# Calculus

## Course Overview

Calculus covers the foundations of differential calculus, including limits, derivatives and integrals. In the beginning of the course, we examine functions and their unique behaviors, and how they relate to concepts such as limits and continuity. The remainder of the course focuses on differentiation, different techniques of differentiation, and applications of derivatives. A wide range of topics are covered, including the limit definition of a derivative, chain rule, product/quotient rules, and implicit differentiation. Optimization is covered extensively, as are related rates problems and Newton’s method. The course concludes with antiderivatives and integration, covering different techniques and their relationships to differentiation techniques. Areas under curves, solids of revolution, and average values are studied in detail.

## Course Content

### Unit 1: A Review of Basic Functions, Limits and Continuity

As Limits and Continuity are the foundations upon which all of Differential Calculus are built, this is perhaps the most important unit for students to have a strong understanding of. Students need to be able to evaluate limits graphically, numerically and algebraically using limit laws. They also need to use limits in combination with the definition of a continuous function to determine if a function is continuous. This unit covers the following topics:

• Review of Mother Functions
• Review of exponential functions from Algebra II
• Review of laws of exponents
• Review of inverse functions from Algebra II and Trigonometry
• Review of Logarithmic functions
• Review of Laws of Logarithms
• Exploration of the problem of coming up with a line tangent to a curve at a point
• Link velocity to tangent line to the position function
• Exploring the basic idea of limits using epsilons and deltas
• Evaluation of limits using graphs and tables
• Evaluating limits using limit laws
• Plug it in, plug it in
• Evaluate one-sided limits
• Determine continuity of a function
• Evaluate limits at +/- infinity using algebra and rules of horizontal asymptotes from Algebra 2
• Determine horizontal and vertical asymptotes using limits at infinity and specific points

### Unit 2: Graphical Derivatives, Limit Definition of Derivatives, and Rules for Differentiation

All of differential Calculus literally is born from the ability to take derivatives. This unit provides students with theoretical/graphical understanding of what a derivative is, as well as Algebraic tools to take derivatives both using Rules of Differentiation and the Limit Definition of Derivatives. This unit covers the following topics:

• Determine derivatives using limit definitions of derivative
• Determine equations of lines tangent using the derivative of a function
• Explore higher order derivatives
• Determine graphs of derivatives of functions based on behavior of “mother” function
• Determine the “mother” function based on the graph of the derivative
• Explore graphs of higher order derivatives
• Find derivatives of functions of the form x^n and a^x
• Find derivatives of products and quotients
• Find the derivatives of all trig functions

### Unit 3: Further Rules of Differentiation

As far as manual, algebraic derivatives are concerned, this is the most important unit. By the end of this unit, students will have all of the skills necessary to take derivatives of any function presented for the rest of the course. This unit includes chain rule, implicit differentiation, logarithmic differentiation and an introduction to applications of derivatives; Related Rates of Change. This unit covers the following topics:

• Find derivatives of functions using the chain rule, and in combination with other rules
• Find derivatives of curves that aren’t necessarily functions using implicit derivatives
• Find derivatives of logarithmic functions
• Use logarithmic differentiation to take derivatives of more complicated functions
• Use differential calculus to solve problems in the sciences
• Use related rates of change to solve applications problems

### Unit 4: Applications of Derivatives

We learn to take derivatives so that we can apply them to solve real-world problems and theoretical problems. This unit involves graph theory, optimization, L’Hospital’s Method and Newton’s Method and represents the pinnacle of differential calculus in two dimensional space. This unit covers the following topics:

• Calculate linearization functions and differential from functions at a given point
• Use linearization to approximate change in functions
• Use differential calculus to determine when functions increase, decrease, have local/relative maxima,
• local/relative minima, and saddle points
• Extreme value theorem
• Ripley Doodle Diagram
• Closed interval method
• Rolle’s Theorem
• Solve for values of a differentiable function that satisfy the mean value theorem
• Use first and second derivatives to determine when a function is increasing, decreasing, concave up/down, max/min, inflections points
• Use L’Hospital’s rule to solve indeterminant forms 0/0 and ∞/∞
• Use L’Hospital’s rule to solve indeterminant forms 0•∞

### Unit 5: An Introduction to Integral Calculus

Much like Differential Calculus, Integral Calculus has far reaching applications in the sciences. This unit provides students with both the theoretical and mechanical/algebraic skills to fully understand how and why integrals are taken. Topics covered in this unit include:

• Summarization of graphing techniques using differential calculus
• Use graphing utilities to confirm graphs found using differential calculus
• Use differential calculus techniques to solve optimization problems
• Solve for x-intercepts of functions
• Find Antiderivatives of functions

### Unit 6: Applications of Integrals

We learn to take integrals so that we can apply them to solve real-world problems and theoretical problems. This unit focuses primarily on areas between curves, volumes of solids of revolution, and volumes of solids of known cross section. Topics covered in this unit include:

• Explore the area between a function and the x-axis
• Relate the area under a velocity curve to the distance traveled
• Define the Definite Integral
• Use rules of series and to evaluate definite integrals
• Use laws of definite integrals to evaluate definite integrals
• Use FTC to evaluate definite integrals
• Determine indefinite integrals of known functions
• Use integration by substitution to evaluate both indefinite and definite integrals

### Unit 7 – Application of Integration

Thus unit looks at various applications of integration, from finding the two-dimensional area to the more difficult three-dimensional volume of a solid formed by the revolution of a function and of a solid formed by cross-sections. This requires students to visualize the dimensions of slices of the area or volumes, and to appreciate how many slices combine to form the entire area or volume. Rather than being given integrals to solve, they will have to use the different techniques, including modeling, to come up with the appropriate integral that represents the quantity being asked for. Students will have to understand how to find the inverse of a function in many cases, as well as many different integration techniques, in order to solve the problems correctly. In the last part of the unit, we look at the average value of a function in a region and how that relates to finding the area underneath a function.

Topics covered in this unit include:

• Determine the area between two functions in the plane using definite integrals
• Determine volumes of solids of rotation around x- and y- axes, horizontal/vertical axes by disk/washer method
• Determine the volume of a solid from a known cross section in the x,y plane
• Determine volumes of solids of rotation around x- and y- axes, horizontal/vertical axes by shell method
• Use the definite integral to find the average value of a function

### Unit 8 – Differential Equations

This unit is so important, both for its applications to the physical sciences as well as giving students a glimpse into the types of course work they might see after calculus. The unit focuses on slope fields, solutions to differential equations, and Euler’s Method for approximations of differential equation solutions. Topics covered in this unit include:

• Determine if a function is a solution to a differential equation
• Sketch slope fields
• Use Euler’s method to approximate the value of a function given a differential equation
• Demonstrate that all exponential growth and decay problems come from the differential equation dy/dx=ky
• Solve exponential growth and decay problems
• Find solutions to differential equations using separation of variables
• Solve population growth problems using the Logistics equation