7 INVERSE FUNCTIONS INVERSE FUNCTIONS 7 3 The

7 INVERSE FUNCTIONS

INVERSE FUNCTIONS 7. 3* The Natural Exponential Function In this section, we will learn about: The natural exponential function and its properties.

THE NATURAL EXPONENTIAL FUNCTION Since ln is an increasing function, it is one-to-one and, therefore, has an inverse function. § We denote this by: exp

NATURAL EXP. FUNCTION Notation 1 Thus, according to the definition of an inverse function,

NATURAL EXP. FUNCTION Equations 2 The cancellation equations are: § In particular, we have: exp(0) = 1 since ln 1 = 0 exp(0) = e since ln e = 1

NATURAL EXP. FUNCTION We obtain the graph of y = exp x by reflecting the graph of y = ln x about the line y = x. § The domain of exp is the range of ln, that is, (-∞, ∞). § The range of exp is the domain of ln, that is, (0, ∞). © Thomson Higher Education

NATURAL EXP. FUNCTION If r is any rational number, the third law of logarithms gives: ln(er ) = r ln e = r § Therefore, by Notation 1, exp(r) = er § Thus, exp(x) = ex whenever x is a rational number.

NATURAL EXP. FUNCTION This leads us to define ex, even for irrational values of x, by the equation ex = exp(x) § In other words, for the reasons given, we define ex to be the inverse of the function ln x.

NATURAL EXP. FUNCTION Notation 3 In this equation, Notation 1 becomes:

NATURAL EXP. FUNCTION Equations 4 & 5 The cancellation Equations 2 become:

NATURAL EXP. FUNCTION E. g. 1—Solution 1 Find x if ln x = 5. § From Notation 3, we see that: ln x = 5 means § Therefore, x = e 5 = x

NATURAL EXP. FUNCTION E. g. 1—Solution 2 Start with the equation ln x = 5 § Then, apply the exponential function to both sides of the equation: e ln x = e 5 § However, Equation 4 says that: e ln x = x § Therefore, x = e 5.

NATURAL EXP. FUNCTION Example 2 Solve the equation e 5 -3 x = 10. § We take natural logarithms of both sides of the equation and use Equation 5:

NATURAL EXP. FUNCTION Example 2 § Since the natural logarithm is found on scientific calculators, we can approximate the solution to four decimal places: x ≈ 0. 8991

NATURAL EXP. FUNCTION The exponential function f(x) = ex is one of the most frequently occurring functions in calculus and its applications. § So, it is important to be familiar with its graph. © Thomson Higher Education

NATURAL EXP. FUNCTION It is also important to be familiar with its properties. § These follow from the fact that it is the inverse of the natural logarithmic function.

PROPERTIES Properties 6 The exponential function f(x) = ex is an increasing continuous function with domain and range (0, ∞). § Thus, e x > 0 for all x

PROPERTIES Properties 6 Also, § So, the x-axis is a horizontal asymptote of f(x) = ex.

PROPERTIES Example 3 Find: § We divide numerator and denominator by e 2 x: § We have used the fact that t = -2 x → -∞ as x → ∞ and so

PROPERTIES Now, we verify that f(x) = ex has the properties expected of an exponential function.

LAWS OF EXPONENTS Laws 7 If x and y are real numbers and r is rational, then 1. e x+y = e xe y 2. e x-y = e x/e y 3. (e x)r = e rx

LAW 1 OF EXPONENTS Proof Using the first law of logarithms and Equation 5, we have: § Since ln is a one-to-one function, it follows that: e xe y = e x+y

LAWS 2 AND 3 OF EXPONENTS Laws 2 and 3 are proved similarly. § See Exercises 95 and 96. § As we will see in Section 7. 4*, Law 3 actually holds when r is any real number.

DIFFERENTIATION Formula 8 The natural exponential function has the remarkable property that it is its own derivative.

DIFFERENTIATION Formula 8—Proof The function y = ex is differentiable because it is the inverse function of y = ln x. § We know this is differentiable with nonzero derivative. To find its derivative, we use the inverse function method.

DIFFERENTIATION Formula 8—Proof Let y = ex. Then, ln y = x and, differentiating this latter equation implicitly with respect to x, we get:

DIFFERENTIATION The geometric interpretation of Formula 8 is: § The slope of a tangent line to the curve y = ex at any point is equal to the y-coordinate of the point. © Thomson Higher Education

DIFFERENTIATION This property implies that the exponential curve y = ex grows very rapidly. § See Exercise 100. © Thomson Higher Education

DIFFERENTIATION Example 4 Differentiate the function y = e tan x. § To use the Chain Rule, we let u = tan x. § Then, we have y = e u. § Hence,

DIFFERENTIATION Formula 9 In general, if we combine Formula 8 with the Chain Rule, as in Example 4, we get:

DIFFERENTIATION Example 5 Find y’ if y = e-4 x sin 5 x. § Using Formula 9 and the Product Rule, we have:

DIFFERENTIATION Example 6 Find the absolute maximum value of the function f(x) = xe -x. § We differentiate to find any critical numbers: f’(x) = xe-x(-1) + e-x(1) = e-x(1 – x)

DIFFERENTIATION Example 6 Since exponential functions are always positive, we see that f ’(x) > 0 when 1 – x > 0, that is, when x < 1. Similarly, f ’(x) < 0 when x > 1.

DIFFERENTIATION Example 6 By the First Derivative Test for Absolute Extreme Values, f has an absolute maximum value when x = 1. § The absolute maximum value is:

DIFFERENTIATION Example 7 Use the first and second derivatives of f(x) = e 1/x, together with asymptotes, to sketch its graph. § Notice that the domain of f is {x | x ≠ 0}. § Hence, we check for vertical asymptotes by computing the left and right limits as x → 0.

DIFFERENTIATION Example 7 As x→ 0+, we know that t = 1/x → ∞. So, § This shows that x = 0 is a vertical asymptote.

DIFFERENTIATION Example 7 As x → 0 -, we know that t = 1/x →-∞. So, As x→ ∞, we have 1/x → 0. So, § This shows that y = 1 is a horizontal asymptote.

DIFFERENTIATION Example 7 Now, let’s compute the derivative. § The Chain Rule gives: § Since e 1/x > 0 and x 2 > 0 for all x ≠ 0, we have f ’(x) < 0 for all x ≠ 0. § Thus, f is decreasing on (-∞, 0) and on (0, ∞). § There is no critical number, so the function has no maximum or minimum.

DIFFERENTIATION Example 7 The second derivative is:

DIFFERENTIATION Example 7 Since e 1/x > 0 and x 4 > 0, we have f ”(x) > 0 when x > -½(x ≠ 0) and f ”(x) < 0 when x < -½. § So, the curve is concave downward on (-∞, -½) and concave upward on (-½, 0) and on (0, ∞). § The inflection point is (-½, e-2).

DIFFERENTIATION Example 7 To sketch the graph of f, we first draw the horizontal asymptote y = 1 (as a dashed line) in a preliminary sketch. § We also draw the parts of the curve near the asymptotes. © Thomson Higher Education

DIFFERENTIATION Example 7 These parts reflect the information concerning limits and the fact that f is decreasing on both (-∞, 0) and (0, ∞). § Notice that we have indicated that f(x) → 0 as x → 0 - even though f(0) does not exist. © Thomson Higher Education

DIFFERENTIATION Example 7 We finish the sketch by incorporating the information concerning concavity and the inflection point. © Thomson Higher Education

DIFFERENTIATION Example 7 We check our work with a graphing device. © Thomson Higher Education

INTEGRATION Formula 10 As the exponential function y = ex has a simple derivative, its integral is also simple:

INTEGRATION Example 8 Evaluate: § We substitute u = x 3. § Then, du = 3 x 2 dx. § So, and

INTEGRATION Example 9 Find the area under the curve y = e-3 x from 0 to 1. § The area is: