Calculus I Midterm Study Guide: Functions, Limits, Derivatives, and Continuity

Functions and Their Properties

Solving for x in Logarithmic Equations

Logarithmic equations often require the use of properties of logarithms to isolate the variable. Recall that:

• Logarithm Properties: $\ln(ab) = \ln a + \ln b$, $\ln(a^k) = k \ln a$

• Solving Example: For $\ln(x - 2) = 3 \ln 2 + \ln(x)$, use properties to combine and solve for $x$.

Example: Solve $\ln(x - 2) = 3 \ln 2 + \ln(x)$.

Composite Functions and Their Domains/Ranges

Given two functions, $f(x)$ and $g(x)$, the composite function $g \circ f$ is defined as $g(f(x))$. The domain of $g \circ f$ consists of all $x$ in the domain of $f$ such that $f(x)$ is in the domain of $g$.

• Definition: $g \circ f(x) = g(f(x))$

• Domain: Set of $x$ such that $x$ is in domain of $f$ and $f(x)$ is in domain of $g$

• Range: Set of all possible values $g(f(x))$ can take

• Example: If $f(x) = \sqrt{x}$ and $g(x) = 2^{x-1}$, find $g(f(x))$ and its domain/range.

Inverse Functions

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. To find the inverse, solve $y = f(x)$ for $x$ in terms of $y$.

• Definition: $f^{-1}(y)$ is such that $f(f^{-1}(y)) = y$

• Domain and Range: The domain of $f^{-1}$ is the range of $f$, and vice versa

• Example: For $f(x) = \ln(8 - 2x)$, solve for $x$ in terms of $y$ to find $f^{-1}(y)$

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as $x$ approaches a particular value. If direct substitution leads to an indeterminate form, algebraic manipulation or L'Hôpital's Rule may be used.

• Definition: $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$

• Example: $\lim_{x \to 2} \frac{13 - x}{x^2 - 6}$

Continuity of Piecewise Functions

A function is continuous at a point if the left and right limits at that point equal the function's value there. For piecewise functions, set the expressions equal at the boundaries and solve for parameters.

• Definition: $f$ is continuous at $x = c$ if $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)$

• Example: For $f(x)$ defined piecewise, find values of $a$ and $b$ for continuity at $x = 1$ and $x = 2$

Graphing and Asymptotes

Horizontal and Vertical Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. Horizontal asymptotes are found by evaluating limits as $x \to \infty$ or $x \to -\infty$.

• Vertical Asymptote: Set denominator to zero and solve for $x$

• Horizontal Asymptote: Evaluate $\lim_{x \to \infty} f(x)$

• Example: For $y = \frac{(x - 9)(3x - 4)}{(x^2 + 3)(x^2 + 9)}$, find all asymptotes

Derivatives and Differentiation Techniques

Basic Differentiation

The derivative of a function measures its rate of change. Use rules such as the power rule, product rule, quotient rule, and chain rule.

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

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

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

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$

• Example: Find $f'(x)$ for $f(x) = \frac{x^2 + 2x - 10}{x^2 + 4x^2}$

Second Derivative and Parameter Dependence

The second derivative $f''(x)$ gives information about the concavity of a function. When a function depends on a parameter, differentiate with respect to $x$ and substitute the parameter as needed.

• Definition: $f''(x) = \frac{d^2}{dx^2} f(x)$

• Example: For $f(x) = \frac{\sin(bx)}{1 + \cos(bx)}$, find $f''(x)$ and evaluate at $x = 0$

Limits and Tangents from First Principles

Derivative from the Definition

The derivative at a point can be defined as a limit:

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Example: For $f(x) = \frac{1}{\sqrt{1 - x}}$, use the definition to find $f'(x)$

Tangent Line to a Curve

The equation of the tangent line to $y = f(x)$ at $(a, f(a))$ is:

• Formula: $y = f(a) + f'(a)(x - a)$

• Example: Find the tangent line at $(2, 1)$ for $f(x) = \frac{1}{\sqrt{1 - x}}$

Bonus: Differentiation of Composite Functions

Chain Rule for Composite Functions

If $h(x) = f(g(x))$, then the derivative $h'(x)$ is given by the chain rule:

• Formula: $h'(x) = f'(g(x)) \cdot g'(x)$

• Example: Given $g(-1) = 2$, $f(-1) = 3$, and $f'(2) = -4$, calculate $M'(-1)$ where $M(x) = f(g(x))$

Summary Table: Differentiation Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Product Rule	$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$	$\frac{d}{dx}(x \sin x)$
Quotient Rule	$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$	$\frac{d}{dx}\left(\frac{x^2}{x+1}\right)$
Chain Rule	$\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$	$\frac{d}{dx} \sin(x^2)$