What is the process to solve this problem? I have no idea how to continue

enter image source here

1 Answer
May 30, 2018

Please see below.

Explanation:

.

color(red)(a))a)

y=2/(2-x)y=22x

P(3,-2)P(3,2), Q(x, 2/(2-x))Q(x,22x)

The slope of line PQ can be calculated for each xx-value using:

m_(PQ)=(y_Q-y_P)/(x_Q-x_P)mPQ=yQyPxQxP

x=2.9, :. y=2/(2-2.9)=2/(-0.9)=-2.222222, :. m_(PQ)=(-2.222222-(-2))/(2.9-3)=(-0.222222)/(-(0.1))=color(red)(2.222222)

x=2.99, :. y=2/(2-2.99)=2/(-0.99)=-2.020202, m_(PQ)=(-2.020202-(-2))/(2.99-3)=(-0.020202)/(-0.01)=color(red)(2.020202)

Much the same way:

x=2.999, :. y=-2.002002, m_(PO)=color(red)(2.002002)

x=2.9999, :. y=-2.00020002, m_(PO)=color(red)(2.000200)

x=3.1, :. y=-1.818181, m_(PQ)=color(red)(1.818181)

x=3.01, :. y=-1.980198, m_(PQ)=color(red)(1.980198)

x=3.001, :. y=-1.998001, m_(PQ)=color(red)(1.998001)

x=3.0001, :. y=-1.999800, m_(PQ)=color(red)(1.999800)

color(red)(b))

As we can see from part a), the values of the slope m converge from both directions to 2. Therefore,

color(red)(m_(PQ)=2)

color(red)(c))

The equation of the tangent line to the curve is:

y=mx+b where m is the slope and b is the y-intercept:

y=2x+b

We can use the coordinates of point P to solve for b:

-2=2(3)+b

-2=6+b

b=-8

Therefore, the equation of the tangent line is:

color(red)(y=2x-8)

The graph below shows the function and the tangent to it at point P:

enter image source here