Calculus I Study Guide: Functions, Limits, Derivatives, and Applications

1. Exponential and Logarithmic Functions

1.1 Exponential Equations

Exponential functions have the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$. Solving exponential equations often involves expressing both sides with the same base or using logarithms.

• Key Point: To solve $a^{f(x)} = a^{g(x)}$, set $f(x) = g(x)$.

• Key Point: If bases differ, take logarithms of both sides.

• Example: Solve $2^{x+1} = 8$.

• Solution: $8 = 2^3 \implies x+1 = 3 \implies x = 2$.

1.2 Logarithmic Functions

The logarithmic function $\log_a(x)$ is the inverse of the exponential function $a^x$. It satisfies $a^{\log_a(x)} = x$ and $\log_a(a^x) = x$.

• Key Point: $\log_a(xy) = \log_a(x) + \log_a(y)$

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

• Key Point: $\log_a(x^r) = r\log_a(x)$

• Example: Find $\log_2(8) = 3$ because $2^3 = 8$.

2. Inverse Trigonometric Functions

2.1 Definitions and Properties

Inverse trigonometric functions, such as $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$, return the angle whose trigonometric value is $x$. Their domains and ranges are restricted to ensure they are functions.

• Key Point: $\arcsin(x)$: domain $[-1,1]$, range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

• Key Point: $\arccos(x)$: domain $[-1,1]$, range $[0, \pi]$

• Key Point: $\arctan(x)$: domain $(-\infty, \infty)$, range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

• Example: $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$

3. Trigonometric Equations and Identities

3.1 Solving Trigonometric Equations

Trigonometric equations can often be solved using identities and by considering the periodicity of the functions.

• Key Point: Use identities such as $\sin^2x + \cos^2x = 1$ and double-angle formulas.

• Example: Solve $\sin(2x) = 0$ for $x$ in $[0, 2\pi]$.

• Solution: $2x = n\pi \implies x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$

4. Trigonometric Functions as Algebraic Functions

4.1 Expressing Trigonometric Functions Algebraically

Trigonometric functions can be expressed in terms of algebraic functions using right-triangle relationships or identities.

• Key Point: $\sin(\arccos(x)) = \sqrt{1 - x^2}$

• Key Point: $\tan(\arcsin(x)) = \frac{x}{\sqrt{1 - x^2}}$

• Example: Express $\sin(\arccos(x))$ in terms of $x$.

5. Inverse Functions and Their Properties

5.1 One-to-One and Inverse Functions

A function is one-to-one if each output is produced by exactly one input. Only one-to-one functions have inverses that are also functions.

• Key Point: The inverse function $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$.

6. Limits and Continuity

6.1 Computing Limits

Limits describe the behavior of a function as the input approaches a certain value. Techniques include direct substitution, factoring, rationalization, and L'Hospital's Rule for indeterminate forms.

• Key Point: $\lim_{x \to a} f(x) = L$ means $f(x)$ approaches $L$ as $x$ approaches $a$.

• Key Point: Indeterminate forms include $\frac{0}{0}$ and $\frac{\infty}{\infty}$.

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

6.2 Continuity

A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$. Piecewise functions require checking continuity at the boundaries.

• Key Point: Discontinuities can be removable, jump, or infinite.

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$ is discontinuous at $x = 1$ but the discontinuity is removable.

7. The Intermediate Value Theorem (IVT)

7.1 Statement and Application

The IVT states that if $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = N$.

• Key Point: Used to show the existence of roots in an interval.

• Example: If $f(1) = -2$ and $f(3) = 5$, then $f(x) = 0$ for some $x \in (1, 3)$.

8. The Derivative: Definition and Computation

8.1 Definition of the Derivative

The derivative of $f$ at $x = a$ is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, representing the instantaneous rate of change or the slope of the tangent line at $x = a$.

• Key Point: The derivative may not exist at points of discontinuity or sharp corners.

• Example: For $f(x) = x^2$, $f'(x) = 2x$.

8.2 Differentiability and Continuity

If a function is differentiable at a point, it is also continuous there, but the converse is not always true.

• Key Point: Differentiability implies continuity, but not vice versa.

• Example: $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$.

9. Piecewise Functions: Continuity and Differentiability

9.1 Analyzing Piecewise Functions

To determine continuity and differentiability for piecewise functions, check the value and derivative from both sides at the boundary points.

• Key Point: Set the left and right limits (and derivatives) equal at the boundary to ensure continuity (and differentiability).

• Example: For $f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x-1 & x > 1 \end{cases}$, check at $x=1$.

10. Asymptotes of Rational Functions

10.1 Horizontal and Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. Horizontal asymptotes are determined by the degrees of the numerator and denominator.

• Key Point: If degree numerator < degree denominator, horizontal asymptote at $y=0$.

• Key Point: If degrees are equal, horizontal asymptote at $y=\frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}}$.

• Example: $f(x) = \frac{2x^2+1}{x^2-3}$ has a horizontal asymptote at $y=2$.

11. Tangent Lines and Applications of the Derivative

11.1 Equation of the Tangent Line

The tangent line to $y = f(x)$ at $x = a$ has the equation $y = f(a) + f'(a)(x - a)$.

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

• Example: For $f(x) = x^2$ at $x=1$, tangent line is $y = 1 + 2(x-1)$.

12. Derivative Rules: Sum, Product, and Quotient

12.1 Basic Derivative Rules

• Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

• Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Example: $\frac{d}{dx}(x^2 \sin x) = 2x \sin x + x^2 \cos x$

12.2 Derivative of Inverse Functions

If $f$ and $f^{-1}$ are inverses, then $[f^{-1}]'(y) = \frac{1}{f'(x)}$ where $y = f(x)$.

• Key Point: The derivative of the inverse at $y$ is the reciprocal of the derivative of the original function at $x$.

• Example: If $f(2) = 3$ and $f'(2) = 5$, then $[f^{-1}]'(3) = \frac{1}{5}$.

13. Table: Domains and Ranges of Inverse Trigonometric Functions

Function	Domain	Range
$\arcsin(x)$	$[-1, 1]$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\arccos(x)$	$[-1, 1]$	$[0, \pi]$
$\arctan(x)$	$(-\infty, \infty)$	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

14. Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, but function not defined or not equal to limit	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Jump	Left and right limits exist but are not equal	Piecewise function with different values at a point
Infinite	Function approaches infinity at a point	$f(x) = \frac{1}{x}$ at $x=0$

Additional info: This study guide covers the foundational topics for a first exam in Calculus I, including functions, limits, continuity, derivatives, inverse functions, and applications such as tangent lines and asymptotes. It is structured to provide both conceptual understanding and practical problem-solving skills.