I, 6) 6. (a) Let (x», yo) be a point of the plane, and let L be the graph of the function f(x) = mx + b. Find the point i such that the distance from (x» yo) to (, f(F)) is smallest. [Notice that minimizing this distance is the same as minimizing its square. This may simplify the computations somewhat.] (b) Also find F by noting that the line from (x», yo) to (F, f(F)) is perpen- dicular to L. (0, e)0 (a, 0) (0, -a)e (c) Find the distance from (x» yo) to L, i.e., the distance from (xa, ya) to (x, f(x)). [It will make the computations easier if you first assume that 6 = 0; then apply the result to the graph of f(x) – mx and the point (xa, yo - 6).J Compare with Problem 4-22. (d) Consider a straight line described by the equation Ax + By +C = 0 (Problem 4-7). Show that the distance from (x, yo) to this line is (Ax, + By, + C)/V + B³. FIGURE 23 Surface area is the sum of these areas 7. The previous Problem suggests the following question : What is the relationship between the critical points of f and those of f? 1 8. A straight line is drawn from the point (0, a) to the horizontal axis, and then back to (1, b), as in Figure 23. Prove that the total length is shortest when the angles a and B are equal. (Naturally you must bring a function into the picture: express the length in terms of x, where (x, 0) is the point on the horizontal axis. The dashed line in Figure 23 suggests an alterna- tive geometric proof; in either case the problem can be solvcd without actually finding the point (x, 0).) FIGURE 24
I, 6) 6. (a) Let (x», yo) be a point of the plane, and let L be the graph of the function f(x) = mx + b. Find the point i such that the distance from (x» yo) to (, f(F)) is smallest. [Notice that minimizing this distance is the same as minimizing its square. This may simplify the computations somewhat.] (b) Also find F by noting that the line from (x», yo) to (F, f(F)) is perpen- dicular to L. (0, e)0 (a, 0) (0, -a)e (c) Find the distance from (x» yo) to L, i.e., the distance from (xa, ya) to (x, f(x)). [It will make the computations easier if you first assume that 6 = 0; then apply the result to the graph of f(x) – mx and the point (xa, yo - 6).J Compare with Problem 4-22. (d) Consider a straight line described by the equation Ax + By +C = 0 (Problem 4-7). Show that the distance from (x, yo) to this line is (Ax, + By, + C)/V + B³. FIGURE 23 Surface area is the sum of these areas 7. The previous Problem suggests the following question : What is the relationship between the critical points of f and those of f? 1 8. A straight line is drawn from the point (0, a) to the horizontal axis, and then back to (1, b), as in Figure 23. Prove that the total length is shortest when the angles a and B are equal. (Naturally you must bring a function into the picture: express the length in terms of x, where (x, 0) is the point on the horizontal axis. The dashed line in Figure 23 suggests an alterna- tive geometric proof; in either case the problem can be solvcd without actually finding the point (x, 0).) FIGURE 24
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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