AP Calculus BC


Semester 2

Click HERE for Semester 1.

Required materials for mathematics classes
Your school-issued laptop, charged!
Two (2) hardbound theme books (i.e. Mead Composition Book) for journaling (no spirals!).  Use one for homework (enter assignments chronologically) and the other for Journal Reflections (this theme book stays in the classroom).
Texas Instruments TI-83+ (or TI-84+) Graphing Calculator + extra AAA cells.  DO NOT expect the school to provide you extra cells!
Basic drawing supplies (ruler that measures in centimeters, compass, protractor).
Several pencils or mechanical pencil with extra lead + a pen for writing JRs.

Click HERE for the Stewart Calculus 7e Website wherein you will find homework hints, "Tools for Enriching Calculus (TEC), additional topics, etc.

Grab THIS document showing the "Essential" and "Recommended" sections of Stewart Calculus 7e.

Linked herein is the topic outline for the course, adapted from the College Board.

  Desmos Graphing Calculator.

Join Wolfram Alpha, which provides step-by-step solutions.

Homework help from Slader.

AP/BC Review Questions (from the end of each chapter).

Chapter 10: Parametric Equations and Polar Coordinates

Entry Task



List what you believe you understand regarding parametric equations.
Introduction of parametric equations.  Note the context of the problem to determine if angles are in degrees or radians!

  • The meaning of Xmin & Xmax, Ymin & Ymax and how to set them.
  • How do assign Tmin, Tmax, Tstep.
  • How to trace specific points.  Identify the input variable.
  • How to proportion the graphing window so a circle does not look like an ellipse.
  • How to write parametric equations for y - 3 = 2(x + 1) and x2 + y2 = 9.
Work through Parametric Equations and Projectile Motion with your table partners to gain a deeper understanding of how Parametric Equations work.

JR:  Why are these equations called parametric?  What makes them that way?
1.  Write parametric equations for the scenario:
The world's latest golfing sensation, Cheetah Forest, selects his sand wedge (56° loft) and will strike the ball at 88 feet per second.
Determine how far the ball will travel horizontally and vertically.
2.  An object moves along the path defined by y = x3 + 3.  The x-coordinate satisfies the equation x(t) = -t + 1.
  a) In what direction is the object moving?
  b) What is the corresponding y(t)?
Download the Course syllabus and save it to your computer's desktop.  Scan through each page and take particular note of required materials, assessment procedures, and assignment requirements
Curves Defined by Parametric Equations.  Skim pp. 660-665.

  Name That Parametrization activity.

JR: Explain in simple language how to convert a set of parametric equations into an equivalent Cartesian equation.
pp. 665 #7, 13, 15.  Graph by hand and provide a table of values you used for the sketch.
An object moves according to the equation y = x2 + 3.  The x-coordinate, x(t), satisfies the equation x = -t + 1. 
  • What direction is the object moving?
  • What is the corresponding y(t)?
Calculus with Parametric Curves. 
  • Derivatives
  • Arc length
  • Surface area
You Gotta Have Heart activity.

JR: Explain in simple language how to take the first and second derivatives of a pair of parametric equations.
p. 675 #7, 11, 17.
Let x(t)  = t3 + t, y(t) = t.  Compute d2y/dx2 at the point (2, 1).  What does it mean?
Polar Coordinates.  Picture Pages activity.

Areas and Lengths in Polar Coordinates.  Polar Propellers activity.

JR: Why does the parametric curve x = f(t), y = g(t) have a horizontal tangent line when dy/dt = 0 and dx/dt ≠ 0, and not the other way around?  Include diagrams.
pp. 686-687 #1, 3, 5, 11; pp. 692-693 #1, 5, 9, 21.
Convert the point (10, π/6) from polar to rectangular coordinates.
Cardioids activity.

JR: Prove ∫ dx/(1 + x2) ≠ ln(x2 + 1) + C.
Chapter10 AP AB/BC Review Questions; AP10-1, 2 (following p. 712).  Do them all and show all work!
Write the polar equation of a circle with radius 3. Write the parametric equations for a circle with radius 3 centered at 1, 2). Problem demonstrations.

JR: Explain in simple language how to convert polar to Cartesian and Cartesian to polar.
Prepare for the test.

Get ready for the test!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.  Place your homework notebook near you for scoring. Test 10.1: Free response.  NO graphing calculator or note card!  Expect questions on
  • Derivatives of polar and parametric functions.
  • Conversion of functions between Cartesian, polar, and parametric forms.
  • Determining tangency to polar and parametric curves.
  • For a polar function: compute area enclosed by a curve; compute arc length.
  • For a parametric function: compute area bounded by the curve and the x-axis; compute arc length.
  • Solve a "real world" problem.
JR: What were the hardest parts of this chapter for you?  What were the easiest parts?
Collect everything you can about vectors (previous notes, homework, etc. along with Websites giving background information).  This is an actual homework assignment you are expected to do!
*Unless otherwise noted, homework is due the next class day.

Chapter 12: Vectors and the Geometry of Space

Entry Task



Write a formula to compute the distance between the given two points in space A and B, for A(a, b, c) and B(d, e, f).  Explain why this formula works.
Three dimensional coordinate system.  Working With Surfaces in Three Dimensional Space activity.  When finished, please check these solutions.

JR:  Explain why y = x is the equation of a plane, not a line, in 3-space.  Sketch the situation.
pp. 814-815 #3, 5, 10, 11, 27.
What are "basis vectors" and how are they used?
Vectors--arithmetic, diagrammatic representations, relationships (e.g. why is i + j not a unit vector even though i and j are unit vectors?).  The Position Vector activity.

JR:  In what ways is arithmetic with vectors like that with real numbers?  In what ways does it differ?
pp. 822-823 #1, 5, 9, 15, 17.
What does it mean to "multiply vectors?"  What result would you expect from the product?
Dot Product.  The Regular Hexagon activity.  HERE are Edge's notes on Dot Product and HERE are Miss Welch's notes from last year.  Consider these online resources
JR:  Explain how to determine if vectors are parallel or perpendicular.
p. 830 #3, 5, 9, 15.

What must be true about the dot product of vectors a and b for them to be orthogonal?  for them to be parallel?  Explain.
pp. 830-831 #19, 23, 33, 35, 39, 41.

JR:  Compute the angle between the vectors <1, 0, -1> and 1, 1, 0>.  Explain the process and the meaning of the result in simple language.
Finish the assigned problems.
Which vector product provides the following?
  • A vector.
  • A scalar.
  • Volume of a rectangular solid.
  • Orthogonality.
  • Parallel-ness.
  • Commutativity.
Cross Product.  Messing with the Cross Product activity.    HERE are Edge's notes on Cross Product.  Consider these online resources

Note that
  • Dot Product essentially applies a vector's “growth in a direction.”  The dot product lets us apply the directional growth of one vector to another: the result is how much we went along the original path (positive progress, negative, or zero).  Example: <5, 0°> • <3, 0°> = 15; <5, 0°> • <3, 90°> = 0; and, <5, 30°> • <3, 60°> = 13.
  • Cross Product results in a vector perpendicular to the plane in which the given two vectors reside.  The  magnitude of the resultant vector is the same as the area of the parallelogram whose sides are derived from the given two vectors (the diagonal of the parallelogram is the sum of the given two vectors).
JR: If a • (b x c) = 0 then why MUST a, b, and c be coplanar?  Explain.
pp. 838 #3, 13, 15.
Review the linked online resources for Cross Product (see yesterday's activity). pp. 838-839 #19, 27, 29, 33, 35, 39.

JR:  Are the Dot Product and the Cross Product commutative in any non-trivial way?  Explain.
Finish the assigned problems.
A line contains the points (-8, 1, 4) and (3, -2, 4).  express the line in each of the following forms.
  • Parametric equations.
  • Symmetric equations.
  • A vector equation (e.g. r = r0 + tv).
Consider using Section 12.5 (pp. 840-847) as a resource.
Lines, planes, and surfaces.  The Match Game activity: You will be handed four index cards, each with equations of four different lines. Your task is, by clever trading with other groups, to wind up with four different descriptions of the same line.

JR:  Explain the relationship between determining the equation of a line based upon
  • The point slope method.
  • Knowing the coordinates of a point and a vector to which the line you want is parallel.
p. 848 #1, 3, 7, 19.
Describe the vertical traces of z = 4x2 + y2.
  • z = x2 + y2 + 1.
  • z2 = x2 + y2.
  • z = (1 - x2 - y2 )^(1/2).
Cylinders and Quadratic Surfaces.  The Matching Game activity.

JR:  Describe and sketch the surface resulting from x2/a2 + y2/b2 + z2/c2 = 1 if a, b, and c are different.
pp. 856-857 #3, 11, 15, 19.
Exercises pp. 859-860 #2,10, 7, 13, 14, 15, 18.
Chapter 12 AP AB/BC Review Questions; AP12-1 (following page 862).
Continue the review problems.

JR:  Explain in simple language why the Cross Product is a vector rather than a scalar, like the result of a Dot Product.
Finish the review problems.
A constant force of 3i + 5j + 10k moves a particle from (1, 0, 2) to (5, 3, 8).  Compute the work done on the particle if the force is in Newtons and the distance is in meters.
In-class review.

JR:  Suppose u * (v x w) = 2.  Compute
  • (u x v) * w
  • u * (w x v)
  • (u x v) * v
  • v * (u x w)
Prepare for the test.

THESE are Dr. Edge's notes on Solving for the equation of the plane containing three given points.

HERE are Miss Welch's Chapter 12 Review notes.
Get ready for the test!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.  Place your homework notebook near you for scoring. Test 12.1: Free Response: you may use one 3 x 5 note card* and YOUR calculator.  Expect questions on
  • Solve for conditions that will make a pair of vectors parallel or orthogonal.
  • Cross and dot product of vectors.
  • Distance, velocity, acceleration, work.
  • Create the equation of a sphere and the equation of a curve resulting from the sphere's intersection with a given plane.
  • Determine the center and radius of the sphere given its equation in terms of x, y, and z.
  • Determine the equation of a plane that contains three given points.
  • Solve a vector problem involving wind effecting a moving object.  Directions will be relative to a magnetic compass.
JR:  For which vector concepts are you still uncertain?
Discovery Project: Geometry of a Tetrahedron (p. 840).
*Unless otherwise noted, homework is due the next class day.

Chapter 13 Vector Functions

Entry Task



3/12 Sketch the curve described by the vector function r(t) = <sin t, t>.
Vector Functions and Space Curves.  The Angle Function activity.

JR:  Use geometry to derive the formula to use to compute the length of a vector in "three-space" (e.g. r(t) = <a(t), b(t), c(t)>.
Pgs. 869-870 #1, 3, 7, 9.
Compute the derivative of the vector function r(t) = <sin t, t>. Derivatives of Vector Functions. Velocity Vectors activity.  Note the theorem on Pg. 872 regarding differentiation rules for vectors and the table of derivatives on Pg. 874.

JR:  In what way does a tangent vector to a curve at a point P and the unit tangent vector to a curve at point P differ?
Pg. 876 #9, 19, 21, 23.
Explain what it means for a curve to be "smooth" and why this is important in determining curvature
Arc Length and Curvature.  The Length of the Reaper activity.

JR:  Examine how curvature is derived (see Pg. 879).  What does this really mean in the context of standard functions in 2-d?
Pg. 884 #1, 13, 17b, 23.
A particle moves along a curve defined by the equation y = sin (πx).  The x-coordinate, x(t), satisfies the equation dx/dt = e2t.  When t = 0 the particle is located at the point (0.5, 1).
  1. Determine x(t) in terms of t.
  2. Determine y(t) in terms of t.
  3. Determine the velocity vector, v.
Back To Start  activity.

JR:  Consider a vector function v = <f(t), g(t), h(t)>. Is it true that |v'(t)| = |v(t)|'? Why or why not?
Pg. 894 #1 (copy the table), 7, 9.

The position vector of an object is r(t) = (t2, sin(t) ).  Determine the velocity, speed, and acceleration of the object. Motion In Space. Tossing A Balloon activity.

A brief overview of Kepler's Laws.

JR:  Explain the meaning of "curvature."
Pg. 894 #7, 9, 15, 19.
What is the length of the curve r(t) = (12t, 5t). –10 ≤ t ≤ 10? Checking Out the Action activity.

JR:  Graph in 3-d the function 8x + 2y + 3z = 0 for –4 ≤ x ≤ 4; –4 ≤ y ≤ 4;  –10 ≤ z ≤ 10.
Pg. 898 #4, 5, 8, 11, 17.

Chapter 13 AP AB/BC Review Questions; AP13-1 (following page 900).  Do them all and show all work! In-class review.  Check answers.

JR:  What does equation 12 (Pg. 893) have to do with Kepler's Laws?  Explain the mathematical relationship!
Prepare for the quiz.
Get ready for the quiz!  Move to a seat where you have ample room, obtain all the materials you need before class starts, seat at most two at the square "cafe tables" and place the paper "blinders" between each pair of people.  Place your homework notebook near you for scoring. Quiz 13.1.  Free Response: you may use one 3 x 5 note card* and YOUR calculator.  Expect Questions on
  • Computing velocity and acceleration vectors given a vector function. 
  • Determine unit tangent vector .
  • Compute curvature.
  • Determine orthogonality.
  • Perform a definite integral of a vector function.
  • Arc length.
JR:  Which calculus topic(s) do you believe the class needs the greatest concentration of preparation before the AP test?
Collect everything you have that pertains to your previous work with series (previous notes, homework, etc.).  Explore at least one of the following resources
*Unless otherwise noted, homework is due the next class day.

Click here for the Reference Pages from Stewart Calculus (7th Edition).

Create and print your own graph paper at THIS Website.

Below are various documents on the operation of the class

Course syllabus

Also note the additional items

    Email: richard.edgerton@highlineschools.org