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tiny-mole [99]
3 years ago
5

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
Mathematics
1 answer:
lukranit [14]3 years ago
7 0

Answer:

See explanation

Step-by-step explanation:

You are given the equation of the curve

y=\dfrac{x}{1+x}

Point P\left(1,\dfrac{1}{2}\right) lies on the curve.

Point Q\left(x,\dfrac{x}{1+x}\right) is an arbitrary point on the curve.

The slope of the secant line PQ is

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

1. If x=0.5, then the slope is

\dfrac{1}{2(0.5+1)}=\dfrac{1}{3}\approx 0.3333

2. If x=0.9, then the slope is

\dfrac{1}{2(0.9+1)}=\dfrac{1}{3.8}\approx 0.2632

3. If x=0.99, then the slope is

\dfrac{1}{2(0.99+1)}=\dfrac{1}{3.98}\approx 0.2513

4. If x=0.999, then the slope is

\dfrac{1}{2(0.999+1)}=\dfrac{1}{3.998}\approx 0.2501

5. If x=1.5, then the slope is

\dfrac{1}{2(1.5+1)}=\dfrac{1}{5}\approx 0.2

6. If x=1.1, then the slope is

\dfrac{1}{2(1.1+1)}=\dfrac{1}{4.2}\approx 0.2381

7. If x=1.01, then the slope is

\dfrac{1}{2(1.01+1)}=\dfrac{1}{4.02}\approx 0.2488

8. If x=1.001, then the slope is

\dfrac{1}{2(1.001+1)}=\dfrac{1}{4.002}\approx 0.2499

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