Chapter 6: Inverse Functions - Exponential, Logarithmic, and Inverse Trigonometric Functions

Section 6.1: Inverse Functions and their Derivatives

• Definition of a One-to-One Function: A function $f$ is called a one-to-one function if it never takes the same value twice; that is, $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$.
• Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once.
• Inverse Functions Definition: Let $f$ be a one-to-one function with domain $A$ and range $B$. Then its inverse function $f^{-1}$ has domain $B$ and range $A$ and is defined by $f^{-1}(y) = x \iff f(x) = y$ for any $y$ in $B$.     - Domain and Range Relationship:         - Domain of $f^{-1} = \text{Range of } f$         - Range of $f^{-1} = \text{Domain of } f$
• Properties of Inverse Functions:     - Cancellation Equations:         - $f^{-1}(f(x)) = x$ for every $x$ in $A$.         - $f(f^{-1}(x)) = x$ for every $x$ in $B$.     - Graphs: The graph of $f^{-1}$ is obtained by reflecting the graph of $f$ about the line $y = x$.
• How to Find the Inverse Function of a One-to-One Function:     - Step 1: Write $y = f(x)$.     - Step 2: Solve this equation for $x$ in terms of $y$ (if possible).     - Step 3: To express $f^{-1}$ as a function of $x$, interchange $x$ and $y$. The resulting equation is $y = f^{-1}(x)$.     - Example: Find the inverse of $y = x^3 + 2$.         - $x = y^3 + 2$         - $x - 2 = y^3$         - $y = \sqrt[3]{x - 2}$         - Hence, $f^{-1}(x) = (x - 2)^{\frac{1}{3}}$.
• Derivatives of Inverse Functions Theorem: If $f$ is a one-to-one differentiable function with inverse function $f^{-1}$ and $f'(f^{-1}(a)) \neq 0$, then the inverse function is differentiable at $a$ and is given by:     - $(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}$
• Example of Derivative Calculation:     - Given $f(y) = 2y + \cos(y)$, find $(f^{-1})'(1)$.     - Step 1: Find $y$ such that $f(y) = 1$. By inspection, if $y = 0$, then $2(0) + \cos(0) = 0 + 1 = 1$. Thus, $f^{-1}(1) = 0$.     - Step 2: Find $f'(y) = \frac{d}{dy}(2y + \cos(y)) = 2 - \sin(y)$.     - Step 3: Use the formula: $(f^{-1})'(1) = \frac{1}{f'(f^{-1}(1))} = \frac{1}{f'(0)} = \frac{1}{2 - \sin(0)} = \frac{1}{2}$.

Section 6.2: The Natural Logarithmic Function

• Definition: The natural logarithmic function is the function defined by $\ln(x) = \int_{1}^{x} \frac{1}{t} \, dt$ for $x > 0$.
• Laws of Logarithms: If $x$ and $y$ are positive numbers and $r$ is a rational number, then:     - $\ln(xy) = \ln(x) + \ln(y)$     - $\ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y)$     - $\ln(x^r) = r \ln(x)$     - Note: Terms like $\ln(x)$ or $\ln(y)$ are undefined if $x \leq 0$ or $y \leq 0$.
• Derivative of the Natural Logarithm:     - $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$     - General Form (Chain Rule): $\frac{d}{dx}(\ln(g(x))) = \frac{g'(x)}{g(x)}$
• Example Derivatives:     - Find $\frac{d}{dx} \ln\left( \frac{x+1}{\sqrt{x-2}} \right)$.     - Simplified using laws: $\ln(x+1) - \ln((x-2)^{\frac{1}{2}}) = \ln(x+1) - \frac{1}{2} \ln(x-2)$.     - Derivative: $\frac{1}{x+1} - \frac{1}{2(x-2)}$.
• Integrals involving the Natural Logarithm:     - $\int \frac{1}{x} \, dx = \ln|x| + C$     - $\int \tan(x) \, dx = \ln|\sec(x)| + C$     - Example with Substitution: Evaluate $\int \frac{2x}{x^2+1} \, dx$.         - Let $u = x^2 + 1$, then $du = 2x \, dx$.         - $\int \frac{1}{u} \, du = \ln|u| + C = \ln(x^2+1) + C$ (absolute value not needed as $x^2+1 > 0$).
• Logarithmic Differentiation: A method to simplify the differentiation of products, quotients, or powers.     - Steps:         1. Take natural logarithms of both sides of $y = f(x)$ and use laws of logarithms to expand the expression.         2. Differentiate implicitly with respect to $x$.         3. Solve for $y'$ and replace $y$ with $f(x)$.     - Example: Differentiate $y = \frac{x^{\frac{3}{4}} \sqrt{x^2+1}}{(3x+2)^5}$.         - $\ln(y) = \frac{3}{4} \ln(x) + \frac{1}{2} \ln(x^2+1) - 5 \ln(3x+2)$.         - $\frac{1}{y} y' = \frac{3}{4x} + \frac{x}{x^2+1} - \frac{15}{3x+2}$.         - $y' = y \left( \frac{3}{4x} + \frac{x}{x^2+1} - \frac{15}{3x+2} \right)$.

Section 6.3: The Natural Exponential Function

• Definition: The natural exponential function $f(x) = e^x$ is the inverse of the natural logarithmic function $\ln(x)$.     - $e^x = y \iff \ln(y) = x$
• Properties:     - $\ln(e^x) = x$ for $x \in \mathbb{R}$     - $e^{\ln(x)} = x$ for $x > 0$     - Limit properties: $\lim_{x \rightarrow -\infty} e^x = 0$ and $\lim_{x \rightarrow \infty} e^x = \infty$
• Differentiation:     - $\frac{d}{dx}(e^x) = e^x$     - Chain Rule: $\frac{d}{dx}(e^{u(x)}) = e^{u(x)} \cdot u'(x)$.
• Integration:     - $\int e^x \, dx = e^x + C$     - Example: Evaluate $\int x^2 e^{x^3} \, dx$.         - Let $u = x^3$, then $du = 3x^2 \, dx$.         - $\frac{1}{3} \int e^u \, du = \frac{1}{3} e^u + C = \frac{1}{3} e^{x^3} + C$.
• Area under the Curve: Find the area under $y = e^{-3x}$ from $0$ to $1$.     - $A = \int_{0}^{1} e^{-3x} \, dx = \left[ -\frac{1}{3} e^{-3x} \right]_{0}^{1} = \left( -\frac{1}{3} e^{-3} \right) - \left( -\frac{1}{3} e^0 \right) = \frac{1}{3} (1 - e^{-3})$.

Section 6.4: General Logarithmic and Exponential Functions

• General Exponential Function: For $b > 0$, we define $b^x = e^{x \ln(b)}$.
• Differentiation of General Exponential:     - $\frac{d}{dx}(b^x) = b^x \ln(b)$     - $\frac{d}{dx}(b^{g(x)}) = b^{g(x)} \ln(b) \cdot g'(x)$
• Exponential Graphs:     - If $b > 1$, $y = b^x$ is increasing. $\lim_{x \rightarrow -\infty} b^x = 0$ and $\lim_{x \rightarrow \infty} b^x = \infty$.     - If $0 < b < 1$, $y = b^x$ is decreasing. $\lim_{x \rightarrow -\infty} b^x = \infty$ and $\lim_{x \rightarrow \infty} b^x = 0$.
• General Exponential Integrals:     - $\int b^x \, dx = \frac{b^x}{\ln(b)} + C$
• The Power Rule versus the Exponential Rule:     1. Variable base, constant exponent: $\frac{d}{dx}(x^n) = nx^{n-1}$.     2. Constant base, variable exponent: $\frac{d}{dx}(b^x) = b^x \ln(b)$.     3. Variable base, variable exponent: Use logarithmic differentiation for functions like $y = (g(x))^{h(x)}$.         - Example: Differentiate $y = (\cos(x))^x$.         - $\ln(y) = x \ln(\cos(x))$         - $\frac{y'}{y} = \ln(\cos(x)) + x \cdot \frac{-\sin(x)}{\cos(x)}$         - $y' = (\cos(x))^x [\ln(\cos(x)) - x \tan(x)]$
• General Logarithmic Function: If $b > 0$ and $b \neq 1$, the inverse of $f(x) = b^x$ is $y = \log_b(x)$.     - $\log_b(x) = y \iff b^y = x$     - $\log_b(x) = \frac{\ln(x)}{\ln(b)}$     - Cancellation Equations:         - $b^{\log_b(x)} = x$         - $\log_b(b^x) = x$
• Derivative of Logarithm with Base $b$:     - $\frac{d}{dx}(\log_b(x)) = \frac{1}{x \ln(b)}$

Section 6.8: Indeterminate Forms and l’Hospital’s Rule

• L’Hospital’s Rule: Suppose $f$ and $g$ are differentiable and $g'(x) \neq 0$ near $a$ (except possibly at $a$). If the limit $\lim_{x \rightarrow a} \frac{f(x)}{g(x)}$ results in an indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:     - $\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}$ provided the right-side limit exists.     - Note: Differentiate the numerator and denominator separately; do not use the Quotient Rule.
• Indeterminate Forms:     - Direct Forms: $\frac{0}{0}, \frac{\infty}{\infty}$.     - Indeterminate Product: $0 \cdot \infty$. Rewrite as $\lim f \cdot g = \lim \frac{f}{1/g}$ to use L’Hospital.     - Indeterminate Difference: $\infty - \infty$. Combine terms or find a common denominator.     - Indeterminate Powers: $0^0, \infty^0, 1^\infty$. Use $y = f(x)^{g(x)}$, then $\ln(y) = g(x) \ln(f(x))$. Find the limit of $\ln(y)$ and then find the limit of $y = e^{\ln(y)}$.
• Example Calculations:     - Limit with ln: Evaluate $\lim_{x \rightarrow 1} \frac{\ln(x)}{x-1}$.         - Form $0/0$. Apply L'Hospital: $\lim_{x \rightarrow 1} \frac{1/x}{1} = 1$.     - Product Example: Evaluate $\lim_{x \rightarrow \infty} x \sin\left(\frac{\pi}{x}\right)$.         - Form $\infty \cdot 0$. Rewrite as $\lim_{x \rightarrow \infty} \frac{\sin(\pi/x)}{1/x}$.         - Let $t = 1/x$, as $x \rightarrow \infty$, $t \rightarrow 0^+$.         - $\lim_{t \rightarrow 0^+} \frac{\sin(\pi t)}{t} = \lim_{t \rightarrow 0^+} \frac{\pi \cos(\pi t)}{1} = \pi(1) = \pi$.     - Power Example: Evaluate $\lim_{x \rightarrow 0^+} x^x$.         - Form $0^0$. Let $y = x^x$, so $\ln(y) = x \ln(x)$.         - $\lim_{x \rightarrow 0^+} x \ln(x) = \lim_{x \rightarrow 0^+} \frac{\ln(x)}{1/x}$.         - Apply L'Hospital: $\lim_{x \rightarrow 0^+} \frac{1/x}{-1/x^2} = \lim_{x \rightarrow 0^+} -x = 0$.         - If $\lim \ln(y) = 0$, then $\lim y = e^0 = 1$.