Course Overview

This course provides a comprehensive introduction to the foundational concepts of Calculus, focusing on limits, derivatives, and integrals. Students will develop both graphical and algebraic skills to analyze and solve problems involving rates of change, optimization, and area under curves.

Learning Objectives

• Evaluate Limits: Use graphical and algebraic methods to determine the limits of functions, including at infinity and for infinite limits.

• Compute Derivatives: Find the derivative of a function using standard rules and techniques.

• Apply Derivatives: Solve problems involving motion, related rates, implicit differentiation, extrema (maxima and minima), graphing, optimization, and L'Hôpital’s Rule.

• Integrate Functions: Compute antiderivatives and solve definite integrals using the Fundamental Theorem of Calculus.

Instructional Objectives by Topic

Limits

• The Idea of Limits: Understand the concept of a limit as a fundamental building block of calculus, describing the behavior of a function as the input approaches a particular value.

• Definitions of Limits: Learn the formal (epsilon-delta) definition of a limit and how it is used to rigorously prove limit statements.

• Techniques for Computing Limits: Apply algebraic manipulation, factoring, rationalization, and special limit laws to evaluate limits.

• Infinite Limits and Limits at Infinity: Analyze the behavior of functions as they approach infinity or as the input grows without bound.

• Continuity: Determine whether a function is continuous at a point or on an interval using the definition of continuity.

• Precise Definitions of Limits: Use the epsilon-delta definition to prove limits and continuity statements.

Derivatives

• The Derivative as a Function: Define the derivative as the instantaneous rate of change and as the slope of the tangent line to a curve at a point.

• Rules of Differentiation: Apply the power, product, quotient, and chain rules to compute derivatives of various functions.

• Derivatives of Trigonometric Functions: Differentiate basic trigonometric functions such as sine, cosine, and tangent.

• Derivatives as Rates of Change: Interpret derivatives in the context of real-world rates, such as velocity and acceleration.

• Implicit Differentiation: Differentiate equations not solved explicitly for one variable in terms of another.

• Related Rates: Solve problems where two or more quantities are related and changing with respect to time.

Applications of the Derivative

• Maxima and Minima: Find local and global extrema of functions using the first and second derivative tests.

• Mean Value Theorem: Understand and apply the Mean Value Theorem for derivatives.

• Graphing Functions: Use derivatives to analyze and sketch the graphs of functions, identifying intervals of increase, decrease, and concavity.

• Optimization Problems: Solve practical problems involving the optimization of quantities such as area, volume, and cost.

• Linear Approximation and Differentials: Use the derivative to approximate function values and estimate errors.

• L'Hôpital’s Rule: Apply L'Hôpital’s Rule to evaluate indeterminate forms of limits.

• Newton’s Method: Use Newton’s Method for approximating roots of equations.

Integration

• Antiderivatives: Find antiderivatives (indefinite integrals) of basic functions.

• Approximating Areas under Curves: Estimate the area under a curve using Riemann sums and other numerical methods.

• Definite Integrals: Compute definite integrals and interpret them as net area.

• Fundamental Theorem of Calculus: Connect differentiation and integration, and use the theorem to evaluate definite integrals.

• Working with Integrals: Apply properties of integrals and use substitution to simplify integration.

• Substitution Rule: Use the substitution method to evaluate more complex integrals.

Key Formulas and Theorems

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

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

• Fundamental Theorem of Calculus: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F'(x) = f(x)$

• Substitution Rule: $\int f(g(x))g'(x)\,dx = \int f(u)\,du$

Example: Evaluating a Limit

• Problem: Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

• Solution: Factor numerator: $x^2 - 4 = (x-2)(x+2)$. Cancel $(x-2)$: $\lim_{x \to 2} x+2 = 4$.

Example: Derivative of a Trigonometric Function

• Problem: Find $\frac{d}{dx}(\sin x)$

• Solution: $\frac{d}{dx}(\sin x) = \cos x$

Example: Definite Integral

• Problem: Compute $\int_0^1 2x\,dx$

• Solution: Antiderivative is $x^2$, so $x^2\big|_0^1 = 1^2 - 0^2 = 1$