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Unit 1: Limits and Continuity

1. Concept of Limits

   • Definition: The limit of a function $f(x)$ as $x$ approaches $c$ equals $L$, written as $\lim_{x \to c} f(x) = L$, if the values of $f(x)$ approach $L$ as $x$ approaches $c$ from both sides.

   • One-Sided Limits:

   • Left-hand limit: $\lim_{x \to c^-} f(x)$

   • Right-hand limit: $\lim_{x \to c^+} f(x)$

   • A limit exists if and only if $\lim<em>{x \to c^-} f(x) = \lim</em>{x \to c^+} f(x) = L$.

   • Evaluating Limits:

   • Direct substitution: Evaluate $f(c)$.

   • Indeterminate form $\frac{0}{0}$: Factor and cancel, rationalize using conjugates, or simplify complex fractions.

   • Special trigonometric limits:

     • $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

     • $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

   • Squeeze Theorem: If $g(x) \le f(x) \le h(x)$ near $c$ (except possibly at $c$) and $\lim<em>{x \to c} g(x) = \lim</em>{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.

2. Continuity and Theorems

   • Definition of Continuity at a Point: $f(x)$ is continuous at $x = c$ if:

   • $f(c)$ is defined.

   • $\lim_{x \to c} f(x)$ exists.

   • $\lim_{x \to c} f(x) = f(c)$.

   • Types of Discontinuities:

   • Removable discontinuity: Limit exists, but $f(c)$ is undefined or does not equal the limit.

   • Jump discontinuity: Left-hand and right-hand limits exist but are not equal.

   • Infinite discontinuity: Function approaches $\infty$ or $-\infty$ (vertical asymptote).

   • Intermediate Value Theorem (IVT): If $f(x)$ is continuous on the closed interval $[a, b]$, and $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one number $c \in (a, b)$ such that $f(c) = k$.

Unit 2: Differentiation: Definition and Fundamental Properties

1. Derivative Definition

   • Limit Definitions of the Derivative:

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

   • At an alternative point $a$: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

   • Differentiability and Continuity:

   • Differentiability implies continuity. Continuity does not guarantee differentiability.

   • A function fails to be differentiable at sharp corners, cusps, vertical tangents, and discontinuities.

2. Basic Derivative Rules

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

   • Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$

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

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

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

3. Derivatives of Standard Functions

   • Trigonometric Functions:

   • $\frac{d}{dx}[\sin(x)] = \cos(x)$

   • $\frac{d}{dx}[\cos(x)] = -\sin(x)$

   • $\frac{d}{dx}[\tan(x)] = \sec^2(x)$

   • $\frac{d}{dx}[\cot(x)] = -\csc^2(x)$

   • $\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$

   • $\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$

   • Exponential and Logarithmic Functions:

   • $\frac{d}{dx}[e^x] = e^x$

   • $\frac{d}{dx}[a^x] = a^x \ln(a)$

   • $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$

   • $\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$

Unit 3: Composite, Implicit, and Inverse Functions

1. Chain Rule

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

   • In Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

2. Implicit Differentiation

   • Applied when $y$ is not explicitly solved for in terms of $x$.

   • Differentiate both sides with respect to $x$, using the chain rule when differentiating terms with $y$ (e.g., $\frac{d}{dx}[y^2] = 2y \frac{dy}{dx}$), then solve for $\frac{dy}{dx}$.

3. Derivatives of Inverses

   • Inverse Function Derivative: If $g(x) = f^{-1}(x)$, then $g'(x) = \frac{1}{f'(g(x))}$.

   • Inverse Trigonometric Functions:

   • $\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}$

   • $\frac{d}{dx}[\arccos(x)] = -\frac{1}{\sqrt{1 - x^2}}$

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

Unit 4: Contextual Applications of Differentiation

1. Straight-Line Motion

   • Position: $s(t)$ or $x(t)$

   • Velocity: $v(t) = s'(t)$

   • Acceleration: $a(t) = v'(t) = s''(t)$

   • Speed: $|v(t)|$

   • Speed is increasing when $v(t)$ and $a(t)$ have the same sign.

   • Speed is decreasing when $v(t)$ and $a(t)$ have opposite signs.

2. Related Rates

   • Establish an equation relating the geometric variables.

   • Differentiate implicitly with respect to time $t$.

   • Substitute known values and rates to solve for the target rate.

3. Linear Approximation and L'Hôpital's Rule

   • Tangent Line Approximation: $L(x) = f(a) + f'(a)(x - a)$

   • If $f''(x) > 0$ (concave up), the tangent line is an underestimate.

   • If $f''(x) < 0$ (concave down), the tangent line is an overestimate.

   • L'Hôpital's Rule: For indeterminate forms $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$, $\lim<em>{x \to c} \frac{f(x)}{g(x)} = \lim</em>{x \to c} \frac{f'(x)}{g'(x)}$ provided the derivative limit exists.

Unit 5: Analytical Applications of Differentiation

1. Theorems on Closed Intervals

   • Extreme Value Theorem (EVT): If $f(x)$ is continuous on a closed interval $[a, b]$, then $f(x)$ attains an absolute maximum and an absolute minimum on $[a, b]$.

   • Mean Value Theorem (MVT): If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists at least one $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

   • Rolle's Theorem: A specific case of MVT where $f(a) = f(b)$, guaranteeing at least one $c \in (a, b)$ such that $f'(c) = 0$.

2. Function Analysis

   • Critical Points: Values of $x$ in the domain of $f(x)$ where $f'(x) = 0$ or $f'(x)$ is undefined.

   • First Derivative Test:

   • $f'(x) > 0$ on an interval implies $f(x)$ is increasing.

   • $f'(x) < 0$ on an interval implies $f(x)$ is decreasing.

   • Relative maximum occurs where $f'(x)$ changes from positive to negative.

   • Relative minimum occurs where $f'(x)$ changes from negative to positive.

   • Second Derivative and Concavity:

   • $f''(x) > 0$ implies $f(x)$ is concave up.

   • $f''(x) < 0$ implies $f(x)$ is concave down.

   • Point of inflection: A point where the concavity changes and the tangent line exists.

   • Second Derivative Test for Local Extrema:

   • If $f'(c) = 0$ and $f''(c) < 0$, there is a local maximum at $x = c$.

   • If $f'(c) = 0$ and $f''(c) > 0$, there is a local minimum at $x = c$.

   • If $f''(c) = 0$, the test is inconclusive.

Unit 6: Integration and Accumulation of Change

1. Riemann Sums and Definite Integrals

   • Approximations: Left Riemann Sum, Right Riemann Sum, Midpoint Rule, and Trapezoidal Rule.

   • Trapezoidal Rule: $\frac{\Delta x}{2}(y<em>0 + 2y</em>1 + 2y<em>2 + \dots + y</em>n)$

   • Limit Definition: $\int<em>a^b f(x)\,dx = \lim</em>{n \to \infty} \sum<em>{i=1}^n f(x</em>i) \Delta x$

2. Fundamental Theorem of Calculus (FTC)

   • FTC Part 1 (Derivative of Accumulation): $\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)$

   • With chain rule: $\frac{d}{dx}\left[\int_a^{g(x)} f(t)\,dt\right] = f(g(x)) g'(x)$

   • FTC Part 2 (Evaluation Theorem): $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F'(x) = f(x)$.

3. Integration Techniques

   • Basic Power Rule: $\int x^n\,dx = \frac{x^{n+1}}{n + 1} + C$ for $n \ne -1$

   • Logarithmic Form: $\int \frac{1}{x}\,dx = \ln(|x|) + C$

   • Exponential: $\int e^x\,dx = e^x + C$

   • Substitution ($u$-substitution): Let $u = g(x)$, then $du = g'(x)\,dx$, transforming $\int f(g(x)) g'(x)\,dx = \int f(u)\,du$.

Unit 7: Differential Equations

1. Slope Fields

   • Graphical representation of solutions to a first-order differential equation $\frac{dy}{dx} = f(x, y)$.

   • Each tick mark shows the tangent slope at that point.

2. Separable Differential Equations

   • Equations of the form $\frac{dy}{dx} = g(x) h(y)$.

   • Rewrite in separated differential form: $\frac{1}{h(y)}\,dy = g(x)\,dx$.

   • Integrate both sides and apply an initial condition to solve for the particular solution constant $C$.

   • Exponential Growth/Decay: If $\frac{dy}{dt} = k y$, then $y(t) = y_0 e^{k t}$.

Unit 8: Applications of Integration

1. Accumulation and Average Value

   • Net Change Theorem: $\int_a^b f'(t)\,dt = f(b) - f(a)$

   • Total Distance Traveled: $\int_a^b |v(t)|\,dt$

   • Displacement: $\int_a^b v(t)\,dt$

   • Average Value of a Function: $f<em>{\text{avg}} = \frac{1}{b - a} \int</em>a^b f(x)\,dx$

2. Area Between Curves

   • Integrated with respect to $x$: $A = \int<em>a^b (y</em>{\text{top}} - y_{\text{bottom}})\,dx$

   • Integrated with respect to $y$: $A = \int<em>c^d (x</em>{\text{right}} - x_{\text{left}})\,dy$

3. Volumes of Solids

   • Known Cross-Sections: $V = \int_a^b A(x)\,dx$

   • Squares: $A(x) = s^2$

   • Semicircles: $A(x) = \frac{\pi}{8} s^2$

   • Equilateral triangles: $A(x) = \frac{\sqrt{3}}{4} s^2$

   • Disk Method (Rotation around axis): $V = \pi \int_a^b (R(x))^2\,dx$

   • Washer Method (Rotation with hollow core): $V = \pi \int_a^b \left((R(x))^2 - (r(x))^2\right)\,dx$