what is the smallest positive integer a such that the intermediate value theorem guarantees a zero exists between 0 and a?
1 answer:
The smallest positive integer that the intermediate value theorem guarantees a zero exists between 0 and a is 3.
What is the intermediate value theorem?
Intermediate value theorem is theorem about all possible y-value in between two known y-value.
x-intercepts
-x^2 + x + 2 = 0
x^2 - x - 2 = 0
(x + 1)(x - 2) = 0
x = -1, x = 2
y intercepts
f(0) = -x^2 + x + 2
f(0) = -0^2 + 0 + 2
f(0) = 2
(Graph attached)
From the graph we know the smallest positive integer value that the intermediate value theorem guarantees a zero exists between 0 and a is 3
For proof, the zero exists when x = 2 and f(3) = -4 < 0 and f(0) = 2 > 0.
<em>Your question is not complete, but most probably your full questions was</em>
<em>Given the polynomial f(x)=− x 2 +x+2 , what is the smallest positive integer a such that the Intermediate Value Theorem guarantees a zero exists between 0 and a ?</em>
Thus, the smallest positive integer that the intermediate value theorem guarantees a zero exists between 0 and a is 3.
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Answer:
Part a) (see the explanation)
Part b) see the explanation
Part c) The graph in the attached figure
Step-by-step explanation:
<u><em>The complete question is</em></u>
With a family bowling pass, families can bowl for $4 per game. The pass costs $10 per year. Use an equation, a table, and a graph to explain the relationship between the total amount of money spent on bowling in a year, a, and the number of games a family plays in a year, g. Part A Use words and an equation to represent this problem. Part B Create a table to show values for g and a. Part C Use the values from your table to draw a graph.
Part A) Use words and an equation to represent this problem
Let
a ----> the total amount of money spent on bowling in a year
g ---> the number of games a family plays in a year
we know that
The linear equation in slope intercept form is equal to
where
m is the slope or unit rate
b is the a-intercept or initial value
In this problem we have
substitute
Part b) Create a table to show values for g and a
Assume different values of g and calculate the corresponding values of a
For g=0 ---->
For g=1 ---->
For g=2 ---->
For g=3 ---->
For g=4 ---->
The table is
Part c) Use the values from your table to draw a graph
we have the ordered pairs
Plot the ordered pairs and join them to graph the line
The graph in the attached figure
