Math 152 Lab 2 - Inverse, Natural Log, and Exponential FunctionsIntroductionIn this notebook, we will explore two functions that have very important Calculus properties: the natural log function ln(x) andthe exponential function ex. As we shall see, they are related to each other as inverse functions. We shall also look more in depthinto this inverse function relationship and how it relates to Calculus.The Exponential FunctionMathematica distinguishes the special number e from the letter e. There are a number of ways of obtaining the number e, andthey are all equivalent.The first and easiest way to get e is to remember that all Mathematica functions and constants start with capital letters:[email protected] could also use palettes, or the escaped shortcut to obtain‰:N@D2.71828You can also use the builtin function Exp and evaluate it at 1 to get the constant e:N@[email protected] here on in, this worksheet will use Exp. Let's see what the graph of the exponential function looks like:Plot@Exp@xD, 8x, − 4, 4<D-4 -2 2 45101520252 Lab 2 - InverseNaturalLogandExponentialFunctions.nbOne question that we may ask is, are there any asymptotes? We can check this with limits:Limit@Exp@xD,x→−InfinityDLimit@Exp@xD,x→ InfinityD0∞The exponential function has a number of special properties, the most relevant of which is that it is the only function which is itsown derivative:Exp@xDD@Exp@xD,xDExp@xD == D@Exp@xD,xDxxTrueThe exponential function also relates addition and multiplication of the exponent to multiplication and exponentiation of thefunction itself:Exp@x ∗ yD == Exp@xD^yExp@x + yD == Exp@xD ∗ Exp@yDxy HxLyTrueThere are a number of special commands for algebraic reduction that Mathematica does not perform by default. The one thatfigures out the relationships with the Exponential function is called PowerExpand:Exp@x ∗ yD == PowerExpand@Exp@xD ^yDTrueIf you forget which special command to use, you can also show they are equal with plots:Lab 2 - InverseNaturalLogandExponentialFunctions.nb 3Plot3D@Exp@x ∗ yD − Exp@xD^y, 8x, −10, 10<, 8y, − 10, 10<DThe Natural Logarithm FunctionThe Natural Logarithm function is much more straightforward than the exponential function. There is only the single commandLog. Using Log with only one argument gives the natural log:N@Log@[email protected] using Log with two arguments allows you to specify the base. Note that the base comes first when you use two arguments.N@Log@2, Exp@1DDDN@Log@Exp@1D,2DD1.44270.693147Let' s see what the graph of the Natural Logarithm looks like :4 Lab 2 - InverseNaturalLogandExponentialFunctions.nbPlot@Log@xD, 8x, − 4, 4<D-4 -2 2 4-2-11Once aagain we can ask if there are any asymptotes :Limit@Log@xD,x→ 0DLimit@Log@xD,x→ InfinityD−∞∞The Natural Log' s special property is that its derivative is the reciprocal function f HxL=1x.D@Log@xD,xD1xIt also relates addition, multiplication, and exponentiation just like the exponential function. Just like before, Mathematica'sdefault solver doesn't check for this and you have to use PowerExpand :Log@xD + Log@yD == PowerExpand@Log@x ∗ yDDy ∗ Log@xD == PowerExpand@Log@x^yDDTrueTrueInverse FunctionsLet us return to an example from the Natural Log section:N@Log@[email protected] number "1" was fed into the exponential function to obtain the number "e." This was then fed into the natural log functionand "1" came back out. This is true for every number in the exponential function's domain. The reverse is also true (but only forthe numbers in the natural log function's domain). Be aware that Mathematica does not do this domain checking.Lab 2 - InverseNaturalLogandExponentialFunctions.nb 5N@Log@Exp@5DDDN@Log@Exp@− 3DDDN@Exp@Log@− 3DDDN@Exp@Log@1DDDN@Exp@Log@5DDD5.− 3.− 3.1.5.We can illustrate this result with a plot:Plot@8Log@xD, x, Exp@xD<, 8x, − 4, 4<,PlotRange → 88− 4, 4<, 8−4, 4<<, AspectRatio → 1D-4 -2 2 4-4-224From the plot, the two functions are reflections of each other about the line y=x. This is called the inverse relationship. Manyother (but not all) functions have inverses. We can find inverses the same way that it is done by hand, solving algebraically:6 Lab 2 - InverseNaturalLogandExponentialFunctions.nbClear@yDf@x_D := 3 ∗ Log@Sqrt@x + 4DDyofx = Solve@y  f@xD,xD@@1DDfinverse@y_D = x ê. yofxPlot@8f@xD, x, finverse@xD<, 8x, − 6, 6<, AspectRatio → 1D9x →−4 +2yê3=− 4 +2yê3-6 -4 -2 2 4 6-10-5510There is an InverseFunction command, but it only works on builtin functions like Sin:f@x_D := 3 ∗ Log@Sqrt@x + 4DDInverseFunction@fDInverseFunction@SinD− 4 +2 13&ArcSinWe can tell whether a function is invertible from its plot. It must pass the horizontal line test. Consider the following two plots:Lab 2 - InverseNaturalLogandExponentialFunctions.nb 7Plot@x^2, 8x, − 4, 4<DPlot@2 ∗ x, 8x, − 4, 4<D-4 -2 2 451015-4 -2 2 4-55The second function is invertible, but the first one is not. If a function is not invertible, we can make it invertible by restricting thedomain. Consider the first function again, but under the assumption of x always being nonnegative:8 Lab 2 - InverseNaturalLogandExponentialFunctions.nbPlot@8x^2, Sqrt@xD,x<, 8x, 0, 4<, PlotRange → 880, 4<, 80, 4<<, AspectRatio → 1D0 1 2 3 401234Knowledge of a function's inverse can give us information about the Calculus properties of the function. This is accomplished bythe formula ddxf-1HxL =1f ' I f-1HxLM. Let's show what is going on with this formula with a plot:Lab 2 - InverseNaturalLogandExponentialFunctions.nb 9Plot@8x^2, Sqrt@xD, x, [email protected],x êH2 [email protected] + [email protected] − [email protected]ê2, x ∗ 2 [email protected] − 1.5<,8x, 0, 2<, PlotRange → 880, 2<, 80, 2<<, AspectRatio → 1, PlotStyle →88Red<, 8Blue<, 8Brown, Dashed<, 8Darker@GreenD, Dotted<, 8Cyan<, 8Gray<<D0.0 0.5 1.0 1.5 2.00.00.51.01.52.0The function x (the blue line of the plot above) at x = 1.5 is 1.5 (the dotted green line). If we go to the spot where theinverse of x , x2 (the red line), is equal to 1.5 , then the function representing the tangent line to x2 at this point (the grayline) is the inverse of the function representing the tangent line to x at x = 1.5 (the cyan line). Because these two tangent linesare inverses, their slopes are reciprocals. This means that the derivatives at the points where x and x2 are both equal to 1.5are reciprocals.If we use other inverse pairs, or other points, the formula says