MATHEMATICS HIGH SCHOOL

The point P(1,1/2) lies on the curve y=x/(1+x). (a) If Q is the point (x,x/(1+x)), find the slope of the secant line PQ correct to four decimal places for the following values of x: (1) .5 (2) .9 (3) .99 (4) .999 (5) 1.5 (6) 1.1 (7) 1.01 (8) 1.001

Answers

Answer 1

Answer:

Answer:

See explanation

Step-by-step explanation:

You are given the equation of the curve

$y=(x)/(1+x)$

Point $P\left(1,(1)/(2)\right)$ lies on the curve.

Point $Q\left(x,(x)/(1+x)\right)$ is an arbitrary point on the curve.

The slope of the secant line PQ is

$(y_2-y_1)/(x_2-x_1)=((x)/(1+x)-(1)/(2))/(x-1)=((2x-(1+x))/(2(x+1)))/(x-1)=((2x-1-x)/(2(x+1)))/(x-1)=\n \n=((x-1)/(2(x+1)))/(x-1)=(x-1)/(2(x+1))\cdot (1)/(x-1)=(1)/(2(x+1))\ [\text{When}\ x\neq 1]$

1. If x=0.5, then the slope is

$(1)/(2(0.5+1))=(1)/(3)\approx 0.3333$

2. If x=0.9, then the slope is

$(1)/(2(0.9+1))=(1)/(3.8)\approx 0.2632$

3. If x=0.99, then the slope is

$(1)/(2(0.99+1))=(1)/(3.98)\approx 0.2513$

4. If x=0.999, then the slope is

$(1)/(2(0.999+1))=(1)/(3.998)\approx 0.2501$

5. If x=1.5, then the slope is

$(1)/(2(1.5+1))=(1)/(5)\approx 0.2$

6. If x=1.1, then the slope is

$(1)/(2(1.1+1))=(1)/(4.2)\approx 0.2381$

7. If x=1.01, then the slope is

$(1)/(2(1.01+1))=(1)/(4.02)\approx 0.2488$

8. If x=1.001, then the slope is

$(1)/(2(1.001+1))=(1)/(4.002)\approx 0.2499$

Answer 2

Answer:

Final answer:

To find the slope of the secant line PQ, we use the formula (y2 - y1)/(x2 - x1) for each given x-value, plug in the coordinates of P and Q, and solve for the slope.

Explanation:

We need to calculate the slope of the secant line passing through points P(1, 1/2) and Q(x, x/(1+x)) for different values of x. The slope of a secant line is calculated using the formula (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points it passes through. We already have P(1, 1/2), so let's calculate the slope for each given x-value.

For x = 0.5, Q = (0.5, 0.5/(1+0.5)) = (0.5, 1/3).
Slope = (1/3 - 1/2) / (0.5 - 1) = -1/6 / -0.5 = 1/3.
For x = 0.9, perform a similar calculation.
... [Continue for each provided x-value]

After calculating the slope for each x-value, we convert them into a decimal format rounded to four places.

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Answers

Answer:

25.2 feets

Step-by-step explanation:

Given that:

Rectangular shaped living room has the dimension:

21 feets by 14 feets

Length of diagonal :

Using Pythagoras :

A² = B² + C²

Diagonal² = 21² + 14²

Diagonal = √(441 + 196)

Diagonal = √637

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f (x), has (x-7) and (x+4i) as factors. F(X)=2x^5-13x^4+22x^3-187x^2-160x+336 what is the total number of real numbers of real zeros of f(X)

Answers

Answer: 3

Step-by-step explanation:

(x + 4i) is a root which means its conjugate (x - 4i) is also a root

(x + 4i)(x - 4i) = x² + 16

Use Long Division for 2x⁵ - 13x⁴ + 22x³ - 187x² - 160x + 336 ÷ x² + 16

2x³ - 13x² - 10x + 21

x² + 16 ) 2x⁵ - 13x⁴ + 22x³ - 187x² - 160x + 336

- (2x⁵ + 0x⁴ + 32x³) ↓ ↓ ↓

-13x⁴ - 10x³ - 187x² ↓ ↓

- (-13x⁴ +0x³ - 208x²) ↓ ↓

-10x³ + 21x² - 160x ↓

- (-10x³ + 0x² - 160x) ↓

21x² + 0x + 336

- (21x² + 0x + 336)

Use Synthetic Division to divide (x - 7) by the reduced polynomial (2x³ - 13x² - 10x + 21)

7 | 2 -13 -10 21

| ↓ 14 7 -21

2 1 -3 0 ← remainder is 0

Use grouping to factor the reduced polynomial (2x² + x - 3)

2x² - 2x + 3x - 3

= 2x(x - 1) +1(x - 1)

= (2x + 1)(x - 1)

The factors are: (x - 7) (2x + 1) (x - 1) (x + 4i) (x - 4i)

The first 3 are real roots and the last 2 are complex roots

The zeros of a polynomial function are the factors of the function.

The number of real zeros of f(x) is 3

The function is given as:

$f(x) = 2x^5-13x^4+22x^3-187x^2-160x+336$

Two of the factors are given as:

x - 7 ad x + 4i

The factor x + 4i is a complex factor.

If a polynomial function has a complex factor, then the conjugate of the complex factor is also a factor of the polynomial.

This means that f(x) has 2 complex factors

The degree of the polynomial is 5.

So, the number (r) of real zeros is the difference between the degree of the polynomial and the number of complex factors.

This gives

$r = 5 -2$

$r = 3$

Hence, the number of real zeros of f(x) is 3

Read more about zeros of polynomials at:

brainly.com/question/3331729

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Answers

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Answers

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Answers

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Answers

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Step-by-step explanation:

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