Dr Thompson's AP Calculus University High School

Exploration of slopes of secant lines and tangent lines. The average and instantaneous rate of change is calculated with the difference quotient. Day 1 is an independent activity located here:

Euler's method to approximate the value of a function given a differential equation and initial point is explored.

We explore the integral test to determine convergence. The harmonic series is revisited.

We explore the comparison test and the ratio test to determine convergence. The area inside polar functions is also explored as well as their slopes.

We explore the error bound for Taylor polynomials.

We explore creating Taylor polynomials. We also center them at other values and create new polynomials from given ones.

We explore creating Taylor polynomials. We also center them at other values and create new polynomials from given ones.

Work done drinking a milk shake

Some example problems involving force and work. Lifting a leaky bucket and pumping to fill a tank.

We explore geometric series. We create new power series from known ones by differentiation and integration. The power series for e^x is explored. We also write pi as an infinite series produced by integrating term by term of a problem producing an inverse tangent.

An example of the shell method

Methods of disks and washers are explored in calculating the volume of a solid. Revolution around a line not an axis is explored. Shell method introduced.

This lesson focuses upon evaluating improper integrals using limits. Using length of the curve, we show that the circumference of a circle of radius pi is 2*pi*r

Methods of disks and washers are explored in calculating the volume of a solid.

This lesson focuses upon evaluating improper integrals using limits. L'Hospital's rule is also discussed.

Several problems are explored finding the area between two curves. Integrating with respect to y is explored as well.

Several problems are explored finding the area between two curves.

This lesson focuses upon sequences. It looks at a formal definition for the limit of a sequence.

Focus upon real life problems and the integral as the net change. Force, Work, Acceleration, Velocity are tied together. Units are discussed. A problem with electricity and Hooke's Law

This lesson focuses upon finding the length of a curve as a function of x and a function of y.

This lesson focuses upon parametric functions and their first and second derivatives.

This lesson focuses upon looking at several examples of partial fraction decomposition.

This lesson focuses upon developing the exponential function A=Pe^(rt) from a separable differential equation. Next Newton's Law of cooling is explored.

Introduction to separable differential equations

Using integration by parts. Investigate problems using this specific method. An "unknown integral" type problem is explored.

Using u-substitution to determine the antiderivative of tangent

2 examples using two methods of u substitution for indefinite integrals.

An introduction to first order differential equations and slope fields. A slope field is graphed using Geometer's Sketchpad. Strategies to investigate slope fields are given.

An introduction to first order differential equations and slope fields. A slope field is graphed using Geometer's Sketchpad. Connections to the fundamental theorem of calculus are made.

Using approximations, the volume of a pool is approximated using trapezoids.

Investigates the second half of the theorem. Time is spent looking at examples as well as finding anti derivatives with specific conditions

The lesson begins with a connection being illustrated between position and velocity. This is a review. However, the area under the velocity graph is calculated and the connection back to position is explored. This lesson also examined a graph and a function defined as the integral of another function. The derivative of the integral is examined.

Exploring through Sketchpad, the connection between a function and a function representing the signed area from a to x is explored. The function [F(x)] that is the signed area from a to x for the function f(x) is revealed to be the anti-derivative. An example using the anti-derivative is completed.

Lesson starts with average value of a function. Exploring through Sketchpad, the connection between a function and a function representing the signed area from a to x is explored. The function [F(x)] that is the signed area from a to x for the function f(x) is revealed to be the anti-derivative.

This section looks at several properties of definite integrals including how to combine or separate them over a given interval, combining functions, and reversing the limits.

A reimann sum is constructed and then developed into the definite integral. A connection to area is explored. Definite integrals are evaluated using knowledge of geometry.

Explore how the area under a curve can connect to a real life situation involving velocity and time. Rectangle methods are explored to approximate the area.

Examine a more complicated example

Examine two related rate questions

Using Differentials to approximate changes in the function

Examines how the slope of the tangent line at a point could aid in the estimation of a function value or in estimation of the x intercepts.

When light reflects off a surface, to another point. This is the shortest distance of any reflection (when the triangles are similar) We consider a calculus proof of the topic.

Examples of optimization of profit and average cost

Using the derivative to maximize or minimize. 2 geometric examples are explored.

Examines the graphical connections between concavity and the first and second derivative

analyzing the connection between pos/neg derivatives and if f' exists or is zero at a specific value

Investigation of the antiderivative of a function. We look at how there are multiple functions that can satisfy a specific derivative.

Investigation of the antiderivative of a function. We look at how there are multiple functions that can satisfy a specific derivative.

Investigate the definition of a critical point. Examine various functions and their extrema.

How fast is a rumor spreading? This question is explored.

Determines the general formula for exponential and natural logarithms. the derivative of e^x is explored. Sketchpad is used to show the connection of slope for e^x.