By Norman Levinson
Big apple 1940 first version. American Mathematical Society Coloquium Vol. XXVI. Hardcover. Sm.4to., 246pp., fabric. proprietor signed. close to nice, no DJ.
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Extra resources for Gap and density theorems
3: Exponential Function and Tangent Line between two Parabolas We used short dashes to draw their graphs. These two parabola touch in the point (1, e). In addition you see the line l(x), which is our candidate for the being the tangent line. On the shown interval, the graphs of f (x) and l(x) are above the graph of q(x) and below the one of p(x). In mathematical notation this is expressed as q(x) ≤ f (x) ≤ p(x) and q(x) ≤ l(x) ≤ p(x). One way of saying that f (x) and l(x) are close to each other near x0 is to require that they are jointly in between two parabolas which touch (and do not cross each other) in the point (x0 , f (x0 )) = (x0 , l(x0 )).
You find a line l which has the same value at x as f (f (x) = l(x)), and the line does not cross the graph of f (near x the graph of f is on one side of the line). This rather practical recipe for finding the tangent line of a differentiable function works for all functions in these notes at almost all points, see Remark 18 on page 164. It works in the previous example as well as in the one we are about to discuss. 1) f (x) = y = 1 − x2 . 2) f (x) = √ −x . 1 − x2 The function describes the upper hemisphere of a circle of radius 1 centered at the origin of the Cartesian coordinate system.
This motivates the following definition. 8. Let a function f be defined on an open interval (a, b). We say that f is differentiable on (a, b) (or differentiable for short) if the derivative f (x) exists for all x ∈ (a, b). In this case we obtain a function f which is defined for all x ∈ (a, b) and which is called the derivative of f . If a function is defined on a union of open intervals, then we say that the function is differentiable if it is differentiable on each of the intervals. 3. 2 in a less elegant but more practical way.