Calculus 220 section 4 1 4 2 Exponential Functions Including e x notes by Tim Pilachowski Take a piece of paper and fold it in half You ve doubled the number of layers from 1 to 2 Fold it in half again and you ve once again doubled the layers from 2 to 4 After the next fold you d have 8 layers After the next 16 Then 32 Then 64 128 256 512 1024 etc After just 10 foldings we have the paperback edition of just the first book of Lord of the Rings that we re attempting to fold in half y 2x When x is the base we have a power function When x is the exponent we have an exponential function The scenario above illustrates the exponential function y 2 x If we compare the graph of y x 2 to the graph of y 2 x we can see that for positive values of x the exponential function grows much more quickly than the power function Thus our difficulty in folding a piece of paper in half successive times We ll be interested in finding a way to describe the slope of an exponential graph i e a way to find its derivative y x2 Exponential functions have many applications because they model many kinds of growth and shrinking e g populations bank deposits radioactive decay Example A You deposit 100 into a certificate of deposit which pays 5 each year on the balance current at the time Find an equation to describe the growth of your money Answer A t 100 1 05 t All of the usual properties apply to exponential functions x b b y b x y bx by b x y 1 by b 0 y b y b x y b xy x x a b ab Examples B We can use these properties to simplify expressions and solve equations 3 Simplify 2 2 x 2 x 1 Answer 2 3 x 3 2 3 2 3 x Simplify 2 x 1 8 x 3 Answer 2 4 x 8 2 8 2 4 x Simplify Simplify 2 x 1 8 x 3 1 2 2x Answer 2 2 x 10 10 2 x 12 6 x 1 x Answer 2 2 x ax x a x b b We ll use functions with the form y Cb kx to model all kinds of applications Solve 2 x 2 3 16 Answer x 1 Solve 2 x 3 x 1 108 Answer x 2 Solve 3 x 271 Solve 5 10 Answer x 2 1 2 x 1 Answer x 6 Functions with the basic form y b x are actually a family of functions We ll consider only values for b that are positive Negative values of b are extremely problematic since even and odd values of x would cause y to fluctuate between positive and negative Consider the functions y 10 x y 5 x y 3 x y 2 x y 1 1 x pictured in the graph to the right Note first the similarities b 0 1 for all values of b 0 so 0 1 makes a good reference point Each of the basic exponential functions has a horizontal asymptote y 0 The graphs also have similar shape the major difference is slope of the curve at specific values of x Note that at x 0 slope of y 10 x is steepest slope of y 1 1x is most shallow Our determination of first derivative will have to reflect this The value of the first derivative at a specific point on a curve can be estimated by f x h f x considering a series of secant lines and using the difference quotient lim h 0 h The text does y 2 x you ll do y 3 x for homework so in class we ll do y 5 x in a process very similar to that of section 1 3 tangent at x 0 tangent at x 1 First consider y 5 x at x 0 where y 1 The slope of the secant line connecting 0 1 to another point on the 5 0 h 5 0 5 h 1 As our second point approaches 0 1 the slope of the h h 5h 1 secant line approaches the slope of the tangent line i e slope of the tangent lim The table below h 0 h provides results from successively smaller values of h curve is given by the formula 5 0 5 1 1 1 1 1 1 1 1 1 5 h 1 0 1 0 01 0 001 0 0001 0 00001 0 000001 0 0000001 0 00000001 h 5 5 1 174618943 1 016224591 1 001610734 1 000160957 1 000016095 1 000001609 1 000000161 1 000000016 h 1 4 0 174618943 0 016224591 0 001610734 0 000160957 0 000016095 0 000001609 0 000000161 0 000000016 h 1 h 4 1 746189431 1 622459127 1 610733753 1 609567434 1 609450864 1 609439208 1 609438043 1 609437916 The slope of the tangent to y 5 x at x 0 and therefore the first derivative of y 5 x at x 0 is approximately 1 61 This is an estimate and is not exact At some later point we ll identify the exact expression of the first derivative of y 5 x at x 0 as ln 5 Next we move to an arbitrary point x 5 x on the graph of y 5 x The slope of the secant line connecting x 5 x to another point on the curve is given by the formula 5 x h 5 x h which can be simplified using the 5 x h 5 x 5 x 5 h 5 x 5 x 5 h 1 As our second point approaches x 5 x the h h h slope of the secant line approaches the slope of the tangent line i e slope of the tangent x d x x 5 x 5h 1 5h 1 x x lim lim 5 5 5 1 61 5 x 5 5 h 0 x 0 x 0 h dx h 0 h properties of exponents Recall now a point made above that in the family of exponential functions the slope of the tangent at x 0 and therefore the first derivative at x 0 ranges from very shallow to very steep Somewhere in that family must be a base which has a first derivative equal to exactly 1 at x 0 This number is e Euler s number Like or 2 e is an irrational number The value of e is approximately 2 7 One decimal place will be sufficient for our purposes The corresponding function y e x is called the natural exponential function dy y 0 1 we can follow a procedure similar dx x 0 to that used above to get a result that is quite interesting If we consider an arbitrary point x e x on the graph of y e x the slope of the secant line Given that for y e x connecting x e x to …