The following graph shows the low temperatures for the past six nights. Which night had a low temperature of 26°F?
2 answers:
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Answer:
The equation does not have a real root in the interval
Step-by-step explanation:
We can make use of the intermediate value theorem.
The theorem states that if is a continuous function whose domain is the interval [a, b], then it takes on any value between f(a) and f(b) at some point within the interval. There are two corollaries:
- If a continuous function has values of opposite sign inside an interval, then it has a root in that interval. This is also known as Bolzano's theorem.
- The image of a continuous function over an interval is itself an interval.
Of course, in our case, we will make use of the first one.
First, we need to proof that our function is continues in , which it is since every polynomial is a continuous function on the entire line of real numbers. Then, we can apply the first corollary to the interval
, which means to evaluate the equation in 0 and 1:
Since both values have the same sign, positive in this case, we can say that by virtue of the first corollary of the intermediate value theorem the equation does not have a real root in the interval . I attached a plot of the equation in the interval
where you can clearly observe how the graph does not cross the x-axis in the interval.
Answer:
Step-by-step explanation:
A = (-3,0) , B = (1,3) , and C = (1,-3)
We're going to use the distance formula to find the length of the sides:
we can use the cosine law to find the angle:
it is to be noted that:
the angle CAB is opposite to the BC.
the angle ABC is opposite to the AC.
the angle BCA is opposite to the AB.
to find the CAB, we'll use:
Although we can use the same cosine law to find the other angles. but we can use sine law now too since we have one angle!
To find the angle ABC
finally, we've seen that the triangle has two equal sides, AB = CA, this is an isosceles triangle. hence the angles ABC and BCA would also be the same.
this can also be checked using the fact the sum of all angles inside a triangle is 180