Answer:
See explanation
Step-by-step explanation:
You are given the equation of the curve
![y=\dfrac{x}{1+x}](https://tex.z-dn.net/?f=y%3D%5Cdfrac%7Bx%7D%7B1%2Bx%7D)
Point
lies on the curve.
Point
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]](https://tex.z-dn.net/?f=%5Cdfrac%7By_2-y_1%7D%7Bx_2-x_1%7D%3D%5Cdfrac%7B%5Cfrac%7Bx%7D%7B1%2Bx%7D-%5Cfrac%7B1%7D%7B2%7D%7D%7Bx-1%7D%3D%5Cdfrac%7B%5Cfrac%7B2x-%281%2Bx%29%7D%7B2%28x%2B1%29%7D%7D%7Bx-1%7D%3D%5Cdfrac%7B%5Cfrac%7B2x-1-x%7D%7B2%28x%2B1%29%7D%7D%7Bx-1%7D%3D%5C%5C%20%5C%5C%3D%5Cdfrac%7B%5Cfrac%7Bx-1%7D%7B2%28x%2B1%29%7D%7D%7Bx-1%7D%3D%5Cdfrac%7Bx-1%7D%7B2%28x%2B1%29%7D%5Ccdot%20%5Cdfrac%7B1%7D%7Bx-1%7D%3D%5Cdfrac%7B1%7D%7B2%28x%2B1%29%7D%5C%20%5B%5Ctext%7BWhen%7D%5C%20x%5Cneq%201%5D)
1. If x=0.5, then the slope is
![\dfrac{1}{2(0.5+1)}=\dfrac{1}{3}\approx 0.3333](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%280.5%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B3%7D%5Capprox%200.3333)
2. If x=0.9, then the slope is
![\dfrac{1}{2(0.9+1)}=\dfrac{1}{3.8}\approx 0.2632](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%280.9%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B3.8%7D%5Capprox%200.2632)
3. If x=0.99, then the slope is
![\dfrac{1}{2(0.99+1)}=\dfrac{1}{3.98}\approx 0.2513](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%280.99%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B3.98%7D%5Capprox%200.2513)
4. If x=0.999, then the slope is
![\dfrac{1}{2(0.999+1)}=\dfrac{1}{3.998}\approx 0.2501](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%280.999%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B3.998%7D%5Capprox%200.2501)
5. If x=1.5, then the slope is
![\dfrac{1}{2(1.5+1)}=\dfrac{1}{5}\approx 0.2](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%281.5%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B5%7D%5Capprox%200.2)
6. If x=1.1, then the slope is
![\dfrac{1}{2(1.1+1)}=\dfrac{1}{4.2}\approx 0.2381](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%281.1%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B4.2%7D%5Capprox%200.2381)
7. If x=1.01, then the slope is
![\dfrac{1}{2(1.01+1)}=\dfrac{1}{4.02}\approx 0.2488](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%281.01%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B4.02%7D%5Capprox%200.2488)
8. If x=1.001, then the slope is
![\dfrac{1}{2(1.001+1)}=\dfrac{1}{4.002}\approx 0.2499](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%281.001%2B1%29%7D%3D%5Cdfrac%7B1%7D%7B4.002%7D%5Capprox%200.2499)