Question

9 Geometry via calculus In this exercise you will see one way to use calculus to do grometry a) Here is one way to find the perpendicsler distance from a point to a line L (no caleulus yet) Let's say L has equation y-3r+2 and the point is (2.1) First, make a graph (picture) of the situation 2Now find an equation for the line AM through (2, 1) perpendicular to L (draw it first, of course). 3. Find the (coordinates of the) point P wbere M istersects L Finally, find the distance from (2,1) to P. This is the perpendicalar distance from (2,1) to L b) Here is the caleulus version Write a formula for the function f with f(x) equal to the squared distance betreen the point (r,3r 2) on L and (2,) (Why squared distane? Look ahead to see what were going to do with f. Does the previous exercise give any insight be?) 2. Find the minimum of f and determine at what point (re 3ro + 2) on L this minimum is achieved. 3. Explain why the line through (2,1) and (o.3ro +2) is perpendicular to L So the perpendicular distance is also the shortest distance from (2. 1) to L. So here caleslas does the same thing as algebra, but gives us more information Let's do the same thing with a "general line and point: using the idea in the previous part, give a formula for the perpendicular distance from a point (z,y) to the line with equation y-mr + (All the ideas necessary are in the previoas part; here you just have to manage to express them in a generic" fashion, so the difficulties are primarily notational It's a good idea to try several examples with umbers in place of r, y,m, b and then check your formula with these examples when you're done.) c) Calculus works the same way for curees Here is an esample Let gla) Explain how to find the closest point P on the graph of g to the point (2, 1) 1. Draw a picture of the situation (now and at every step below) 2. Show that the line joining the point P to (2, 1) is perprndicular to the tangent line to the graph of g at P. (Hint: what does the function f(a) (r-2-12 have to do with anything?) 3. Show that if J(a) for some a, then the line through (o,a and (21) is perpendicular to the tangent line to the graph of g at (o,a) (draw a picture) 4. How many points (a, a2) have f'(a)0 What is their geometric interpretation? That is how are they distinguished relative to the graph of g and the distance to (2,1)? How are they distinguished relative to the graph of f? (Note: I'm not asking you to find the point P explicitly, but to explain how to find it and to show that it satisfies a condition just like in the case of lines.)

Answer 1

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: y=3x+ 2 Ause turo, 3 3