Suppose that f is a differentiable function defined on the whole real number line R and that the graph of f contains a point Q(x, y) closest to the point P(xo, yo) not on the graph. Show that X-XO y - yo at Q. Conclude that the segment PQ is perpendicular to the line tangent to the curve at Q. [Suggestion: Minimize the square of the distance PQ.] Use the result of Problem 92 to show that the minimum dis- tance from the point (xo, yo) to a point of the straight line Ar+By+C =0 is f(x) = - | Axo + Byo + CL √A²+ B²
Suppose that f is a differentiable function defined on the whole real number line R and that the graph of f contains a point Q(x, y) closest to the point P(xo, yo) not on the graph. Show that X-XO y - yo at Q. Conclude that the segment PQ is perpendicular to the line tangent to the curve at Q. [Suggestion: Minimize the square of the distance PQ.] Use the result of Problem 92 to show that the minimum dis- tance from the point (xo, yo) to a point of the straight line Ar+By+C =0 is f(x) = - | Axo + Byo + CL √A²+ B²
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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solve 93 using sytem of equations
![Suppose that f is a differentiable function defined on the
whole real number line R and that the graph of f contains a
point Q(x, y) closest to the point P(xo. yo) not on the graph.
Show that
f(x) = -
X-XO
y- yo
at Q. Conclude that the segment PO is perpendicular to the
line tangent to the curve at Q. [Suggestion: Minimize the
square of the distance PQ.]
Use the result of Problem 92 to show that the minimum dis-
tance from the point (xo. yb) to a point of the straight line
Ax+By+C =0 is
| Axo + Byo + CI
√A²+ B²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7c1e554-035d-49e2-a1d1-507d99d18d03%2Fc1a248b1-0161-45c1-9d65-85be0b9f2030%2F3g32hi6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that f is a differentiable function defined on the
whole real number line R and that the graph of f contains a
point Q(x, y) closest to the point P(xo. yo) not on the graph.
Show that
f(x) = -
X-XO
y- yo
at Q. Conclude that the segment PO is perpendicular to the
line tangent to the curve at Q. [Suggestion: Minimize the
square of the distance PQ.]
Use the result of Problem 92 to show that the minimum dis-
tance from the point (xo. yb) to a point of the straight line
Ax+By+C =0 is
| Axo + Byo + CI
√A²+ B²
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Step 1: Given information
VIEWStep 2: Solution
VIEWStep 3: To prove that PQ is normal to tangent
VIEWStep 4: The minimum gap between the line Ax+by+C=0 and point P
VIEWStep 5: Co-ordinates of Intersection point Q of line Ax+By+C=0 and line through point P
VIEWStep 6: length PQ , minimum gap from P to line Ax+By+C
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