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lakkis [162]
3 years ago
5

If n lines are drawn in a plane, how many regions do they separate the plane into? induction

Mathematics
1 answer:
grandymaker [24]3 years ago
6 0
Hello,

Let's assume i the number of lines,
 u(i) the number of regions:

u_0=1\\\\
u_1=2=1+1=1+u_0\\\\
u_2=4=2+u_1\\\\
u_3=7=3+u_2\\\\
u_4=11=4+u_3\\\\
u_5=16=5+u_4\\\\
...\\\\
\boxed{u_{n+1}=(n+1)+u_{n}}\\\\

u_{n+1}=(n+1)+n+(n-1)+...+1+u_0\\\\

\boxed{u_{n+1}= \dfrac{(n+1)*(n+2)}{2}+1 }
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mylen [45]
Problem 1

With limits, you are looking to see what happens when x gets closer to some value. For example, as x gets closer to x = 2 (from the left and right side), then y is getting closer and closer to y = 1/2. Therefore the limiting value is 1/2

Another example: as x gets closer to x = 4 from the right hand side, the y value gets closer to y = 4. This y value is different if you approach x = 0 from the left side (y would approach y = 1/2)

Use examples like this and you'll get the results you see in "figure 1"

For any function values, you'll look for actual points on the graph. A point does not exist if there is an open circle. There is an open circle at x = 2 for instance, so that's why f(2) = UND. On the other hand, f(0) is defined and it is equal to 4 as the point (0,4) is on the function curve.

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Problem 2

This is basically an extension of problem 1. The same idea applies. See "figure 2" (in the attached images) for the answers. 

6 0
3 years ago
Last month Maria hiked the 5-mile mountain trail a number of times and she hiked the 10-mile canal trail several times. Let x re
professor190 [17]
Number of times she climbed 5-mile mountain trail = x
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total distance of the 10-mile mountain trail will be  = 10*y = 10y
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pantera1 [17]

Answer:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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