Finding distance...and function questions.?

Question

So the first question is

a) Find the distance between point (3, 5) and the line y=x-10. (can use catesian coordinate system i think?)

b)determine whether f is even, odd or neither.justify answer:

i) x^2cos(x)/x^4+1

ii) (1+3x^3-x^5) |x|

Answer 1

a. Note that by distance, they mean the shortest distance. To find that, you would need to draw a perpendicular line from the point to the line.

y = x - 10 has a slope of 1. A line perpendicular to it will have a slope of -1.

So, the equation will be y = -x + b

We know that this line goes through (3, 5) so we can use it to solve for b.

5 = -3 + b

b = 8

So the equation of the line through (3,5) perpendicular to y = x-10 is

y = -x + 8

Now, we want to find where these two lines intersect. This is the point where the point (3,5) is closest to the line.

Solving the two equation gives (11, 1).

So, we need to find the distance between (3,5) and (11, 1) which is root(80)

bi) f(-x) = f(x) so it is even. By that, I mean, replace all the x with -x

    (-x)^2 is x^2

   cos(-x) is cos(x) ...it is just a property

  (-x)^4 = x^4, ....

  so...that is how you show its even.

bii) neither..

    f(-x) does not equal f(x) so it is not even

    f(-x) does not equal -f(x) so it is not odd

    b is a little hard to explain, best to ask someone walk you through one example. But, the general idea is to replace all x with -x and see if you get something like the original equation back, or if you get -original equation back.

Hope this helps.