# High School AP Calculus Course

**This page includes information about the material covered in the High School AP Calculus course. While we no longer offer AP Calculus, we do offer high school Pre-Calculus. You can find that course description and overview here.**

Time4Learning offers an online, interactive high school curriculum that correlates to state standards. It can be used as a primary homeschool curriculum, a supplement to your current curriculum and as an afterschool or summer skill building program. At the high school level, Time4Learning is organized by courses rather than grade levels, so parents have the option of choosing any four as part of membership. AP Calculus is taught using a combination of multimedia lessons, instructional videos, worksheets, quizzes, tests and both online and offline projects.

### Chapter 1 – The Basics

**The materials in this chapter introduce and cover The Basics. **It is organized into sections that teach, reinforce and test students on the concepts of an overview of calculus and precalculus review.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to Thinkwell’s Calculus, the two questions of calculus, the average rates of change, and the basic math concepts needed for calculus.
- Students will learn the math functions, graphing lines, parabolas, and some non-eudidean geometry.

### Chapter 2 – Limits

**The materials in this chapter introduce and cover the Limits. **It is organized into sections that teach, reinforce and test students on the concepts of a limit and how to evaluate limits.

**Lessons in this chapter are organized into the following sections:**

- Students will learn about finding the rate of change over an interval, how to find limits graphically, the formal definition of a limit, The Limit Laws, One-Sided Limits, and the continuity and discontinuity of limits.
- Students will evaluate limits, learn techniques for evaluating limits, be introduced to limits and indeterminate forms, as well as working on a general overview of limits.

### Chapter 3 – An introduction of Derivatives

**The materials in this chapter introduce and cover an Introduction of Derivatives. **It is organized into sections that teach, reinforce and test students on the concepts understanding the derivative, using the derivative and some special derivatives.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to concepts of the rates of change, secants, and tangents, instanttaneous velocity, the derivative, and differentiability.
- Students will discuss the slope of a tangent line, instantaneous rate, and the equation of a tangent line.
- Students will learn the derivative of reciprocal function and square root function.

### Chapter 4 – Computational Techniques

**The materials in this chapter introduce and cover Computational Techniques. **It is organized into sections that teach, reinforce and test students on the concepts of the Rules; The Power Rule, The Product and Quotient Rules, and The Chain Rule.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to a shortcut for finding derivatives, a quick proof of the power rule, and how to use the power rule.
- Students will be introduced to the product rule and the quotient rule.
- Students will be introduced to the chain rule, how to use the chain rule, and how to combine computational techniques.

### Chapter 5 – Special Functions

**The materials in this chapter introduce and cover Special Functions. **It is organized into sections that teach, reinforce and test students on the concepts of a review of trigonometry, graphing trigonometric functions, the derivatives of trignonmetric functions, the number Pi, graphing exponential functions, derivatives of exponential functions, logarithmic functions, the derivatives of natural log function, and the use of the derivative rules with transcendental function.

**Lessons in this chapter are organized into the following sections:**

- Students will cover a review of trigonometry, graphing trigonometric functions, the derivatives of trigonometric functions, and the number Pi.
- Students will be introduced to graphing exponential functions, derivatives of exponentail functions, and the music of math.
- Students will discuss the evaluating of logarithmic functions, the derivative of the natural log function, and how to use the derivative rules with transcendental function.

### Chapter 6 – Implicit Differentiation

**The materials in this chapter introduce and cover Implicit Differentiation. **It is organized into sections that teach, reinforce and test students on the concepts of an introduction of implicit differentiation, how to find the derivative implicity, and the use and applying of implicit differentiation.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to implicit differentiation and finding the derivative implicity.
- Students will be introduced to using and applying implicit differentiation.

### Chapter 7 – Application of Differentiation

**The materials in this chapter introduce and cover the Application of Differentiation. **It is organized into sections that teach, reinforce and test students on the concepts of position and velocity, linear approximation, related rates and optimization.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to acceleration and the dericative and solving word problems involving distance and velocity.
- Students will learn about higher-order derivatives and linear approximation, using the tangent line and approximation formula, and Newton’s Method.
- Students will be introduced to problems including the Pebble Problem, the Ladder Problem, the baseball Problem, the Blimp Problem, and math anxiety.
- Students will learn the connection between slope and optimization, the Fence Problem, the Box Problem, the Can Problem, and the Wire-Cutting Problem.

### Chapter 8 – Curve Sketching

**The materials in this chapter introduce and cover Curve Sketching. **It is organized into sections that teach, reinforce and test students on the concepts of an introduction to curve sketching, critical points, concavity, graphing using the derivative, and asymptotes.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to curve sketching, the three big theorems, and morale moment.
- Students will be introduced to critical points, the regions where a function increases or decreases, the first derivative test, and math magic.
- Students will be introduced to concavity and inflection points,the second derivative to examine concavity, and the mobius band.
- Students will be introduced to graphs of polynomial functions, cusp points and the derivative, domain-restricted functions and the derivative, and the second derivative Students will be test.
- Students will be introduced to vertical asymptotes, horizontal asymptotes ad infinite limits, graphing functions with asymptotes, functions with asymptotes, and functions with asymptotes and critical points.

### Chapter 9 – The Basics of Integration

**The materials in this chapter introduce and cover The Basics of Integration. **It is organized into sections that teach, reinforce and test students on the concepts of antiderivatives, integration by substitution, illustrating integration by substitution, and the fundamental theorem of calculus.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to antidifferentiation, antiderivatives of powers of x, and antiderivatives of trig and exponential functions.
- Students will learn about undoing the chain rule, and integrating polynomials of substitution.
- Students will be introduced to integrating composite trig functions by substitution, integrate composite expo and rational function by substitution, and choosing effective function decompositions.
- Students will be introduced to approximating areas of plane regions, areas, riemann sums, and definite integrals, the fundamental theorem of calculus, illustring the fundamental theorem of calculus, and evaluating definite integrals.

### Chapter 10 – Applications of Integration

**The materials in this chapter introduce and cover Applications of Integration. **It is organized into sections that teach, reinforce and test students on the concepts motion, finding the area between two curves, and integrating with respect to Y.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to antiderivatives and motion, gravity and vertical motion, and solving vertical motion problems.
- Students will be introduced to antiderivatives and motion, gravity and vertical motion, and solving vertical motion problems.
- Students will find areas by integrating with respect to Y, area, and integration by substitution and trigonometry.

### Chapter 11 – Calculus I Review

**The materials in this chapter introduce and cover Calculus I Review. **It is organized into sections that teach, reinforce and test students on the concepts the close of calculus I.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced into Calculus II.

### Chapter 12 – Math Fun

**The materials in this chapter introduce and cover Math Fun. **It is organized into sections that teach, reinforce and test students on the concepts of Paradoxes and Sequences.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to paradoxes, paradoxes and air safety, Newconb’s Paradox, and Zeno’s Paradox.
- Students will be introduced to Fibonacci Numbers.

### Chapter 13 – An Introduction to Calculus II

**The materials in this chapter introduce and cover An Introduction to Calculus II. **It is organized into sections that teach, reinforce and test students on the concepts basic introduction to Calculus II.

**Lessons in this chapter are organized into the following sections:**

- Students will be welcomed to Calculus II and learn about Calculus I in 20 minutes.

### Chapter 14 – L’Hopital’s Rule

**The materials in this chapter introduce and cover L’Hopital’s Rule. **It is organized into sections that teach, reinforce and test students on the concepts of indeterminate quotients and other indeterminate forms.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to indeterminate forms, L’Hopital’s ruel, and exotic examples of indeterminate forms.
- Students will continue to work on L’Hopital’s Rule and indeterminate products, L’Hopital’s Rule and indeterminate differences, and L’Hopital’s Rule and one to the infinite power.

### Chapter 15 – Elementary Function and Inverses

**The materials in this chapter introduce and cover Elementary Functions and Inverses. **It is organized into sections that teach, reinforce and test students on the concepts of inverse functions, the calculus of inverse functions, inverse trigonometric functions, calculus of inverse trigonometric functions and the hyperbolic functions.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to the exponential and natural lof functions, the basics of inverse functions, and finding the inverse of a function.
- Students will be introduced to the derivatives of inverse functions.
- Students will be introduced to the inverse sine, cosine, and tangent functions, the inverse secant, cosecant, and cotangent functions, and evaluating inverse trigonometric functions.
- Students will be introduced to derivatives of inverse trigonometric functionsand more calculus of inverse trionometric functions.
- Students will be introduced to defining the hyperbolic functions, hyperbolic identities, and derivatives of hyperbolic functions.

### Chapter 16 – Techniques of Integration

**The materials in this chapter introduce and cover Techniques of Integration. **It is organized into sections that teach, reinforce and test students on the concepts integration using tables, integrals involving powers of sine and cosine, integrals involving power of other trigonometric functions, introductions to integration by partial fractions, integration by partial fractions with repeat factors, integration by parts, an introduction to trigonometric substitution, trigonometric substitution strategy, and numerical integration.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to the integral table and making u-substitutions.
- Students will be introduced to integrals with powers of sine and cosine and integrals with even and odd powers od sine and cosine.
- Students will lean about integrals of other trigonometric functions, integrals with odd powers of tangent and any powers of secant and integrals with even powers of tangent and secant.
- Students will be introduced to finding partial fraction decompositions, partial fractions, and long division.
- Students will be introduced to repeated linear fractors, distinct and repeated quadratic factors, and partial fractions of transcendental functions.
- Students will be introduced to interation by parts, applying integration by parts to natural log function, inspirational examples of of integration by parts, repeated applications of integration by parts, and algebraic manipulation and integration by parts.
- Students will be introduced to converting radicals into trogonometric expressions, using trigonometric substitution to integrate radicals, and trigonometric substitutions on rational powers.
- Students will discuss an overview of trigonometric substition strategy and trigonometric substitution involving a definite integral.
- Students will be introduced to deriving the trapezoidal rule and an example of the trapezoidal rule.

### Chapter 17 – Improper Integrals

**The materials in this chapter introduce and cover Improper Integrals. **It is organized into sections that teach, reinforce and test students on the concepts Improper Integrals.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to the first type of improper integral, the second type of improper type of improper integral, and infinite limits of integration, converge and diverge.

### Chapter 18 – Application of Integral Calculus

**The materials in this chapter introduce and cover Application of Integral Calculus. **It is organized into sections that teach, reinforce and test students on the concepts of the average value of a function, finding volumes using cross-sections, disks and washers, shells, arc lengths and functions, work, and moments and centers of mass.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to finding the average value of a function.
- Students will be introduced to finding volumes using cross-sectional slices and an example of finding cross-sectional volumes.
- Students will be introduced to solids of revolution, the disk method along the y-Axis, a transcendental example of the disk method,the washer method across the X-Axis, and the washer method across the Y-Axis.
- Students will be introduced to the shell method, why shells can be better than washers, and the shell methd integrating with respect to Y.
- Students will be introduced to arc ength, and finding arc lengths of curves given by functions.
- Students will be introduced to work, calculating work, and hooke’s Law.
- Students will be introduced to center of mass, and the center of mass of a thin plate.

### Chapter 19 – Sequences and Series

**The materials in this chapter introduce and cover Sequences and Series.** It is organized into sections that teach, reinforce and test students on the concepts of sequences, monotonic and bounded sequences, infinite series, convergence and divergence the integral test, the direct comparison test, the limit comparison test, the alternating series, absolute and conditional convergences, the ratio and root tests, polynomial approximations of elementary functions, Taylor and Maclaurin polynomials, Taylor and Maclaurin series, power series, and power series representations of functions.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to the limit of a sequence, determining the limit of a sequence, and the squeeze and absolute value theorems.
- Students will be introduced to monotonic and bounded sequences.
- Students will be introduced to infinite series, the summation of infinite series, geometric series, and telescoping series.
- Students will learn about the properties of convergent series and the nth-term test for divergence.
- Students will be introduced to the integral test, examples of the integral test, and defining p-series.
- Students will be introduced to the direct comparison test and using the direct comparison test.
- Students will be introduced to the alternating series, the alternating series test, and estimating the sum of an alternating series.
- Students will be introduced to absolute and conditional convergence.
- Students will be introduced to the ratio test, examples of the ratio test, and the root test.
- Students will be introduced to polynomial approximation of elementary functions and higher-degree approximations.
- Students will be introduced to the Taylor Polynomials, Maclaurin Polynomials, the remainder of a Taylor Polynomial, and approximating the value of a function.
- Students will be introduced to the Taylor series, examples of the Taylor and Maclaurin series, the new Taylor series, and the convergence of Taylor series.
- Students will be introduced to the definition of power series, the interval and radius of convergence, and finding the inteval and radius of convergence.
- Students will be introduce to differentiation and integration of power series, finding power series representations by differentiation, finding power series representations by integration, and integrating functions using power series.

### Chapter 20 – Differential Equations

**The materials in this chapter introduce and cover Differential Equations. **It is organized into sections that teach, reinforce and test students on the concepts separable differential equations,, solving a homogeneous differential equation, growth and decay problems, solve first-order linear differential equations.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to differential equations, solving separable differential equations, finding a particular solution, and direction fields.
- Students will learn to separate homogeneous differential equations and change of variables.
- Students will learn about exponential growth and redioactive decay.
- Students will be introduced to first-order linear equations and using integrating factors.

### Chapter 21 – Parametric Equations and Polar Coordinates

**The materials in this chapter introduce and cover Parametric Equations and Polar Coordinates.** It is organized into sections that teach, reinforce and test students on the concepts of understanding parametric equations, calculus and parametric equations, understanding polar coordinates, polar functions and slope, and polar functions and area.

**Lessons in this chapter are organized into the following sections:**

- Students will be introduced to parametric equations, the cycloid, and eliminating parameters.
- Students will be introduced to derivatives of parametric equations, graphing the elliptic curve, the arc length of a parameterized curve, and find arc length of curves given by parametric equations.
- Students will be introduced to the polar coordinate system, converting between polar and cartesian forms, spirals and circles, and graphing some special polar functions.
- Students will be introduced to calculus and the rose curve and finding the slopes of tangent lines in polar form.
- Students will be introduced to heading toward the area of a polar region and finding the area of a polar region,area of a region bounded by two polar curves.