By Brezis H.
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19. Let be a differentiable function and let to the quotient ❨ ❨ ✰ ✥ be a small positive number. Give a physical meaning ✜ ❞ ✜ ✲ ❇ ❨ ✰ ✥ ✲ ✜ in the following cases. (a) ✥ ✲ ❨ ✰ ✥ ✲ ❨ ✰ ✥ ✲ (b) ✰ (c) is the height of a missile ❨ seconds after liftoff. ✥ is the population of a city at time . ✥ is the cost in dollars of producing units of a product. 2 RULES FOR DIFFERENTIATION We have identiﬁed the derivative of a differentiable function as one of its important properties. 1 on page 44, which is often a cumbersome process.
As we have mentioned earlier, ﬁnding the range of a function can be difﬁcult. In the case of rational functions with quadratic terms, we can sometimes use our knowledge of quadratic equations to ﬁnd the range. A rational function is one that can be expressed as the quotient of two polynomials. 17 Find the range and natural domain of the function ✥ ❨ ✰ ✥ ✲ ✓ ✡ ✥ ❇ ✼ ✑ It is easy to see that the natural domain is the set of all real numbers range, we put for which ✥ ✥ ✡ ❦ ➤ . To ﬁnd the ❈ ✥ ④ ✡ ✥ ❇ ✼ ✑ and rearrange the equation to get .
We could, for instance, deﬁne a function ‘step’ by the rule ❨ ✜ ❛ ✈ step ✙ ✰ ✢ ✲ ✪ ✢ ✪ ✢ ◗ ✙ ✡ ✖ ▼ ✙ ✓ Some functions have names which are universally agreed upon. You have probably met some of these before. Common examples are , and , usually written as , and . We don’t even have to use letters, for example the square root function is denoted by the special symbol and the output number corresponding to the input number is denoted by . Many simple functions do not have universally agreed names and are deﬁned by algebraic formulas, for example polynomials such as ò ❰ ó ✰ ✥ ✲ ô õ ò ✰ ✥ ✲ ❮ õ ö ✰ ✥ ✲ ò ❰ ó ✥ ô õ ò ✥ ✥ ❮ õ ö ✥ ✥ ➋ ➋ ✞ ❨ ✰ ✥ ✲ ✡ ✥ ❇ ✥ ❞ ❈ ✥ ❇ ✖ ✓ ✑ Indeed, we often only refer to the formula in speaking of the function.