Answer:
a. The value of the slope at P(3,-3) is then m = 3.
b. The equation of the tangent line to the curve at P(3,- 3) is [tex]y=3x-12[/tex].
Step-by-step explanation:
The slope of a line is given by
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Let the point P be [tex](x_2=3,y_2=-3)[/tex] and the point Q be [tex](x_1=x,y_1=\frac{3}{2-x} )[/tex]. So
[tex]m=\frac{-3-\frac{3}{2-x} }{3-x} =\frac{\frac{3x-9}{2-x}}{3-x}=\frac{3\left(x-3\right)}{\left(2-x\right)\left(3-x\right)}\\\\m=-\frac{3}{-x+2}[/tex]
Next, substitute the value of x in the formula of the slope
[tex]m=-\frac{3}{-2.9+2}=3.333333[/tex]
Do this for the other values of x.
Below, there is a table that shows the values of the slope.
a. From the table, as x approaches 3 from the left side (2.9 to 2.9999), the slopes are approaching to 3 and as x approaches 3 from the right side (3.1 to 3.0001), the slopes are approaching to 3. The value of the slope at P(3,-3) is then m = 3.
b. Use the point-slope form of a line [tex]y-y_1=m(x-x_1)[/tex], where m = 3 and [tex](x_1,y_1) = (3,-3)[/tex]. Then solve for y:
[tex]y+3=3(x-3)\\y=3x-12[/tex]
We can check our results with the graph of the function.
Slope m for the required x.
[tex]\\\begin{aligned} m\ &for\ x\\3.333333\ &for\ 2.9\\3.030303\ &for\ 2.99\\3.003003\ &for\ 2.999\\3.00030003\ &\ for\ 2.9999\\2.727272\ &for\ 3.1\\2.970297\ &for\ 3.01\\2.997002\ &for\ 3.001\\2.999700\ &for\ 3.0001\\\end{aligned}[/tex]
It is the angle of a line from the x-axis.
[tex]P(3,-3)\ and\ Q(x,\dfrac{3}{2-x} )[/tex]
[tex]Slope\ (m) = \dfrac{y_{2} -y_{1} }{x_{2} -x_{1}}[/tex]
Then
[tex]\\\begin{aligned} m\ &for\ x\\3.333333\ &for\ 2.9\\3.030303\ &for\ 2.99\\3.003003\ &for\ 2.999\\3.00030003\ &\ for\ 2.9999\\2.727272\ &for\ 3.1\\2.970297\ &for\ 3.01\\2.997002\ &for\ 3.001\\2.999700\ &for\ 3.0001\\\end{aligned}[/tex]
Thus, slope m for the required x.
More about the slope link is given below.
https://brainly.com/question/2514839
