AP Calculus BC
Chapter 10A - Parametric Functions
Lesson Plans
#1
Parametric Functions - Precal stuff
p. 703 Sec. 10.2 #5, 11, 15, 18, 21, 25, 29, 49,
(A) Find two sets of parametric equations for the rectangular equation x= y2 + 2
Critical Thinking Journal: (CTJ 40 pts.) This CTJ is due ________.
Discuss the following and work problems:
(1) How and why do you eliminate the parameter - 2 ways? (Work 1 example for each method not like anyone else’s ex.) (5 pts.)
(2) What are arrows of orientation and how do you find them? (Discuss 2 ways to find arrows
of orientation - with Calculus (derivatives) and without Calculus (helping graphs - not tables of
values). Give an example. In your example use calculus to find arrows of orientation and draw 2
helping graphs to find arrows.) (5 pts.)
(3) Which is the independent and dependent variables, how do you find domain and ranges,
and how might this affect the graph? (5 pts.)
(4) How do you use Calculus to determine if the parametric function is a smooth curve? Why
does “derivative” mean “smooth curve?” (5 pts.)
The following problems are also part of the CTJ.
(5) p. 703 #32a, c, d (15 pts.)
For each pair of parametric equations in #32,
(1) eliminate the parameter (to a recognizable function)
(2) draw a large graph with orientation arrows
(3) state exact domain of t or θ
(4) state ranges of x and y
(5) In a paragraph discuss their three similarities and differences.
(6) Sketch the following graphs in your CTJ and match to equations. Give three reasons without
eliminating the parameter. (5 pts.)
#2 Parametric Functions - Calculus stuff
p. 711 Sec. 10.3 #1-7 odd, 11, 15, 19, 31, 35, 46, 49, 65, 67, 79, 89, 97, 98
(A) Find the area enclosed by one loop of the cycloid with parametric equations
x = Q - sin Q and y = 1 - cos Q (Ans: 3p)
(B) Find the volume of rotating one loop of the curve in (A) above about the x-axis.
(Ans: 49.348)
TEST I: All formulas below. For 10 pts on HW Curve, you must have HW #1 and 2.
Parametric Functions
1. 1st and 2nd derivatives
2. How to find equations of horizontal and vertical tangents to parametric curves
3. Arc length
4. surface area of solid of revolution (about x and y-axes)
5. area enclosed by parametric curve (to x- and y- axes)
6. volume of solid of revolution using disks of area enclosed by parametric curves
Chapter 10A - BC Calculus Problem Set - Parametric Functions
Do not work on this sheet. Show all your work and explanations in your homework notebook.
(1)
The graph in the xy plane represented by x = sec t and y = tan t for −
(A) circle
(B) semicircle
(C) ellipse
d2y
3
(2) x = t + 1, y = t , then
= ( A)
2
dx
4t
2
(3)
3
(D) hyperbola
(B)
3
2t
( C ) 3t
π
π
< t < is:
2
2
(E) one branch of a hyperbola
( D ) 6t
(E)
3
2
The length of the curve determined by the equations x = t2 and y = t from t = 0 to t = 4 is
4
( A)
∫
4
4t + 1 dt
0
0
4
(D)
∫
0
( B ) 2 ∫ t + 1 dt
2
4t 2 + 1 dt
( E ) 2π
4
(C )
∫
2t 2 + 1 dt
0
4
∫
4t 2 + 1 dt
0
(4)
Consider the curve in the xy-plane represented by x = et and y = te−t for t > 0. Find the
slope of the line tangent to the curve at the point where x = 3. (calculator)
(A) 20.086 (B) 0.342 (C) −0.005 (D) −0.011 (E) −0.033
(5)
A curve in a plane is defined parametrically by the equation x = t3 + t and y = t4 + 2t2. An
equation of a line tangent to the curve at t = 1 is
(A) y = 2x (B) y = 8x (C) y = 2x − 1 (D) y = 4x − 5 (E) y = 8x + 13
(6)
Section II Problem: (calculator allowed but show work)
x1(t) = 3sin t − 3
y1(t) = t − 1 for 0 < t < 2p
(a) Sketch the graph of the system of parametric equations indicating the direction of increasing t.
(b) Find the position of the particle at t = 3.
(c) Find the slope of the line tangent to the graph at t = 3.
(d) Find the area enclosed by the parametric equations and the graph x2(t) = -3, y2(t) = t - 1.
(e) Find the volume of the solid of revolution formed by revolving the first curve around the
second curve from t = 0 to t = p.
Answers: (1) E, (2) A, (3) D, (4) D, (5) C, (6) (b) (−2.577, 2), (c) −0.337 (d) 12 (e) 44.413
Little Green Book of Calculus BC Properties
Ch. 10A – Parametric Functions
10A-1 Parametric Equations (definition, parameter, orientation of curve,
independent and dependent variables, domain, ranges)
10A-2 Eliminating Parameter by Solving Simultaneously (how, adjusting the
domain, ex)
10A-3 Eliminating Parameter by Using Trig Identities (how, adjusting the
domain, ex)
10A-4 First and Second derivative of Parametric Equations (formulas and ex.)
10A-5 Tangent Lines to Parametrically Defined Curves (horizontal, vertical and
oblique, example finding equation of line of each)
10A-6 Area inside Parametrically Defined Curves (formulas and ex.)
10A-7 Volume of Solid of Revolution of Parametrically Defined Curves
(formulas and ex.)
10A-8 Arc Length of Parametrically Defined Curves (formulas and ex.)
10A-9 Surface Area of Solid of Rev. of Parametrically Defined Curves (formulas
& ex.)
10B-10 Disher Philosophy – “80% rule – you should like at least 80% of all aspects
of your life as long as the other 20% is not physically, emotionally or
spiritually harmful.” (Comment on how you see this philosophy in my life and
how you can use it in your life.)