Business Calculus Study Guide: Applications of Derivatives, Exponential & Logarithmic Functions, and First Derivative Test

Section 1.5 Applications of Derivatives

Finding the Slope of the Tangent Line

The derivative of a function at a particular point gives the slope of the tangent line to the graph at that point. This is a fundamental concept in calculus, used to analyze instantaneous rates of change.

• Key Point: The slope of the tangent line at $x = a$ is $f'(a)$.

• Example: For $f(x) = x^2$, the slope at $x = 3$ is $f'(3) = 2 \times 3 = 6$.

Equation of the Tangent Line

The equation of the tangent line to $f(x)$ at $x = a$ can be written using the point-slope form.

• Formula: $y = f'(a)(x - a) + f(a)$

• Example: For $f(x) = x^2$ at $x = 3$, $y = 6(x - 3) + 9$

Instantaneous vs. Average Rate of Change

The instantaneous rate of change is given by the derivative, while the average rate of change is calculated over an interval.

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Instantaneous Rate of Change: $f'(a)$

• Example: For $f(x) = x^2$ from $x = 2$ to $x = 4$, average rate is $\frac{16 - 4}{4 - 2} = 6$; instantaneous at $x = 2$ is $f'(2) = 4$.

Section 1.6 Product and Quotient Rules

Product Rule

The Product Rule is used to differentiate products of two functions.

• Formula: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Example: If $u(x) = x^2$, $v(x) = \sin x$, then $\frac{d}{dx}[x^2 \sin x] = 2x \sin x + x^2 \cos x$

Quotient Rule

The Quotient Rule is used to differentiate quotients of two functions.

• Formula: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Example: If $u(x) = x^2$, $v(x) = x + 1$, then $\frac{d}{dx}\left[\frac{x^2}{x+1}\right] = \frac{2x(x+1) - x^2(1)}{(x+1)^2}$

Section 1.7 Chain Rule

Chain Rule for Composite Functions

The Chain Rule is used to differentiate composite functions, where one function is nested inside another.

• Formula: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

• Example: For $f(x) = (3x + 2)^5$, $f'(x) = 5(3x + 2)^4 \cdot 3 = 15(3x + 2)^4$

Section 2.1 Exponential and Logarithmic Functions

Converting Between Exponential and Logarithmic Equations

Exponential and logarithmic equations are inverses of each other. Understanding their relationship is essential for solving equations and modeling growth or decay.

• Exponential Form: $a^x = y$

• Logarithmic Form: $x = \log_a y$

• Example: $2^3 = 8$ is equivalent to $3 = \log_2 8$

Properties of Logarithms

Logarithms have several important properties that simplify expressions and solve equations.

• Sum Property: $\log_a (xy) = \log_a x + \log_a y$

• Difference Property: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• Power Property: $\log_a (x^k) = k \log_a x$

Solving Exponential and Logarithmic Equations

• Exponential Equations: Use logarithms to solve for the exponent.

• Logarithmic Equations: Convert to exponential form to solve for the variable.

• Example: Solve $e^x = 5$; $x = \ln 5$

• Example: Solve $\ln x = 2$; $x = e^2$

Section 2.2 Derivatives of Exponential Functions with Base e

Derivative of Exponential Functions

The derivative of the natural exponential function $e^x$ is itself. For more general exponential functions, the chain rule may be needed.

• Formula: $\frac{d}{dx}[e^x] = e^x$

• Formula: $\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)$

• Example: $\frac{d}{dx}[e^{3x}] = 3e^{3x}$

Equation of Tangent Line to Exponential Function

• Formula: $y = f'(a)(x - a) + f(a)$, where $f(x)$ is exponential.

• Example: For $f(x) = e^x$ at $x = 0$, $y = 1(x - 0) + 1 = x + 1$

Section 2.3 Derivatives of Natural Log Functions

Derivative of Logarithmic Functions

The derivative of the natural logarithm function $\ln x$ is a fundamental result in calculus.

• Formula: $\frac{d}{dx}[\ln x] = \frac{1}{x}$

• Formula: $\frac{d}{dx}[\ln g(x)] = \frac{g'(x)}{g(x)}$

• Example: $\frac{d}{dx}[\ln(x^2 + 1)] = \frac{2x}{x^2 + 1}$

Equation of Tangent Line to Logarithmic Function

• Formula: $y = f'(a)(x - a) + f(a)$, where $f(x)$ is logarithmic.

• Example: For $f(x) = \ln x$ at $x = 1$, $y = 1(x - 1) + 0 = x - 1$

Section 2.2/2.3 Applications of Derivatives: Growth, Decay, and Other Models

Exponential Growth and Decay Models

Exponential functions model processes where the rate of change is proportional to the current value, such as population growth or radioactive decay.

• Growth Model: $y = y_0 e^{kt}$, where $k > 0$

• Decay Model: $y = y_0 e^{-kt}$, where $k > 0$

• Derivative: $\frac{dy}{dt} = k y_0 e^{kt} = k y$

• Application: Use the original function for predicting future values; use the derivative for instantaneous rate of change.

• Example: If $y_0 = 100$, $k = 0.05$, after $t = 10$ years: $y = 100 e^{0.5}$

Logarithmic Models

Logarithmic functions are used in models such as pH in chemistry, sound intensity, and decibel scales.

• Formula: $y = a \ln x + b$

• Derivative: $\frac{dy}{dx} = \frac{a}{x}$

Section 3.1 First Derivative Test

Finding Critical Numbers

Critical numbers are values of $x$ where $f'(x) = 0$ or $f'(x)$ does not exist. These are candidates for local maxima or minima.

• Step: Solve $f'(x) = 0$ or find where $f'(x)$ is undefined.

• Example: For $f(x) = x^3 - 3x^2 + 2$, $f'(x) = 3x^2 - 6x$, set $3x^2 - 6x = 0$.

First Derivative Test (Sign Chart)

The First Derivative Test uses the sign of $f'(x)$ around critical numbers to determine relative extrema and intervals of increase/decrease.

• Relative Extrema: If $f'(x)$ changes from positive to negative at $x = c$, $f$ has a local maximum at $c$. If $f'(x)$ changes from negative to positive, $f$ has a local minimum.

• Intervals of Increase/Decrease: $f$ is increasing where $f'(x) > 0$, decreasing where $f'(x) < 0$.

• Example: For $f(x) = x^3 - 3x^2 + 2$, critical points at $x = 0, 2$; use sign chart to determine behavior.

Test Point	$f'(x)$	Function Behavior
x < 0	+	Increasing
0 < x < 2	-	Decreasing
x > 2	+	Increasing

Additional info: These notes cover key Business Calculus topics, including differentiation techniques, applications to exponential and logarithmic functions, and the use of derivatives to analyze function behavior. Practice problems are suggested for further study.