Question

3. What point on the line y = 7 - 3x is closest to the origin? a. Sketch the line carefully and mark the point on the line that you think is closest to the origin. b. Write the distance between the origin and a point (x,y) in the plane. If you don't know, think of a triangle with base x and height y. 8 7 6 c. The point must be on the line, so you can write the distance Din terms of just x or just y. 5 d. Turns out, it is easier to minimize D2, so write S = D. S stands for the square of the distance. st e. Use calculus to find the value of x (or of y, whatever you used) that minimizes S. 3 2. 2 3 f. How can you be certain that you have found a local minimum of S 0 instead of a local maximum? 1 B. Find the (x,y) coordinates of the point on the line that is closest to the origin. Add that point to your sketch, and draw the line segment from the origin to the point. h. Find the slope of the line from the origin through the point you found on the line. i. What is the angle between the line y = 7 - 3x and the line from the origin through the nearest point? How can you tell?

Answer 1

7* 6+ 5+ 4 3 2 tly -0)2 (x,y) (3.520) D - + + y=4-3x stront we have sketched the line beside. #the distance D = D=562-x)² + y2-yi)2 D = f (x-02 # D2 = ge² ty2 3)92 = ge? +17-38) 2 * D2 = ge2 +49 +96² - 42 x →D? = 1082 to find minimum value D or D², will derivation du fle) ziox2 420 +49. id fix) 20g -42. ☺ 0 1 2 colo) 3 14 5 -42 x +49. of apply da d2 f(a) = 20 dze 2 € since this a positive value in its parent diled fire) will return a numum value dre would d f(x) =0 da 208-42=0 r = 42 20 X=2-1 ::Dioge2_423 +49 ID = 2.21 also y = 7-32 = 9-362.0 7- 6.3 y = 0.7 CSScanned with CamScanner

0 the point (211,017) will have uncumum distance from the origin. and slope of the line line connec connecting from origin to this point will be m'a ya - y (1)-0 542-44 3 n=1/3 3 also i angle between this connecting line and the line y = 7-3x us qo? as product of their slopes = -1 mxen CScanned with CamScanner -341 3