How Can I Solve These Two Calculus Problems on Parabolas and Tangents?

[JustinJS]

I'm having a little trouble with 2 problems because the day that they went over the problems i was out of school and my freakin teachers is never there after school to help me.

1) A particle moves along the parabola y = x^2 from the point (1,1) to the point (x,y) Show that (change in)y/(change in)x = x +1

2) Consider a circle of a radius 5, centered at (1,2). Find an equation of the line tangent to the circle at the point (-2, 6). Describe th procedure that you used to get your answer.

 

[HallsofIvy]

The particle will move from (1, 1) to (x, x²) since y= x².

Okay, y has changed from 1 to x². What is the "change in y"? How much does y change?

x has changed from 1 to x. What is the "change in x"? How much does x change?

If you can answer those, just divide!

There are two ways I can think of two answer (2). One of them involves writing the equation of the circle and using "implicit differentiation" to find the slope of the tangent line- you may not be ready for that.

The other is- find the slope of the line from (1,2) to (-2, 6), a radius of the circle. The tangent line at (-2, 6) must be perpendicular to that radius. Do you know how to find the slope of a line perpendicular to a given line (and you know the slope of that line)? After you know the slope of the tangent line and that it goes through (-2, 6) it should be easy to find the equation.