Download Precalculus with Calculus Curriculum Map

Precalculus with Calculus
A one-year course that includes an extensive introduction to Elementary Differential Calculus
Time
Frame
3-4
Days
General Topic
Section(s) Content
Comments
Solving Equations: Linear, Quadratic,
Rational, Radical, Absolute Value,
Higher Degree
P1
Various methods of solving quadratics; use of a dummy
variable to solve a “pseudo-quadratic;” notion of extraneous
roots.
2-3
Days
Solving inequalities
P2
Linear, Absolute-value, quadratic, and rational inequalities;
interval notation; use of a number line and test points to find
intervals of solution
2 day
Rudiments of graphing
P4
Graphing a line; finding x and y intercepts; tests for
symmetry; graphs of circles
Spend one
day on
modeling
problems
“and” vs.
“or”
statements;
spend one
day on
modeling
problems
Include
finding
intercepts
of circles;
emphasize
writing
linear
equations
given
certain
information
2 day
Linear Equations
P5
Point-slope form of a line; Slope as a rate of change; yintercept as an initial conditions, alternate forms of the linear
equation e.g.,
(x/a + y/b = 1)
“Quest” on P1, 2, 4, and 5
1 day
8-10
Days
Functions
1 day
1 day
8-10
days
1 day
Theory of Polynomial Equations
1.1-1.6;
2.1
Definition of a relation; function; domain/codomain/range;
independent/dependent variable; methods of representing
functions (mappings, rule, table, verbal); graph of a function;
vertical line test, intervals of increase/decrease, local max/min,
monotonicity, symmetry tests (even/odd functions), basic
functions (constant, linear, quadratic, cubic, absolute value,
exponential, reciprocal, semi-circle, greatest integer, etc.),
transformations of functions, difference quotient/avg rate of
change, 1-1 functions, onto functions, restricting a domain to make
a function 1-1, composition of two or more functions (include
discussion of commutativity and associativity as they relate to
composition of functions), inverse functions, modeling problems
leading to linear and quadratic functions, intercepts as initial/end
conditions, optimization with quadratic equations or equations
containing quadratic expressions (e.g., quadratic expression under
a square root); modeling and variation problems
Ch 1
Review
Exercises
One Flex/Review Day
2.2-2.5
Major Test #1 (Chapters P, all of 1; 2.1)
Polynomial functions of higher degree; end behavior (introduce
limit language here); finding zeros by factoring (where possible);
sketching a probable graph; multiplicity and behavior near zeros
of the function; polynomial long division and synthetic division;
remainder and factor theorems; complex numbers: operations,
properties, notion of conjugate; fundamental theorem of algebra
and corollaries; rational zeros theorem; DesCartes’ rule of signs;
special theorems involving conjugates; linear factor theorem;
finding remaining zeros of a polynomial given a partial set; solving
a polynomial equation completely; sketching the graph of a
polynomial function; constructing a polynomial from given
information (assuming it’s sufficient).
Quest on Polynomial functions
Spend at
least three
days
exclusively
on
modeling
problems
3-4
days
Rational Functions
1 day
1 day
7-9
days
1 day
10-13
days
1 day
1 day
Intro to Conics; definitions; four
standard forms (include here
discussion of features such as
center, foci, asymptotes, major
and minor radii and axes, focal
radii, focal chords, directrices,
eccentricity, etc); translations of
the standard forms; writing the
equation given certain
information; general form of a
conic: completing the square to
rewrite in standard form;
classifying conics; graphing;
writing an equation from a graph.
Limits and Continuity
2.6
Find x and y intercepts, method of finding vertical and horizontal asymptotes,
asymptotic behavior, using symmetry to facilitate graphing, oblique
asymptotes, special cases (e.g., graphs with a hole, finding the coordinates of
the hole, graphs which cross their horizontal asymptotes, finding the point of
intersection of a graph with its horizontal asymptote).
Ch 2
Review
Exercises
One Flex/Review Day
12.1-12.3
3.1-3.5;
5.5
Ch 3
Review
Exercises
Major Test #2 (Chapter 2, excluding 2.1)
Only discuss directrix/directrices in relation to parabola, except for passing
mention; omit rotation of axes
Quest on Conics
Intro to Calculus: tangent line problem; area problem, evaluating limits
numerically, evaluating limits analytically (various techniques); basic limits;
properties of limits; limits of composite functions; limit evaluation strategies;
squeeze theorem; one-sided limits; continuity at a point and on an open
interval; continuity of piece-wise defined functions; intermediate-value
theorem; infinite limits; limits at infinity (revisit graphing rational functions)
Note: omit epsilon-delta and Trig limits
One Flex/Review Day
Major Test #3 (Chapter 3, include 5.5 also)
13-17
days
Basic Differentiation Rules
1 day
1 day
5-7
days
4.1-4.5
Ch 4
Review
Exercises
(omit
related
rate
problems)
Related Rates
4.6
1 day
2 days
0 days
3-4
days
Extrema on an Interval
5.1
The tangent line problem (more depth than in
Ch. 3); definition of derivative (show use of
both h and delta-x), finding the derivative
using the definition; alternate definition forms;
derivative as a rate of change; derivative at a
point; non-differentiable points (infinite
discontinuities, corners, cusps);
differentiability implies continuity; Basic
differentiations rules: constant multiple rule,
power rule (prove this); rewriting to use
power rule; sum and difference; product and
quotient; chain rule; implicit differentation;
tangent line problems; velocity, falling object,
and rectilinear motion problems; other
applications.
One Flex/Review Day
Major Test #4 (Chapter 4, omit 4.6 also)
Meaning of rate/related rates; steps for solving
related rate problems; various types of related
rate problems: e.g., cone (upright and
inverted), cylinder, sliding ladders, inflated
balloons, aviation/rocket, expanding solids,
rate of change of distance from a point to a
curve, non-routine problems
Quest on Related Rates (Optional)
Review for Midterm Exam
MIDTERM EXAM: REGENTS WEEK (TBA)
END OF SEMESTER ONE
Definition of extrema; terminology:
local/relative max/min, absolute max/min;
extreme value theorem; critical numbers;
finding extrema on a closed interval
Use proof judiciously; use
graphing calculator, Desmos,
and other demonstration tools
for graphic representations.
Apply implicit differentiation
techniques to conic sections;
include tangent line and other
application problems with
conics (revisit 12.1-12.3).
Use other sources besides text:
Anton book; AP test questions
Omit problems involving
Trigonometric functions
2 days
Rolle’s Theorem and the Mean
Value Theorem
5.2
Rolle’s Theorem; Mean Value Theorem
Include at least an informal
proof of Rolle’s Theorem;
Explain MVT as a “rotated”
version of Rolle.
Emphasize that on the AP exam, students
must state their conclusions verbally
and/or with mathematically concise
language (a number line argument alone is
not acceptable); Be sure to include two
important cases:
1) Problems where ‘c’ is a critical
number, even though f ’(c) fails to
exist
2) Problems where ‘c’ is a critical
number but f (c) is neither a local
max nor a local min.
Use “necessary” and “sufficient” language
here.
Present at least one case where the second
derivative test fails to yield a definitive
conclusion.
2 days
Increasing and Decreasing
Functions and the First Derivative
Test
5.3
Definitions of increasing and
decreasing; test for increasing and
decreasing; first derivative test;
general guidelines for finding
intervals of increase/decrease;
applying the first derivative test
(using a number line argument)
2 days
Concavity and the Second
Derivative Test
5.4
Definition of concavity; test for
concavity; definition of points of
inflections; finding points of
inflection; second derivative test
Present at least one case where a point of
inflection exists at x = c even though f ”(c)
does not exist (e.g., f ( x)  3 x )
1 day
Quest on 5.1-5.4
2-3
days
Curve Sketching
5.6
Analyze and sketch the graph of a
function without the aid of a
calculator (except for checking
purposes). Graph polynomial,
rational, and radical functions.
General guidelines for analyzing
the graph of a functions
3-4
days
Optimization Problems
5.7
Use Calculus to solve applied
minimum and maximum
problems: max volume,
minimizing distance, area, length
Use the Mnemonic ISEMICA to help sketch a
reasonable graph
I = Intercepts
S = Symmetry
E = Extent (domain and range)
M = Mini/Max
I = Inflection Points
C = Concavity
A = Asymptotes
Compare the graphs of f, f ’, and f ”
Be sure to include problems with endpoint
optimums.
Review general guidelines for setting up
and solving applied optimization problems.
Connect to prior optimization work done
without Calculus
2 days
1 day
1 day
Differentials; Local Linearity
5.8
Ch 5
Review
Exercises
Tangent line approximations;
differentials; difference between
dy and y ; estimation of error;
calculating a differential; finding
the differential of a composite
function; approximating function
values.
One Flex/Review Day
Major Test #6 (Chapter 5)
2 days
Exponential Functions
7.1
3-4
days
Logarithmic
Functions
7.2-7.3
2-3
days
Equations, Applications, and
Modeling with Exponential and
Logarithmic Functions
7.4-7.5
Graphing exponential functions;
domain and range of exponential
functions; graphing
transformations of exponential
functions, the natural base (e);
compound interest problems,
radioactive decay; use of graphing
utility; application problems.
Definition of log function; log
function as the inverse of the
exponential; the natural
logarithmic function (specifically
relate y = ex and y = ln(x)); domain
and range of the log function
(relate this to domain and range of
exponential); graphing
transformations of log functions;
properties of logarithms;
rewriting expressions using
properties; using log properties to
write equivalent expressions;
properties of the natural log.
Solving exponential and
logarithmic equations; using
exponential and logarithmic
models to solve application
problems (e.g., Finance,
Economics, Physics, and
Chemistry).
2-3
days
Derivative of exponential and
exponential functions
1 day
1 day
5-7
days
8.1-8.2
Derivatives of exponential and
logarithmic functions (general);
derivatives of y = ex and y = ln(x);
derivatives of composite
expressions that contain
exponential and logarithmic forms
(chain rule); logarithmic
differentiation.
Chapter 7
and 8
Review
Exercises
Review of Trigonometry
Chapter 9
all;
Chapter
10 all;
13.1-13.2
(omit integration problems)
Major Test #7 (Chapters 7 and 8)
Trigonometry of the general angle,
radian and degree measure,
graphing, solving triangles,
proving identities, solving trig
equations, use of formulas
(sum/diff, double angle, half angle,
etc.), inverse trig functions, law of
sines and cosines, curve sketching.
New: Sum to product formulas
(and vice versa), more complex
application problems and
modeling (e.g., problems involving
bearing and heading).
2-3
days
2-3
days
Limits of Trigonometric
Functions; Special Trig Limits
Derivatives of Trigonometric
Functions
11.1
11.2
Where applicable, revisit other
differentiation topics (e.g., optimization,
curve sketching, etc.) and apply those
techniques to exponential and logarithmic
functions.
Derivative of the six trig functions.
Revisit the Chain Rule and Implicit
differentiation and apply them to
problems that include trig
functions.
N.B.—Do not reteach Trig “from scratch.”
Many of these topics were covered in Trig,
but not in enough depth. Focus on giving
the students more depth of understanding
and choose more challenging examples in
the same topics. Also demonstrate proofs of
more important identities. The best
approach is to prove one or two basic
identities and direct the students to come
up with the rest.
Omit DeMoivre’s Theorem until later in the
course.
2-3
days
Applications of Derivatives of Trig
Functions
Not in text
2 days
Derivatives of Inverse
Trigonometric Functions
11.4
1 day
1 day
Revisit related rate, curve
sketching, and optimizations
topics and include problems that
involve trigonometric functions.
Rewrite y = arcsin x
as x = sin y and use implicit
differentiation and right triangle
trig to rewrite the expression. Use
a similar strategy for the other
functions.
Select
review
exercises
from Ch’s
9, 10, 11,
and 13
Use of the Anton text or other
supplementary texts may be needed here.
(omit integration problems)
Major Test #8 (Trig)
Min Days: 122
Max Days: 151
Avg: 136.5
Next: Parametric; Polar; DeMoivre’s Theorem; Matrices and Determinants/Cramer’s Rule, Vectors; Sequences and Series