MCV4U Course Notes
Calculus And Vectors - University
Preparation (Ursula Franklin
Academy)
, Calculus and Vectors Notes
UNIT 1: Introduction to Calculus and Derivatives
Rate of Change and Limit Notation
Def.: 1. The average rate of change of a function over a given interval [a,b] is given by 𝑓(𝑏)−𝑓(𝑎)
. In
𝑏−𝑎
other words, it is the slope of the secant that joins the points (𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏)).
2. The tangent to a function, f, at 𝑥 = 𝑎, is the line that touches f at (𝑎, 𝑓(𝑎)), and best
approximates the function near x a .
3. For a given a, the slopes of the 𝑓(𝑎+ℎ)−𝑓(𝑎)
, approach the slope of the tangent to f at a,
secants, as h approaches 0. ℎ
Def.: The number (slope of the tangent) 𝑓(𝑎+ℎ)−𝑓(𝑎)
approaches as h approaches 0, from both the
ℎ
that left and from the right, is called the 𝑓(𝑎+ℎ)−𝑓(𝑎)
as h approaches 0 and is denoted
ℎ
limit of
𝑓(𝑎+ℎ)−𝑓(𝑎)
𝑚 = lim
𝑇𝐴𝑁 ℎ→0 ℎ
● Notice that 𝑓(𝑎+ℎ)−𝑓(𝑎)
is not defined at ℎ = 0,. We are only interested in what occurs as h
ℎ
approaches 0
● One can approximate the value of the limit by choosing several values approaching 0 from both
sides and looking for a pattern. Or choose a single nearby point and use the slope of the secant as
an estimate. We are seeking a more analytical approach.
Def.: A limiting process is one in which successive approximations are calculated for a measure
by making a variable approach a particular number or infinity.
Ex.: Approximating the slope of a tangents at a given point with the slope of secants using a
second point which approaches the given point
Approximating the area under a curve using rectangles whose bases are approaching 0
Approximating the perimeter/area of a circle with n-sided regular polygons where n is
approaching ∞.
Limits
While we have used the idea of a limit in the context determining the slope of a tangent (the number that
slopes of secants approach to give the slope of the tangent), we have seen that limiting processes are used in
other contexts (areas under curves, perimeter/area of circles, and others). For this reason, we define the
𝑓(𝑎+ℎ)−𝑓(𝑎)
concept of a limit for any function, and not just the limit of the expression apply the concept
,of limits in the other contexts mentioned. , so that we can
ℎ
, Def.: A function f has a limit, L∈ 𝑅, at x a if the function values, 𝑓(𝑥), approach L as x approaches a
from the right and from the left (The behaviour of the function at 𝑥 = 𝑎 is of no consequence). We
write lim 𝑓(𝑥) = 𝐿..
𝑥→𝑎
Def.: When a function has a limit at 𝑥 = 𝑎, we say that the limit exists at a.
Note: Scenarios where a function f does not have a limit at 𝑥 = 𝑎:
● The function approaches a different number from the left as it does from the right (a jump)
● The function approaches ±∞ from the left and/or right (vertical asymptote)
● You cannot approach 𝑥 = 𝑎 from the right and/or left (e.g. 𝑦 = √𝑥 at 𝑥 = 0)
● (Note that a function with a hole at 𝑥 = 𝑎 does have a limit - 𝑓(𝑥) approaches the same value as x
approaches a from the left and right; what happens at 𝑥 = 𝑎 is of no concern)
Recall:
Def.: A piecewise function is a function which is defined with different equations for different parts of its
domain.
Ex.: 𝑥, 𝑥 ≥ 0
𝑓(𝑥) = |𝑥| = {
−𝑥, 𝑥 < 0
Since the behaviour of a function might be different on either side of a point or may not even be defined to
one side of a point, we introduce the idea of right-handed and left-handed limits.
Def.: A function f has a right-handed limit, L, at 𝑥 = 𝑎 if 𝑓(𝑥) approaches L as x approaches a from the
right. We write 𝑥lim+ 𝑓(𝑥) = 𝐿.
→𝑎
A function f has a left-handed limit, L, at 𝑥 = 𝑎 if 𝑓(𝑥) approaches L as x approaches a from the
left. We write lim 𝑓(𝑥) = 𝐿.
𝑥→𝑎−
Note: A function, f, has a limit L at 𝑥 = 𝑎 if lim 𝑓(𝑥) = lim
− 𝑓(𝑥) = 𝐿
𝑥→𝑎+ 𝑥 →𝑎
Note: A function with an absolute value term can be written as a piecewise function by separating the
domain based on where the input to the absolute value is positive or negative. Where the input is positive, it
remains unchanged, and where it is negative, multiply it by −1 to make it positive.
𝑦 𝑥 + 3 , 𝑥 ≥ −3
E.g. = |𝑥 + 3| can be written as: 𝑦={
−(𝑥 + 3), 𝑥 < −3
𝑥 − (𝑥2 − 1) , 𝑥 < −1
𝑦 = 𝑥 − |𝑥2 − 1| can be written as: 𝑦 = {𝑥−[−(𝑥2 − 1)] , −1 ≤ 𝑥 ≤ 1
𝑥 − (𝑥 − 1) , 𝑥 > 1
2
Calculus And Vectors - University
Preparation (Ursula Franklin
Academy)
, Calculus and Vectors Notes
UNIT 1: Introduction to Calculus and Derivatives
Rate of Change and Limit Notation
Def.: 1. The average rate of change of a function over a given interval [a,b] is given by 𝑓(𝑏)−𝑓(𝑎)
. In
𝑏−𝑎
other words, it is the slope of the secant that joins the points (𝑎, 𝑓(𝑎)) and (𝑏, 𝑓(𝑏)).
2. The tangent to a function, f, at 𝑥 = 𝑎, is the line that touches f at (𝑎, 𝑓(𝑎)), and best
approximates the function near x a .
3. For a given a, the slopes of the 𝑓(𝑎+ℎ)−𝑓(𝑎)
, approach the slope of the tangent to f at a,
secants, as h approaches 0. ℎ
Def.: The number (slope of the tangent) 𝑓(𝑎+ℎ)−𝑓(𝑎)
approaches as h approaches 0, from both the
ℎ
that left and from the right, is called the 𝑓(𝑎+ℎ)−𝑓(𝑎)
as h approaches 0 and is denoted
ℎ
limit of
𝑓(𝑎+ℎ)−𝑓(𝑎)
𝑚 = lim
𝑇𝐴𝑁 ℎ→0 ℎ
● Notice that 𝑓(𝑎+ℎ)−𝑓(𝑎)
is not defined at ℎ = 0,. We are only interested in what occurs as h
ℎ
approaches 0
● One can approximate the value of the limit by choosing several values approaching 0 from both
sides and looking for a pattern. Or choose a single nearby point and use the slope of the secant as
an estimate. We are seeking a more analytical approach.
Def.: A limiting process is one in which successive approximations are calculated for a measure
by making a variable approach a particular number or infinity.
Ex.: Approximating the slope of a tangents at a given point with the slope of secants using a
second point which approaches the given point
Approximating the area under a curve using rectangles whose bases are approaching 0
Approximating the perimeter/area of a circle with n-sided regular polygons where n is
approaching ∞.
Limits
While we have used the idea of a limit in the context determining the slope of a tangent (the number that
slopes of secants approach to give the slope of the tangent), we have seen that limiting processes are used in
other contexts (areas under curves, perimeter/area of circles, and others). For this reason, we define the
𝑓(𝑎+ℎ)−𝑓(𝑎)
concept of a limit for any function, and not just the limit of the expression apply the concept
,of limits in the other contexts mentioned. , so that we can
ℎ
, Def.: A function f has a limit, L∈ 𝑅, at x a if the function values, 𝑓(𝑥), approach L as x approaches a
from the right and from the left (The behaviour of the function at 𝑥 = 𝑎 is of no consequence). We
write lim 𝑓(𝑥) = 𝐿..
𝑥→𝑎
Def.: When a function has a limit at 𝑥 = 𝑎, we say that the limit exists at a.
Note: Scenarios where a function f does not have a limit at 𝑥 = 𝑎:
● The function approaches a different number from the left as it does from the right (a jump)
● The function approaches ±∞ from the left and/or right (vertical asymptote)
● You cannot approach 𝑥 = 𝑎 from the right and/or left (e.g. 𝑦 = √𝑥 at 𝑥 = 0)
● (Note that a function with a hole at 𝑥 = 𝑎 does have a limit - 𝑓(𝑥) approaches the same value as x
approaches a from the left and right; what happens at 𝑥 = 𝑎 is of no concern)
Recall:
Def.: A piecewise function is a function which is defined with different equations for different parts of its
domain.
Ex.: 𝑥, 𝑥 ≥ 0
𝑓(𝑥) = |𝑥| = {
−𝑥, 𝑥 < 0
Since the behaviour of a function might be different on either side of a point or may not even be defined to
one side of a point, we introduce the idea of right-handed and left-handed limits.
Def.: A function f has a right-handed limit, L, at 𝑥 = 𝑎 if 𝑓(𝑥) approaches L as x approaches a from the
right. We write 𝑥lim+ 𝑓(𝑥) = 𝐿.
→𝑎
A function f has a left-handed limit, L, at 𝑥 = 𝑎 if 𝑓(𝑥) approaches L as x approaches a from the
left. We write lim 𝑓(𝑥) = 𝐿.
𝑥→𝑎−
Note: A function, f, has a limit L at 𝑥 = 𝑎 if lim 𝑓(𝑥) = lim
− 𝑓(𝑥) = 𝐿
𝑥→𝑎+ 𝑥 →𝑎
Note: A function with an absolute value term can be written as a piecewise function by separating the
domain based on where the input to the absolute value is positive or negative. Where the input is positive, it
remains unchanged, and where it is negative, multiply it by −1 to make it positive.
𝑦 𝑥 + 3 , 𝑥 ≥ −3
E.g. = |𝑥 + 3| can be written as: 𝑦={
−(𝑥 + 3), 𝑥 < −3
𝑥 − (𝑥2 − 1) , 𝑥 < −1
𝑦 = 𝑥 − |𝑥2 − 1| can be written as: 𝑦 = {𝑥−[−(𝑥2 − 1)] , −1 ≤ 𝑥 ≤ 1
𝑥 − (𝑥 − 1) , 𝑥 > 1
2
