Lecture for Week 8 Secs 4 1 4 2 Exponentials and Inverse Functions 1 Let s get right to the point The main reason for studying exponential functions is to solve problems like those in Sec 4 5 where the growth of some stuff is proportional to the amount of stuff already present dP t kP t dt Compound interest population growth radioactive decay all fall into this category 2 Suppose we knew a function whose derivative is itself d exp x exp x dx Then P t C exp kt would solve P kP for any constant C Normalization Suppose also that exp 0 1 And suppose that we know P 0 Then P t P 0 exp kt is the correct solution of for our problem 3 With this we have solved our first nontrivial differential equation The trouble with this approach to exp x is that it may not be obvious that such a function exists or that there are not more than one of them satisfying exp 0 1 Here is a different approach summarized from Stewart 1 From elementary algebra we understand what ax means for any a 1 and any rational number x 4 2 We can define ax for irrational x by continuity filling in the holes in the graph 3 Define e as the number such that eh 1 1 lim h 0 h d x 4 Prove that e ex dx 5 Observe that a0 1 for all a hence e0 1 5 So we can define exp x as ex You can think of ex either as a number e raised to a power or as a special function exp x analogous to sin x and the other trig functions The numerical value of e is a famous transcendental number like e 2 7182818 6 Algebraic properties of exponentials the laws of exponents ex y ex ey ex y exy 1 e x x e e0 1 e1 e 7 These laws also hold with a in place of e everywhere Also ab x ax bx And we also have ex 0 and lim ex x lim ex 0 x x These also hold for a if a 1 if 0 a 1 the limits reverse since ax 1 a x 8 Exercise 4 1 29 Differentiate y xe2x Exercise 4 1 51 If f x e 2x find f 8 x 9 y xe2x Use the product rule the chain rule and the bad u sic exponential derivative formula du e eu d 2x e dx e2x 2xe2x y e2x x 10 y e 2x What is its 8th derivative Every time I differentiate I just get a factor 2 So f 8 x 2 8 e 2x 256e 2x Exponential and logarithmic functions are inverses of each other so at this point we digress to discuss inverse functions in general 11 Remember the formula for the volume of a sphere V 43 r 3 f r Solve it to get a formula for the radius as a function of the volume r 3 3V r g v f 1 V 4 The functions f and g are inverse to each other 1 Note that f 1 does not mean in this conf text 12 In a physical application like that the variables have natural names r and V because they represent physical quantities But in generic math we usually write y f x Should we then write x f 1 y g y or y f 1 x g x Both Stewart and Maple insist on the latter so that x is always the independent variable and y the dependent one To avoid confusion I try to use neutral letters say u f w and w f 1 u 13 In obtaining an inverse for a given function two complications may arise the domain problem and the branch problem You know that they both occur for the square root the inverse of a very simple function u w2 f w 1 f 1 u may not be defined for some values of u Example If u 0 then u is not equal to w2 for any real w So the domain of the square root function contains only nonnegative numbers 14 2 To make f 1 single valued as required by the definition of a function we may need to exclude some values of w that satisfy f w u We must choose just one w for each u This is called choosing a branch of the inverse function Example We define u to be the nonnegative square root The graph of u is the right hand half of the graph of w2 flipped over so that the u and w positive axes are interchanged 15 A condition that assures that the branch problem does not arise is the horizontal line test which says that the graph of the inverse will pass the vertical line test without our having to throw part of the graph away The original function is then called one to one We don t have two points w mapping into the same u The domain problem does not arise if the function is onto R that is every u R appears as f w for some w which will be f 1 u 16 Exercise 4 2 13 Show that f x 4x 7 is one to one and find its inverse function 17 We need to solve y 4x 7 That is elementary y 7 1 x y 7 4 4 4 This makes sense for all y so the function f is onto And the solution for x is unique so f is one to one Alternatively you could sketch the graph of f and see that every horizontal line crosses it exactly once Therefore we could write f 1 y 18 y 4 7 4 but to match the textbook s notation we must switch the variables f 1 x x 7 4 4 Finally we get to the main point What is the derivative of f 1 x We re assuming we know the derivative of f 19 This question could have been answered back in the section on implicit differentiation To say that w f 1 u g u is to say that u f w and that a branch has been chosen if necessary So 1 du dw f w f w g u du du So g u f w 1 20 1 f f 1 u Back in our original example we might want to write this relation as dr dV dV dr 1 Here the exponent 1 does mean to take the reciprocal one over the number Like the chain rule dV dV dC dr dC dr 21 the theorem looks trivial in the Leibniz notation But be careful where the functions are evaluated We need f f 1 u not f u Exercise 4 2 31 Suppose g f 1 and f 4 5 f 4 g 5 22 2 3 Find We want g 5 Since g is the inverse of f g is the reciprocal of f So we look for f …
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