Question

Prove that there exists a point (a, b) inside the circle (x − 3)^2 + (y − 2)^2 = 13 such that a < 0 and b > 3.5. (recall: the point (a, b) is inside the circle means that (a − 3)^2 + (b − 2)^2 ≤ 13)

Answer #1

Prove that every point on the line y = 6 − x is outside the
circle with equation (x + 3)^2 + (y − 1)^2 = 16 (recall that a
point (a, b) is outside the circle means that (a + 3)^2 + (b − 1)^2
≥ 16).
Proof from scratch please!

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

Find an equation for the line tangent to the circle x^(2)+y^(2)=
25 at the point(3,−4).

3)
a) Find a polar equation for the circle x^2 + (y -2)^2 = 4.
b)Find the arc length of the polar curve r =
3^θ from θ=0 to θ=2.

Given: equation of circle k (x - 3)2 + (y -
2)2 = 36 and equation of the line g: y = mx + 8.
What are the conditions of m under which: a) m does not touch
the circle b) m intersects the circle in one point c) m intersects
the circle in two points?

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

Exercise1.2.1: Prove that if t > 0 (t∈R),
then there exists an n∈N such that 1/n^2 < t.
Exercise1.2.2: Prove that if t ≥ 0(t∈R), then
there exists an n∈N such that n−1≤ t < n.
Exercise1.2.8: Show that for any two real
numbers x and y such that x < y, there exists an irrational
number s such that x < s < y. Hint: Apply the density of Q to
x/(√2) and y/(√2).

Let y = x 2 + 3 be a curve in the plane.
(a) Give a vector-valued function ~r(t) for the curve y = x 2 +
3.
(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint:
do not try to find the entire function for κ and then plug in t =
0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)|
dT~ dt (0) .]
(c) Find the center and...

a circle has the equation: (x+5)2 + (y-3)2
= 37. What is the equation of the line tangent to the circle at the
point whose coordinates are (1,4)? please write in ax+ by = C form
.

Circle A with the equation (x+1)2 + (y‒4)2 = 20 was transformed
to Circle B with an equation of (x‒5)2 + (y‒2)2= 5. Which statement
describes the dilation that would be performed to enlarge Circle B
to the same size as Circle A? A. Multiply the radius by 4. B. Add 2
units to the radius. C. Multiply the radius by 2. D. Add 15 to the
radius.

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