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Homework Statement
2 problems, i solved both of them but I am not 100 % I am right
Find all points on the curve y=x-2cosx where the tangent line to the curve is parallel to the line y=x and write an equation of the tangent line at such point
Let f^-1 be the inverse of the one -to- one function f(x)=x^3+x.
Find (f^-1)'(10).
Homework Equations
slope of tangent line=f'(of the point)
The Attempt at a Solution
1st.
This seemed tricky and I did not figure it out till I had a good 11 hour sleep, what I did:
y=x , parallel lines have same slopes=>slope of tangent line is 1
y'=1+2sinx
1+2sinx=1
x=0+2pi*n
we found all points on the curve where tangent is parallel to y=x
lets find y coordinate
y=0-2cos(0)=-2
equation of tangent line
y+2=1(x-0)
y=x-2
2nd.
I think this has a twist in it but I can't find it :( , here's what I did
f^-1(x)=1/(f'(f^-1(x))
we need to find f^-1(10)
since we got f(x) its not hard to do so
10=x^3+x
x=2
f^-1(x)=1/(f'(2))
f'=3x^2+1
f^-1(x)=1/13
this is the answer?
Please check what I have written, I apologize for such a mess t.t