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.
Learn more about intermediate value theorem here:
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Answer:
<u>Volume</u>
For the rectangle, h = 3cm, l = 8cm, w = 6cm
V = length x width x height
V = 8cm x 6cm x 3cm
V = 144cm^3
For the semi circle, we need to find the radius. The radius is width/2, so 6cm/2 = 3cm. r = 3cm, = 3.14
V = radius^2 x height x
V = 3cm^2 x 3cm x 3.14
V = 84.8 cm^3/2 (because the cylinder needs to be divided to form a semi-circle)
V= 42.4cm^3 (there are two cylinders though so we will multiply this by 2 in the total volume)
Total volume:
V = 144cm^3 + 42.4cm^3(2)
V = 186.4cm^3
<u>Surface Area</u>
Rectangular prism:
A = 2[w(l) + h(l) + h(w)]
A = 2[6cm(8cm) + 3cm(8cm) + 3cm(6cm)]
A = 180cm^2
But there are two sides that are covered by the semi-circular prisms, so we will have to calculate those sides and remove them.
A = l x w
A = 6cm x 3cm
A = 18cm^2(2) (2 being the two faces)
A = 36cm^2
A = 180cm^2 - 36cm^2
A = 144cm^2 (the area of the rectangle)
Semi-circular prism:
A = 2rh + 2
r^2
Earlier, we found out that the radius of the circle is 3cm, so we will plug that in.
A = 2(3.14)(3cm)(3cm) + 2(3.14)(3cm)^2
A = 113.09cm^2
Total surface area:
A = 144cm^2 + 133.09cm^2
A = 277.09cm^2
Therefore the total volume of the prism is 186.4cm^3 and the total surface area is 277.09cm^2.