there.

What occurs between the two pieces is called a jump discontinuity.

Therefore, A is the correct answer.

The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.

To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):

(3√2 / 3√6) * (-√6 / -√6)

This simplifies to:

-3√12 / (-3√36)

Further simplifying, we have:

-3√12 / (-3 * 6)

-3√12 / -18

Finally, we can cancel out the common factor of 3:

- 3√9 / - 6.

Simplifying further, we get:

3√9 / 6.

Option C is the correct answer.

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Step-by-step explanation:

a = k/m    or       ma = k    

using 4 and 9     4* 9 = k = 36

then the equation becomes:

    ma = 36    

  using a = 6

      6 * m = 36     shows m = 6 kg

The measure of the intercepted arc SU in the circle is 112 degrees.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

From the diagram:

Inscribed angle = 56 degrees

Intercepted arc SU= ?

Plug the given value into the above formula and solve for the intercepted arc.

Inscribed angle = 1/2 × intercepted arc

56 = 1/2 × arc SU

Multiply both sides by 2:

56 × 2 = 1/2 × 2 × arc SU

112 = arc SU

Arc SU = 112°

Therefore, the intercepted arc measure 112 degrees.

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Answer:

-33x√2

Step-by-step explanation:

[tex]-5x\sqrt{98}+2\sqrt{2x^2}\\\\= -5x\sqrt{2*7^{2} } + 2(x\sqrt{2} )\\\\= -5x(7)(\sqrt{2} ) + 2x\sqrt{2} \\\\= -35x\sqrt{2} +2x\sqrt{2} \\\\= (-35+2)x\sqrt{2}\\ \\=-33x\sqrt{2}[/tex]

Step-by-step explanation:

Y = 2 sin 3x            '2' is the amplitude

                      ( 'sin x' usually has amplitude of '1'...then you multiply it by '2' )

                                '3' changes the period

Answer:

Step-by-step explanation:

he amplitude of the sinusoid graph y=2sin3x is 2.

The solution to the polynomial division in quotient and remainder form is: (x + 2) + (-5x + 4)/(x² - 2x + 1)

The polynomials we want to divide are:

x³ - 8x + 6 by x² - 2x + 1 and as such we can write it as:

                x + 2

x² - 2x + 1|x³ - 8x + 6

             -  x³ - 2x² + x

                     2x² - 9x + 6

                  -  2x² - 4x + 2

                            -5x + 4

Thus, the solution expressed in quotient and remainder form is:

(x + 2) + (-5x + 4)/(x² - 2x + 1)

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The central angles for the categories with the numbers 40, 100, 50, and 50 are 60 degrees, 150 degrees, 75 degrees, and 75 degrees, respectively.

To calculate the central angles for each category based on the given numbers 40, 100, 50, and 50, we need to find the proportion of each value to the total sum of all the values. Let's proceed with the following steps:

Step 1: Calculate the total sum of the given numbers: 40 + 100 + 50 + 50 = 240.

Step 2: Find the proportion of each value by dividing it by the total sum and multiplying it by 360 (since a full circle has 360 degrees).

Central angle for the first category: (40/240) * 360 = 60 degrees.

Central angle for the second category: (100/240) * 360 = 150 degrees.

Central angle for the third category: (50/240) * 360 = 75 degrees.

Central angle for the fourth category: (50/240) * 360 = 75 degrees.

The central angles for each category based on the given numbers are 60 degrees, 150 degrees, 75 degrees, and 75 degrees, respectively.

These central angles represent the relative proportions of each category's spending in relation to the total spending. They can be used to create a pie chart or visualize the distribution of expenses in a circular graph.

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The surface area of the triangular pyramid is approximately 331.93 square yards.

To find the surface area of the triangular pyramid, we need to calculate the areas of its individual components and then sum them up.

The triangular pyramid has a base shaped like an equilateral triangle. The legs of the equilateral triangle are all 11 yards long, and the height of the equilateral triangle is 9.5 yards. The formula to calculate the area of an equilateral triangle is:

Area = (√3/4) * [tex]side^2[/tex]

Plugging in the values, we get:

Area of the base equilateral triangle = (√3/4) * 11^2 ≈ 52.43 square yards

The triangular pyramid also has three triangular faces. Each face is an isosceles triangle, with two sides measuring 11 yards (same as the sides of the base equilateral triangle) and a slant height of 17 yards. We can use the formula for the area of an isosceles triangle:

Area = (1/2) * base * height

Since the base of the isosceles triangle is 11 yards and the height is 17 yards, the area of each triangular face is:

Area of each triangular face = (1/2) * 11 * 17 = 93.5 square yards

Now, we can calculate the total surface area of the triangular pyramid by summing up the areas of the base and the three triangular faces:

Surface area = Area of the base equilateral triangle + 3 * Area of each triangular face

Surface area = 52.43 + 3 * 93.5

Surface area ≈ 331.93 square yards

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Lim as x approaches 0: (sin3x)/(2x-Sinx)

Taking the derivative of the numerator and denominator separately:

Numerator: d/dx(sin3x) = 3cos3x

Denominator: d/dx(2x - sinx) = 2 - cosx

Now, evaluate the limit using L'Hôpital's Rule:

Lim as x approaches 0: (3cos3x)/(2 - cosx)

Plugging in x = 0:

Lim as x approaches 0: (3cos(0))/(2 - cos(0))

= 3/2

Therefore, the limit as x approaches 0 of (sin3x)/(2x-Sinx) is 3/2.

Lim as x approaches infinity: x^-1 lnx

Taking the derivative of the numerator and denominator separately:

Numerator: d/dx(x^-1 lnx) = (1/x)lnx

Denominator: d/dx(1) = 0

Since the denominator is 0, we cannot apply L'Hôpital's Rule. However, we can still evaluate the limit:

Lim as x approaches infinity: x^-1 lnx

As x approaches infinity, the natural logarithm (lnx) grows without bound, so the overall limit is 0.

Therefore, the limit as x approaches infinity of x^-1 lnx is 0.

Lim x approaches infinity: x/ e^x

Taking the derivative of the numerator and denominator separately:

Numerator: d/dx(x) = 1

Denominator: d/dx(e^x) = e^x

Now, evaluate the limit using L'Hôpital's Rule:

Lim as x approaches infinity: 1/ e^x

As x approaches infinity, the exponential function e^x grows without bound, so the overall limit is 0.

Therefore, the limit as x approaches infinity of x/ e^x is 0.

Answer:

The overall average nickel content of the rock samples is approximately 7.97%.

Step-by-step explanation:

To find the overall average nickel content of the rock samples, we need to take into account the number of samples from each location. Since we know the average nickel content of each set of samples, we can use a weighted average formula:

overall average nickel content = (total nickel content from first location + total nickel content from second location) / (total weight of samples from both locations)

To calculate the total nickel content from each location, we need to multiply the average nickel content by the number of samples:

total nickel content from first location = 8.43% x 1 sample = 8.43%

total nickel content from second location = 7.81% x 6 samples = 46.86%

To calculate the total weight of the samples from both locations, we need to add up the number of samples:

total weight of samples from both locations = 1 + 6 = 7

Now we can substitute these values into the formula and calculate the overall average nickel content:

overall average nickel content = (8.43% + 46.86%) / 7 ≈ 7.97%

Therefore, the overall average nickel content of the rock samples is approximately 7.97%.

The function for which f(x) is equal to f⁻¹(x) is: C. [tex]f(x) = \frac{x+1}{x-1}[/tex]

In this exercise, you are required to determine the inverse of the function f(x) with an equivalent inverse function f⁻¹(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;

[tex]f(x) = y = \frac{x+6}{x-6} \\\\x=\frac{y+6}{y-6}[/tex]

x(y - 6) = y + 6

y = xy - 6x - 6

f⁻¹(x) = (-6x - 6)/(x - 1) ⇒ Not equal.

Option B.

[tex]f(x) = y = \frac{x+2}{x-2} \\\\x=\frac{y+2}{y-2}[/tex]

x(y - 2) = y + 2

y = xy - 2x - 2

f⁻¹(x) = (-2x - 2)/(x - 1) ⇒ Not equal.

Option C.

[tex]f(x) = y = \frac{x+1}{x-1} \\\\x=\frac{y+1}{y-1}[/tex]

x(y - 1) = y + 1

y - xy = x + 1

f⁻¹(x) = (x + 1)/(x - 1) ⇒ equal.

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Answer:

  f(3) = 1.432

Step-by-step explanation:

You want to know the third term of the sequence defined by ...

The terms of the sequence can be found one at a time by evaluating the recursive relation. The attached calculator output shows the first three terms are ...

  f(1) = 0.2 . . . . . . . given

  f(2) = 0.4(0.2) +1 = 1.08

  f(3) = 0.4(1.08) +1 = 1.432

The third term of the sequence is 1.432.

__

Additional comment

The explicit form of the function is ...

  f(n) = 5/3 -11/3(2/5)^n

Terms will asymptotically approach a value of 5/3.

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1. 1/3 of a full rotation: The coordinates of point P after rotating 1/3 of a full rotation counterclockwise are approximately (0.5, 0.866).

2. 1/2 of a full rotation: The coordinates of point P after rotating 1/2 of a full rotation counterclockwise are (-1, 0).

3. 2/3 of a full rotation: The coordinates of point P after rotating 2/3 of a full rotation counterclockwise are approximately (-0.5, -0.866).

1/3 of a full rotation:

To find the coordinates of point P after rotating 1/3 of a full rotation counter clockwise, we need to determine the angle of rotation.

A full rotation around the unit circle is 360 degrees or 2π radians.

Since 1/3 of a full rotation is (1/3) [tex]\times[/tex] 360 degrees or (1/3) [tex]\times[/tex] 2π radians, we have:

Angle of rotation = (1/3) [tex]\times[/tex] 2π radians

Now, let's use the properties of the unit circle to find the new coordinates.

At the initial position, point P is located at (1, 0).

Rotating counterclockwise by an angle of (1/3) [tex]\times[/tex] 2π radians, we move along the circumference of the unit circle.

The new coordinates of point P after the rotation will be (cos(angle), sin(angle)).

Substituting the angle of rotation into the cosine and sine functions, we get:

New coordinates of P = (cos((1/3) [tex]\times[/tex] 2π), sin((1/3) [tex]\times[/tex] 2π))

Calculating the values:

cos((1/3) [tex]\times[/tex] 2π) ≈ 0.5

sin((1/3) [tex]\times[/tex] 2π) ≈ 0.866

Therefore, the coordinates of point P after rotating 1/3 of a full rotation counterclockwise are approximately (0.5, 0.866).

1/2 of a full rotation:

Following a similar process, when rotating 1/2 of a full rotation counterclockwise, we have an angle of (1/2) [tex]\times[/tex] 2π radians.

New coordinates of P = (cos((1/2) [tex]\times[/tex] 2π), sin((1/2) [tex]\times[/tex] 2π))

Calculating the values:

cos((1/2) [tex]\times[/tex] 2π) = cos(π) = -1

sin((1/2) [tex]\times[/tex] 2π) = sin(π) = 0

Therefore, the coordinates of point P after rotating 1/2 of a full rotation counterclockwise are (-1, 0).

2/3 of a full rotation:

For a rotation of 2/3 of a full rotation counterclockwise, the angle is (2/3) [tex]\times[/tex] 2π radians.

New coordinates of P = (cos((2/3) [tex]\times[/tex] 2π), sin((2/3) [tex]\times[/tex] 2π))

Calculating the values:

cos((2/3) [tex]\times[/tex] 2π) ≈ -0.5

sin((2/3) [tex]\times[/tex] 2π) ≈ -0.866

Therefore, the coordinates of point P after rotating 2/3 of a full rotation counterclockwise are approximately (-0.5, -0.866).

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