Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 37E
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Can you please answer this question
![Consider the function f(x) = arctan(6x).
Show that the graph y = f(x) and its tangent line
y = g(x) at x = 0) intersect only 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) =
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 [1, 0]. Get a contradiction.
4) Now assume that we have ₂ > 0 where
f(x₂):
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7634a6a-5ecc-40e2-9892-46ba7a4cefd4%2Fb684faa1-d74c-4c60-a741-ee277beea674%2Fiaquoda_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the function f(x) = arctan(6x).
Show that the graph y = f(x) and its tangent line
y = g(x) at x = 0) intersect only 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) =
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 [1, 0]. Get a contradiction.
4) Now assume that we have ₂ > 0 where
f(x₂):
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).
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