Consider the function f(x) = arctan(3x). Show that the graphy = f(x) and its tangent line y = g(x) at (0,0). Intermediate steps: 1) The line tangent to y = f(x) at x =0 is y = g(x) where g(x) = 2) Let H(x) = f(x) - g(x). The derivative of H (x) is H'(x) = = 0) intersect only at which is zero only when x = 3) Now assume that we have ₁ <0 where f(x₁) = g(x₁). Apply Rolle's theorem to H (x) on the interval [x1, 0]. Get a contradiction. 4) Now assume that we have 2 > 0 where f(2)= g(x2). Apply Rolle's theorem to H(x) on the interval [0, ₂]. Get a contradiction. 5) Conclude that the graph of f(x) and its tangent line intersect only at (0,0).
Consider the function f(x) = arctan(3x). Show that the graphy = f(x) and its tangent line y = g(x) at (0,0). Intermediate steps: 1) The line tangent to y = f(x) at x =0 is y = g(x) where g(x) = 2) Let H(x) = f(x) - g(x). The derivative of H (x) is H'(x) = = 0) intersect only at which is zero only when x = 3) Now assume that we have ₁ <0 where f(x₁) = g(x₁). Apply Rolle's theorem to H (x) on the interval [x1, 0]. Get a contradiction. 4) Now assume that we have 2 > 0 where f(2)= g(x2). Apply Rolle's theorem to H(x) on the interval [0, ₂]. Get a contradiction. 5) Conclude that the graph of f(x) and its tangent line intersect only at (0,0).
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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![Consider the function f(x) = arctan(3x). Show that the graph y = f(x) and its tangent line y = g(x) at x =
(0,0).
Intermediate steps:
1) The line tangent to y = f (x) at x = () is y = g(x) where
g(x) =
2) Let H(x) = f(x) — g(x). The derivative of H (x) is
H'(x) =
O intersect only at
which is zero only when x =
3) Now assume that we have X1
O where f(x₁) = g(x₁). Apply Rolle's theorem to H(x) on the interval [₁, 0]. Get a contradiction.
4) Now assume that we have 2> 0 where f(x₂) = g(₂). Apply Rolle's theorem to H(x) on the interval [0, x₂]. Get a contradiction.
5) Conclude that the graph of f(x) and its tangent line intersect only at (0,0).
A GRAPH OF THE SITUATION. JUSTIFY EACH APPLICATION OF ROLLE'S THEOREM.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7634a6a-5ecc-40e2-9892-46ba7a4cefd4%2F9bc46097-e99f-410c-8e28-e1634da63afc%2Ffsrczi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the function f(x) = arctan(3x). Show that the graph y = f(x) and its tangent line y = g(x) at x =
(0,0).
Intermediate steps:
1) The line tangent to y = f (x) at x = () is y = g(x) where
g(x) =
2) Let H(x) = f(x) — g(x). The derivative of H (x) is
H'(x) =
O intersect only at
which is zero only when x =
3) Now assume that we have X1
O where f(x₁) = g(x₁). Apply Rolle's theorem to H(x) on the interval [₁, 0]. Get a contradiction.
4) Now assume that we have 2> 0 where f(x₂) = g(₂). Apply Rolle's theorem to H(x) on the interval [0, x₂]. Get a contradiction.
5) Conclude that the graph of f(x) and its tangent line intersect only at (0,0).
A GRAPH OF THE SITUATION. JUSTIFY EACH APPLICATION OF ROLLE'S THEOREM.
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