1)

Compute the value of x such that \(\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots.\)

2)

The graph of the parabola x = 2y^2 - 6y + 3 has an x-intercept (a,0) and two y-intercepts (0,b) and (0,c). Find a + b + c.

3)

Circles with centers of (2,2) and (17,10) are both tangent to the x-axis. What is the distance between the closest points of the two circles?

4)

Given the function y=x^2+10x+21, what is the least possible value of y?

5)

What real values of x are not in the domain of \(f(x)=\frac{1}{|x^2+3x-4|+|x^2+9x+20|}?\)

6)

Define the operation \(a\nabla b = b^a - 2\). What is the value of \((2\nabla 3) \nabla 2\)?

daddypig May 5, 2020

#1

#3**+2 **

5) x^2 + 10x + 21

The x coordinate of the vertex = -10/[ 2 * 1 ] = -5

The y coordinate of the vertex is the min =

(-5)^2 + 10(-5) + 21 =

-4

CPhill May 5, 2020

#4**+2 **

3)

Circles with centers of (2,2) and (17,10) are both tangent to the x-axis. What is the distance between the closest points of the two circles?

The distance between the centers is

sqrt [ (17-2)^2 + (10-2)^2] = sqrt [15^2 + 8^2] =sqrt 289 = 17

The radius of the first circle is 2 and the radius of the second circle is 10

So.....the distance between the two closest points = 17 - 2 - 10 = 5

CPhill May 5, 2020