# Lecture 24: Course Review

## Calculus I — Differential Calculus

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Identify the major themes and theorems of Calculus I
2. Recall key definitions, formulas, and techniques from the entire course
3. Recognize which technique to apply to a given problem
4. Connect the concepts of limits, derivatives, and integrals into a unified framework
5. Prepare effectively for the final examination

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## Lecture Content

### I. Limits and Continuity (Lectures 1-3)

The course began with **functions** -- their domains, ranges, graphs, transformations, compositions, and piecewise definitions -- and then moved to the concept at the heart of calculus: the **limit**. The notation lim_{x -> a} f(x) = L means that f(x) can be made arbitrarily close to L by taking x sufficiently close to a. One-sided limits (lim_{x -> a^-} and lim_{x -> a^+}) refine this idea, and a two-sided limit exists if and only if both one-sided limits exist and are equal.

The **limit laws** (sum, difference, product, quotient, power) allow limits to be computed algebraically. When direct substitution fails, techniques such as factoring and canceling, rationalizing by multiplying by the conjugate, and the **Squeeze Theorem** (if g(x) <= f(x) <= h(x) and the outer limits agree, so does the middle one) come into play. Two key limits underpin the trigonometric derivatives: lim_{x -> 0} (sin x)/x = 1 and lim_{x -> 0} (cos x - 1)/x = 0.

**Continuity** at a point means that lim_{x -> a} f(x) = f(a). Discontinuities come in four types: removable, jump, infinite, and oscillatory. The **Intermediate Value Theorem** guarantees that continuous functions on [a, b] attain every value between f(a) and f(b), which is particularly useful for proving the existence of roots.

### II. The Derivative (Lectures 4-6)

The **derivative** f'(a) = lim_{h -> 0} [f(a+h) - f(a)] / h captures two complementary ideas: geometrically, it is the slope of the tangent line at (a, f(a)); physically, it is the instantaneous rate of change of f at x = a. A central theorem states that **differentiability implies continuity**, though the converse is false.

The major **differentiation rules** were developed systematically. The **Power Rule** gives d/dx [x^n] = nx^{n-1}. The **Product Rule** states d/dx [fg] = f'g + fg'. The **Quotient Rule** gives d/dx [f/g] = (f'g - fg')/g^2. The **Chain Rule**, arguably the most important of all, states d/dx [f(g(x))] = f'(g(x)) * g'(x).

### III. Derivatives of Special Functions (Lectures 7-8)

The **trigonometric derivatives** complete a natural cycle: d/dx [sin x] = cos x, d/dx [cos x] = -sin x, d/dx [tan x] = sec^2 x, d/dx [sec x] = sec x tan x, d/dx [cot x] = -csc^2 x, and d/dx [csc x] = -csc x cot x.

The **exponential and logarithmic derivatives** center on the special properties of e: d/dx [e^x] = e^x, d/dx [a^x] = a^x ln a, d/dx [ln x] = 1/x, and d/dx [log_a x] = 1/(x ln a). The **inverse trigonometric derivatives** include d/dx [arcsin x] = 1/sqrt(1 - x^2) and d/dx [arctan x] = 1/(1 + x^2).

**Logarithmic differentiation** -- taking the natural logarithm of both sides and differentiating implicitly -- is essential for functions of the form f(x)^{g(x)}.

<image>A comprehensive one-page derivative reference sheet organized in a table format. Column headers: "Function" and "Derivative." Sections: Power Functions, Trigonometric, Inverse Trigonometric, Exponential/Logarithmic, and Rules (Product, Quotient, Chain). All formulas listed compactly. Title: "Calculus I — Complete Derivative Reference Sheet."</image>

### IV. Applications of the Derivative (Lectures 9-11)

**Implicit differentiation** handles equations where y is not explicitly solved for: differentiate both sides with respect to x, treat y as y(x), and solve for dy/dx. **Related rates** problems involve quantities linked by an equation that all change with time. The strategy is to draw a diagram, identify variables, write the connecting equation, differentiate with respect to t, substitute known values, and solve.

**Linear approximation** uses the tangent line to estimate function values near a point: f(x) is approximately f(a) + f'(a)(x - a). **Differentials** formalize this: dy = f'(x) dx approximates the actual change Delta y, and they are invaluable for estimating propagated errors in measurements.

### V. Curve Analysis (Lectures 12-15)

**Critical numbers** (where f'(c) = 0 or f'(c) does not exist) are the candidates for extrema. The **Extreme Value Theorem** guarantees that a continuous function on [a, b] attains absolute maximum and minimum values, and the **Closed Interval Method** finds them by evaluating f at critical numbers and endpoints.

The **Mean Value Theorem** asserts that there exists c in (a, b) with f'(c) = [f(b) - f(a)]/(b - a). The **First Derivative Test** classifies critical numbers by checking whether f' changes sign (positive-to-negative for a local max, negative-to-positive for a local min). The **Second Derivative Test** offers an alternative: at a critical point where f'(c) = 0, f''(c) > 0 implies a local minimum and f''(c) < 0 implies a local maximum.

**Concavity** is determined by the second derivative: f'' > 0 means concave up, f'' < 0 means concave down. **Inflection points** occur where concavity changes. **L'Hopital's Rule** resolves 0/0 and infinity/infinity indeterminate forms by replacing lim f/g with lim f'/g', and other indeterminate forms (0 * infinity, infinity - infinity, 0^0, 1^infinity, infinity^0) must be converted first.

<image>A curve-sketching checklist as a visual flowchart. Step 1: Domain. Step 2: Intercepts. Step 3: Symmetry (even/odd/periodic). Step 4: Asymptotes (vertical, horizontal, oblique). Step 5: First derivative — increasing/decreasing, local extrema. Step 6: Second derivative — concavity, inflection points. Step 7: Plot key points and sketch. Each step has an icon or mini-diagram. Title: "Curve Sketching Checklist."</image>

### VI. Optimization and Numerical Methods (Lectures 16-17)

**Optimization** problems require translating a word problem into maximizing or minimizing a function. The key steps are identifying the objective function and constraints, reducing to one variable using the constraint, finding critical numbers, checking endpoints, and verifying the nature of the extremum.

**Newton's Method**, given by the iteration x_{n+1} = x_n - f(x_n)/f'(x_n), approximates roots of f(x) = 0 with remarkable efficiency. It exhibits quadratic convergence when it works, meaning the number of correct digits roughly doubles each iteration, though it can fail with a poor initial guess.

### VII. Integration (Lectures 18-23)

**Antiderivatives** reverse differentiation: F'(x) = f(x), and the indefinite integral is written as integral of f(x) dx = F(x) + C. The **definite integral** integral from a to b of f(x) dx = lim_{n -> infinity} sum f(x_i*) Delta x represents the signed area under the curve.

The **Fundamental Theorem of Calculus** unites the two branches. FTC1 states d/dx [integral from a to x of f(t) dt] = f(x), showing differentiation undoes integration. FTC2 states integral from a to b of f(x) dx = F(b) - F(a) where F' = f, providing a practical evaluation method.

The **Substitution Rule** reverses the chain rule: integral of f(g(x)) g'(x) dx = integral of f(u) du where u = g(x). For definite integrals, the limits must be changed to u-values when substituting.

**Areas between curves** are computed as integral from a to b of [f(x) - g(x)] dx (top minus bottom), or as integral from c to d of [f(y) - g(y)] dy (right minus left) when integrating with respect to y. **Volumes of revolution** use three methods: the disk method (V = pi * integral of R^2 dx), the washer method (V = pi * integral of (R^2 - r^2) dx), and the shell method (V = 2*pi * integral of (radius)(height) dx).

### VIII. Exam Preparation Tips

Effective preparation requires knowing the major theorems (EVT, IVT, MVT, FTC) along with their hypotheses and conclusions. All derivative rules and basic antiderivatives should be memorized. Practice technique selection: given a problem, identify which tool to use, whether it is a limit law, a differentiation rule, or an integration method. Always show all work, justify steps, state theorems when used, and include +C for indefinite integrals. Finally, check answers by differentiating antiderivatives, verifying signs, checking units, and plugging in endpoints.

<image>A concept map of Calculus I showing the connections between major topics. At the top: "Functions." Branching down to "Limits" (with sub-nodes: continuity, IVT, Squeeze Theorem). From Limits, an arrow leads to "Derivatives" (with sub-nodes: rules, applications — related rates, optimization, curve sketching, linear approximation). From Derivatives, a two-way arrow (labeled "FTC") connects to "Integrals" (with sub-nodes: Riemann sums, substitution, areas, volumes). The MVT is shown connecting Derivatives to both Limits and Integrals. Title: "Calculus I Concept Map: How Everything Connects."</image>