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"Find the equation of a tangent line to the graph of f(x) = cos x that can be used to approximate the value of cos(π/6 + 0.1). Then, find an approximation of cos(π/6 + 0.1)."
Tangent-line equation: f'(x) = -sin x
f(x0) ≈ f'(c)(x0 - c) + f(c)
where x0 = the number whose f( ) value we're trying to approximate = π/6 + 0.1, and
c = a convenient number close to x0 on the tangent line = π/6
cos (π/6 + 0.1) ≈ -(sin(π/6))(π/6)(π/6 + 0.1 - π/6) + cos(π/6)
cos (π/6 + 0.1) ≈ -(π/6)(1/2)(0.1) + (√3)/2
cos (π/6 + 0.1) ≈ -π/120 + (√3)/2
cos (π/6 + 0.1) ≈ -(π + 60)/120
But somehow it seems a little weird to have π, though it's a cornerstone of radian notation, in a y-value ... Am I on the right track? Thanks, anon.
Where does the expression pi origionate from? —Preceding unsigned comment added by 207.224.29.240 ( talk) 03:12, 16 January 2008 (UTC)
In my Metric Spaces examination this morning, the following question was worth 4 marks out of 60:
Suppose that f and (S,p) [are a function and metric space fulfilling the requirements of] Banach's Contraction Mapping Principle and suppose that g:S->S is a function with the property that f(g(x)) = g(f(x)) for all x in S. Show that g has a unique fixed point.
Now, I was easily able to show that g has at least one fixed point (it being the same fixed point as the one f has, as guaranteed by B's CMP). However, I got stuck on the uniqueness part until I realised this:
Unless I'm missing something... was the question wrong? If so, what would ensure g had a unique fixed point?
Rawling 4851 12:06, 16 January 2008 (UTC)
In the graph of a rational function, I understand that the vertical asymptotes can never be crossed, and the one possible non-vertical asymptote can be crossed "near" the y-axis.
Is there a limit to number of times the non-vertical asymptote can be crossed?
I'm hoping that there is such a limit and that it is related to the degree of the denominator...or at least the degree of the denominator once all common factors have been eliminated.
Thanks, Stableyr —Preceding unsigned comment added by 66.100.0.42 ( talk) 13:12, 16 January 2008 (UTC)
Mathematics desk | ||
---|---|---|
< January 15 | << Dec | January | Feb >> | January 17 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
"Find the equation of a tangent line to the graph of f(x) = cos x that can be used to approximate the value of cos(π/6 + 0.1). Then, find an approximation of cos(π/6 + 0.1)."
Tangent-line equation: f'(x) = -sin x
f(x0) ≈ f'(c)(x0 - c) + f(c)
where x0 = the number whose f( ) value we're trying to approximate = π/6 + 0.1, and
c = a convenient number close to x0 on the tangent line = π/6
cos (π/6 + 0.1) ≈ -(sin(π/6))(π/6)(π/6 + 0.1 - π/6) + cos(π/6)
cos (π/6 + 0.1) ≈ -(π/6)(1/2)(0.1) + (√3)/2
cos (π/6 + 0.1) ≈ -π/120 + (√3)/2
cos (π/6 + 0.1) ≈ -(π + 60)/120
But somehow it seems a little weird to have π, though it's a cornerstone of radian notation, in a y-value ... Am I on the right track? Thanks, anon.
Where does the expression pi origionate from? —Preceding unsigned comment added by 207.224.29.240 ( talk) 03:12, 16 January 2008 (UTC)
In my Metric Spaces examination this morning, the following question was worth 4 marks out of 60:
Suppose that f and (S,p) [are a function and metric space fulfilling the requirements of] Banach's Contraction Mapping Principle and suppose that g:S->S is a function with the property that f(g(x)) = g(f(x)) for all x in S. Show that g has a unique fixed point.
Now, I was easily able to show that g has at least one fixed point (it being the same fixed point as the one f has, as guaranteed by B's CMP). However, I got stuck on the uniqueness part until I realised this:
Unless I'm missing something... was the question wrong? If so, what would ensure g had a unique fixed point?
Rawling 4851 12:06, 16 January 2008 (UTC)
In the graph of a rational function, I understand that the vertical asymptotes can never be crossed, and the one possible non-vertical asymptote can be crossed "near" the y-axis.
Is there a limit to number of times the non-vertical asymptote can be crossed?
I'm hoping that there is such a limit and that it is related to the degree of the denominator...or at least the degree of the denominator once all common factors have been eliminated.
Thanks, Stableyr —Preceding unsigned comment added by 66.100.0.42 ( talk) 13:12, 16 January 2008 (UTC)