By H.F. Baker
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Additional info for Abel's theorem and the allied theory, including the theory of the theta functions, Edition: New title
15. The function g(x) = bx + c is defined for all real numbers x for which bx + c ≥ 0. This function is differentiable for all x for which bx + c > 0, and the derivative is b g (x) = √ . 9 the graph of its derivative g (x) = 2√1x+1 . 16. Find the domain and the derivative of the function f (x) = √ 5x − 3. Solution: The function is defined whenever 5x − 3 ≥ 0, and this means that x ∈ [3/5, ∞). The derivative of the function is 5 f (x) = √ . 2 5x − 3 The expression for the derivative holds for x ∈ (3/5, ∞).
69), then we need that f (x) = (bx + c)1/α is differentiable for those x for which bx + c > 0. 15. 32) will not be as generally applicable as we may want it to be. Still, we can take care of this matter when time comes. 4. CHAPTER 2. THE DERIVATIVE 44 for all x ∈ (−π/4, π/4). So, setting x0 = 0, l(x) = 1 and A = 1/2 we see that |f (x) − l(x)| ≤ A(x − x0 )2 for all x ∈ (−π/4, π/4). The slope of the line l(x) is zero. So, according to our definition, f is differentiable at x0 = 0 and f (0) = 0. ♦ Exercise 34.
So, according to our definition, f is differentiable at x0 = 0 and f (0) = 0. ♦ Exercise 34. Let f (x) = sin x. Show that f (0) = 1. 30). 5) |f (x) − l(x)| ≤ A(x − x0 )2 for all x in some open interval around x0 , expresses that the function f (x) is close to its tangent line l(x) on some interval around x0 . It does not hurt to take a numerical example. Suppose A = 1, it is merely a scaling factor anyway. If x is close to x0 , then |x − x0 | is small, and (x − x0 )2 is very small. 01. 000001. Our first example is relevant and easy.