Math 3, Section 3
Winter, 2017
January 9, 2017
Math 3, Section 3
Plan for Today:
● Tangent Line Problem
● Group Work
● Limits
● More Group Work
● Left and Right Limits
● Assignments for Wednesday
Math 3, Section 3
Tangent Line Problem
Math 3, Section 3
Tangent Line Problem
Math 3, Section 3
Tangent Line Problem
Math 3, Section 3
Tangent Line Problem
Math 3, Section 3
Tangent Line Problem
Math 3, Section 3
Equations of Secant Lines
Suppose I have two points (x1 , y1 ) and (x2 , y2 ). Then, the slope of
1
the line passing through these points is yx22 −y
−x1 .
The equation of the line going through the point (x0 , y0 ) with
slope m is y − y0 = m(x − x0 ).
Example
Let P = (2, 3) and Q = (4, 2). Then, the slope of the line PQ
(denoted mPQ ) is 2−3
4−2 = −1/2.
The equation of the line, in point-slope form is y − 3 = −1/2(x − 2).
Math 3, Section 3
Group Work
Let f (x) = x 2 .
1. What is the slope of the secant line passing through (1, 1) and
(2, 4)?
2. What is the slope of the secant line passing through (1, 1) and
(a, a2 ), when a =/ 1?
3. What is the slope of the secant line passing through (1, 1) and
(1 + h, (1 + h)2 ), where h =/ 0?
4. What is the slope and equation of the tangent line at (1, 1)?
5. What is the slope and equation of the tangent line at (a, a2 )?
Math 3, Section 3
Example: f (x) =
x 2 −1
x−1
Suppose that we have the
2 −1
function f (x) = xx−1
. It is not
defined at x = 1, but we can
figure out what the value should
be at x = 1 based on the value
of the points nearby.
Math 3, Section 3
x
2
1.5
1.25
1.125
1.0625
f(x)
3
2.5
2.25
2.125
2.0625
Intuitive Definition of a Limit
Definition
Suppose f (x) is defined when x is near the number a. (This
means that f is defined on some open interval that contains a,
except possibly a itself.) Then, we write limx→a f (x) = L if we can
make the values f (x) arbitrarily close to L (as close to L as we
like) by restricting x to be sufficiently close to a (on either side of
a) but not equal to a. Then, we say
lim f (x) = L.
x→a
Math 3, Section 3
More Group Work
6. Determine whether the following limits exist and if so, calculate
the value:
i. The limit as x → 1:
ii. limx→2 2x 3 + 1
iii. limx→0 sin(1/x)
iv. limx→0 1/x
v. limx→1
Math 3, Section 3
x 3 −6x 2 +3x+10
x−2
Left and Right Limits
What if I had a function such as:
⎧
⎪
⎪x if x ≥ 0
f (x) = ⎨
⎪
⎪
⎩2 if x < 0.
What is the limit of f as x goes to 0?
Math 3, Section 3
Intuitive Definition of One-sided limits
Definition
The Left-hand limit of f (x) as x approaches a is equal to L if we
can make the values of f (x) arbitrarily close to L by taking x to be
sufficiently close to a with x < a. We write
lim f (x) = L.
x→a−
Math 3, Section 3
Intuitive Definition of One-sided limits
Definition
The Right-hand limit of f (x) as x approaches a is equal to L if we
can make the values of f (x) arbitrarily close to L by taking x to be
sufficiently close to a with x > a. We write
lim f (x) = L.
x→a+
Math 3, Section 3
Formal Definition of a Limit
Definition
We say that limx→a f (x) = L if for every ε > 0, there exits a δ > 0
such that if ∣x − a∣ < δ then ∣f (x) − L∣ < ε.
Math 3, Section 3
Assignments for Wednesday:
1. Read 1.6-1.7 in Stewart.
2. Khan Academy problems (will be posted on Canvas soon!)
3. Finish Group Work if you haven’t already.
Math 3, Section 3