http://www.dpcdsb.org/CAMPI/Learning/Departments/Mathematics/Grade+12+Calculus+and+Vectors.htm
Unit 1: Introduction to Calculus
1
Days
Learning Goals
Introduction to Course
Success Criteria
p. 2 -3, #1-4, 7 -8
2
1.1 Rationalizing the Denominator
p. 9 # 1-7
3
1.4 –The Limit of a Function: One Sided Limits
p. 38 #5, 6 -11
exclude 8, *12, *14
-graphs and equations
-left and right handed limits
- piecewise functions
-limit exists if : limit of f(x) as x approaches a- exists, limit of f(x) as x
approaches a+ exists, and these limits are equal to each other
-limit can exist even if it doesn’t equal f(a)
*look at lim 1  x as an example where one-sided limited doesn’t exist
x 1
4
1.5 – Properties of Limits
Strategies for evaluating a limit
-direct substitution
-factoring (indeterminate quotient/removable discontinuity)
-rationalizing
-one-sided limits
-change of variable method
p. 45 #1-4,6-10, 12,
13, 16, 17
5
1.6 – Continuity
p. 51 #4 – 8, 10-12,
14, 16
-definition of continuity: three conditions
a function continuous at x= a, if:
1. a is in domain of f(x),
2. limit f(x) as x  a exists
3. lim f(x) = f(a).
6
1.2 – Slope of a Tangent
lim
xa
f(x) - f(a)
xa
lim
h 0
f(x  h) - f(x)
h
p.18 #4, 8-11, 15-17,
*21, *24
-slope and equation of line
-one rational and one polynomial example
7
1.3 – Rates of Change
p. 29 #1,4, 7-14, 16,
20, 21, *22
8
Review
p. 56 # 1, 2, 4, 7, 11,
14, 15, 16– 19 (ace)
Unit 2: Derivatives
Day
1
Learning Goals
2.1 – The Derivative Function
-definition of derivative: slope of tangent/rate of change
-derivative at specific value a VS derivative of function
2
Success Criteria
p.73 #2, 5ad, 6bd,
7, 9, 10, TRY #14
2.1b – The Derivative Function, con’t
-the normal line
-differentiability
3
2.2 – Derivatives of Polynomial Functions
-constant function rule (prove using power (algebraically) and
using points where tangent is horizontal (graphically))
-power rule
-constant multiple rule
-sum/difference rule
4
2.3 – The Product Rule
5
(power of a function rule)
2.4 – The Quotient Rule
6
2.5 – The Derivatives of Composite Functions (Chain Rule)
-simple example and changing function into u=g(x) and y=f(x)
7
2.5b – Derivatives of Composite Functions, con’t
8
Derivatives Practice
Review
Test
p. 82 #2 –12, 1517
p. 90 #1-2, 5-7,
*12, *14
p. 97 #4 – 5 (ace),
6, 7, 13
p. 105 #4 –7
p. 106 #8, 9, 11-14,
17-19
Handout
p. 110 #2 – 12
Unit 3: Curve Sketching
Day
1
Learning Goals
3.1 – Higher Order Derivatives, Velocity and
Acceleration
Success Criteria
p. 127 #2 – 14, *16
-speed (absolute value of velocity)
-velocity
-acceleration/jerk or jolt
Word problems:
-max height/direction change/stops moving (v=0)
-return to original position (s=0)
-positive/negative direction: solving inequalities using
interval chart
2
4.1 – Increasing and Decreasing Functions
p. 169 #1- 9, 11
-graph function given derivative
3
4.2 – Critical Points, Local Maxima and Local Minima
4
-graph of derivative given function
-first derivative test
4.4 – Concavity and Points of Inflection
p. 178 #2, 3ac, 5ac, 7-11, 1315
p. 205 #1, 2, 3, 5, 8, 9
-concavity test/ second derivative test
- if f “(c) = 0 then it is not a max/min but a point of
inflection
5
4.3 – Vertical and Horizontal Asymptotes
p. 194 #1-5, 7, 9, 11, 14
6
4.5 – An Algorithm for Curve Sketching
p. 213 #4abd, 6, 7
Polynomial Functions
7
4.5b – An Algorithm for Curve Sketching
Rational Functions
8
Review
9
Test
p. 213 #4 c – i
p. 194 #10
p. 213 #4
p. 216 #3ac, 4, 5, 7, 8, 10,
13-16
Unit 4: Applications of Derivatives
Day
1
Learning Goals
3.2 – Minimum and Maximum on an Interval (Extreme Values)
Success Criteria
p. 135 #2 – 4,
6- 9, 11
-testing critical numbers (i.e. when derivative is zero) and endpoints
= abs max/min value is the largest/smallest
2
3.3 – Optimization Problems
p. 145 #1-7
1. Set up Optimization equation and Constraint Equation
2. Differentiate Op. Eq’n
-maximizing area/volume
-minimizing distance
3
3.3b – Optimization Problems, con’t
p. 145 #8-16, 19, 20
4
3.4 – Optimization Problems in Economics and Science
p. 151 # 1-15, *17, *18
*19
C ( x)
-avg cost:
x
-marginal cost: C’(x)
-marginal revenue: R’(x)
To max revenue: marginal rev = mar cost
To min avg cost: mar cost = avg cost
5
Problem Solving Practice
6
Ch 3 In Class Assignment # 1
7
*Implicit Differentiation (Appendix p. 561)
p. 156 # 14, 17, 18, 20,
21, 23
p. 564 #2ace, 3ac, 5, 9
-explicit (solved for y) VS implicit
-tangent to a circle/ellipse
-finding normal
8
*Related Rates (Appendix p.565)
-rate of change of area/volume VS rate of change of
radius/height/length/width (i.e. rate of “flow” problems)
10
*Related Rates, con’t
p. 569
# 1ab, 3-7, 12, 13, 15, 16,
*20
p. 569 #1cd, 8, 9, 11, 18
-rate of change of shadow’s length (similar triangles)
-rate of change of distance between two moving objects (Pyth. Th’m)
11
*Related Rates Practice
12
Ch 3 In Class Assignment #2
Worksheet
p. 170 -171 #1, 7, 8, 13,
14,
p. 178 – 179 # 2, 3, 4, 11
Unit 5: Derivatives of Exponential and Trigonometric Fns
Days
Learning Goals
Success Criteria
1
5.1 – Derivatives of Exponential Functions, y  e
2
5.2 – The Derivative of the General Exponential Function,
y b
x
p. 232 #1 –5, 7-13
p. 240 #1 – 6, *7
x
3
5.3 – Optimization Problems Involving Exponential Functions
4
Trigonometry Review
5
Trigonometry Review, con’t
6
5.4 – The Derivatives of y = sin x and y = cos x
p. 256 # 1 – 6, *9
7
5.5 – The Derivative of y = tan x
p. 260 # 1- 8, 10, 11
8
Review
p. 263 #1 – 3 (ace),
5, 7, 9, 11, 12, 13 –
15 (ace), 16
9
Test
10
Derivative of the Natural Logarithm (Appendix)
p. 575 #3-6, 8, 10
OPTIONAL
Derivative of general logarithmic functions (Appendix)
p. 578 #1-5, 7
Logarithmic Differentiation
p. 582 #1-5, 9, 10
(lna)(logax) = lnx
𝑙𝑜𝑔3
𝑙𝑜𝑔𝑥
Ex. (ln3)(log3x)=𝑙𝑜𝑔𝑒 × 𝑙𝑜𝑔3
𝑙𝑜𝑔𝑥
=𝑙𝑜𝑔𝑒
=logex
=lnx
p. 245 #3-8, 14, 16
*for #8 use t+60 for
growth function
p. 224 #5, 7, 10, 11
Unit 6: Introduction to Vectors
Days
Learning Goals
Success Criteria
1
Prerequisite Skills
p.273 #2 – 8
2
6.1 – Introduction to Vectors
p. 279 #1, 2, 4, 5 –
8, 11
-vectors VS. scalars
-geometric vectors
-parallel, opposite, equivalent, coincident?
3
p. 290 #1 – 7, 9,
11- 13
4
6.2 – Vector Addition
Subtraction, plane question (true/quad bearing)
6.3 – Multiplication of a Vector by a Scalar
-collinear, parallel, scalar multiple
1
-unit vector (vector with magnitude of 1): |𝑣⃗| 𝑣
5
6.4 – Properties of Vectors
p. 306 #5 – 11
6
6.5 – Vectors in R 2 and R 3
p. 316 #3, 5, 6, 8,
10, 15
7
6.6 – Operations with Algebraic Vectors in R 2
p. 324 #1 – 3, 5, 6 –
8(ac), 9, 10-13, 15,
16
-unit vector form vs. component form
-add, subtract, multiply
-vector between two points
-magnitude
p. 298 #2,4,5,7,8,9,
13,14
8
6.7 – Operations with Vectors in R 3
p. 332 #1- 8, 10-12,
15
10
Review
p. 344 #2 – 4, 7, 8,
11, 14, 15(not d), 16,
17, 23
TRY #21
11
Test
Unit 7: Applications of Vectors
Days
1
Learning Goals
7.1 - Vectors as Forces
-equilibrium system of 3 vectors
(triangle inequality: sum of any 2 sides must be great than or
equal to third side)
Success Criteria
p. 362 #5,6,8-18,
*19
-tension problem/ramp problem
2
7.2 - Velocity
3
7.3 - The Dot Product of Two Geometric Vectors
4
7.4 - The Dot Product of Algebraic Vectors
5
7.5 - Scalar and Vector Projections
-direction cosines
6
7.6 - The Cross Product of Two Vectors
7
7.7 - Applications of the Dot Product and Cross Product
p. 369 #2-12, *13,
14
p. 377 #2,3,6 -7
(left) 9, 11
p. 385 #3-16
p. 398 #1, 4, 5, 6,
7, 8, *11, *14
p. 407 #3, 4(left),
5, 7, 8, 9, 13
p. 415 #3, 5-9
-work, torque, triple scalar product, area of parallelogram
8
Review
9
Test
p. 418 #1-37
Unit 8: Equations of Lines and Planes
Days
1
2
Learning Goals
8.1 – Vector and Parametric Equations of a Line in R
-finding a point on the line
-checking to see if point is on line
-equation given two pts/ pt and another line (parallel/perp.)
8.2 – Cartesian Equation of a Line
(scalar equation of the line)
-vertical lines:
𝑥 = 𝑥0
𝑥 − 𝑥0 = 0
𝑟⃑ = (𝑥0 , 𝑦0 ) + 𝑡 (0, 𝑏)
2
Success Criteria
p. 433 #2-9
p. 443 #1, 3, 4, 6, 7,
8
-horizontal lines:
𝑦 = 𝑦0
𝑦 − 𝑦0 = 0
𝑟⃑ = (𝑥0 , 𝑦0 ) + 𝑡 (𝑎, 0)
3
8.3 – Vector, Parametric and Symmetric Equations of a Line
in R 3
4
-given two points
6.8 – Linear Combinations and Spanning Sets
5
8.4 – Vector and Parametric Equations of a Plane
6
8.5 – The Cartesian Equation of a Plane
7
-converting vector to scalar and vice versa
-angle between two lines
Review
8
Test
p. 449 #1–8, *13
p. 340 #9, 11, 13
p. 459 # 1, 4-6 , 911. 13. 14
(see ex 3, p. 457)
p. 468 #1, 5, 6, 7, 8
*11
p. 480 #1 - 34
Unit 9: Intersections
1
9.1 – The Intersection of a Line with a Plane and the
Intersection of Two Lines:
0
-infinite solutions 0t=0 -->𝑡 = 0
p. 496 #1, 4, 5, 7, 8,
9, 11
TWO DAYS
𝑛
-no solutions 0t=n --> 𝑡 = 0
*4 cases (lines)
See p. 156 #6a, 5ab, 9a (for skew LS not equal to RS)
*3 cases (line and plane)
See p. 159 #4cde
2
3
4
9.2 – Systems of Equations
*eliminate x, y or z in two equations and then use backward
sub
*use sub to eliminate a variable in other two equations
9.3 – Intersection of 2 Planes (3 cases)
9.4 – Intersection of 3 planes
-consistent (4 cases)
p. 507 # 3, 7, 9, 11,
12
p. 516 #1-11
p. 530 #1- 13
See p. 163 #7d, 5d, 5b, 6c
-inconsistent (4 cases)
See p. 163 #7a, 6ab
*for intersections of planes, lines, etc ALWAYS check parallel and
coincident first