By Knut Aas-Jakobsen, Mathis Grenacher (auth.)

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**Extra info for Analysis of Slender Reinforced Concrete Frames**

**Sample text**

The correct definition is a). The idea of an asymptotic behaviour is getting closer to a (non-vertical) straight line but this doesn’t exclude touching or crossing it. 5 -1 3. The tangent line to a curve at a certain point that touches the curve at infinitely many other points cannot be a non-vertical asymptote to this curve. Counter-example. sin 2 x at x S touches the x curve at infinitely many other points and is a non-vertical asymptote to this curve. The tangent line y = 0 to the curve y 38 8 -6 -4 0 -2 sin 2 x x y 2 0 2 4 6 8 -2 -4 4.

The derivative of the function y x 3 is zero at the point x = 0 but the function is increasing at this point. 59 y x3 2 8 -6 -4 0 -2 0 2 4 6 8 -2 -4 7. If a function is differentiable and decreasing on (a,b) then its gradient is negative on (a,b). Counter-example. x 3 is differentiable and decreasing on R but its The function y gradient is zero at the point x = 0. y x3 2 8 -6 -4 0 -2 0 2 4 6 8 -2 -4 8. If a function is continuous and decreasing on (a,b) then its gradient is non-positive on (a,b).

Both functions y = f(x) and y = g(x) are differentiable and f(x) > g(x) on the interval (a,b) but f c( x ) g c( x ) on (a,b). f(x) g(x) a 2. b If a non-linear function is differentiable and monotone on (0, f ) then its derivative is also monotone on (0, f ) . Counter-example. The non-linear function y x sin x is differentiable and monotone on (0, f ) but its derivative y c 1 cos x is not monotone on (0, f ) . 56 x sin x y 30 20 10 0 0 10 20 30 40 50 60 -10 -20 30 8 1 cos x yc 6 4 2 0 0 5 10 15 20 25 -2 -4 -6 3.