Question

which statement is always true? a. a ray consists of two lines. b. a line has length and width. c. a line segment has only two points. d. two skew lines never intersect each other. e. two opposite rays make up a line segment. 45 consider the function f(x)=x² - 4. which of these statements is true about the average change of this function? a. the average rate of change of the function over the interval 3,4 is 7. b. the average rate of change of the function over the interval 0,3 is 1. c. the average rate of change of the function over the interval 0,4 is - 2. d. the average rate of change of the function over the interval 1,2 is - 1.5. e. the average rate of change of the function over the interval 3,4 is 1/7.

Explanation:

Step1: Recall geometric definitions

A ray is a part - of a line with one endpoint, not two lines (so A is false). A line has only length and no width (so B is false). A line - segment has two endpoints but infinitely many points on it (so C is false). Two opposite rays make up a line, not a line - segment (so E is false). Skew lines are non - coplanar lines and by definition, they never intersect, so D is true for the first part.

Step2: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$.
For $f(x)=x^{2}-4$:

• Over the interval $[3,4]$:
• $f(3)=3^{2}-4=9 - 4 = 5$, $f(4)=4^{2}-4=16 - 4 = 12$.
• The average rate of change is $\frac{f(4)-f(3)}{4 - 3}=\frac{12 - 5}{1}=7$.
• Over the interval $[0,3]$:
• $f(0)=0^{2}-4=-4$, $f(3)=3^{2}-4 = 5$.
• The average rate of change is $\frac{f(3)-f(0)}{3 - 0}=\frac{5-(-4)}{3}=\frac{9}{3}=3$.
• Over the interval $[0,4]$:
• $f(0)=-4$, $f(4)=12$.
• The average rate of change is $\frac{f(4)-f(0)}{4 - 0}=\frac{12-(-4)}{4}=\frac{16}{4}=4$.
• Over the interval $[1,2]$:
• $f(1)=1^{2}-4=-3$, $f(2)=2^{2}-4 = 0$.
• The average rate of change is $\frac{f(2)-f(1)}{2 - 1}=\frac{0-(-3)}{1}=3$.

Answer:

For the first question: D. Two skew lines never intersect each other.
For the second question: A. The average rate of change of the function over the interval $[3,4]$ is 7.