Download Limit worksheets - Canvas

Unit: Integration
MATH 3
Scott Pauls
Department of Mathematics
Dartmouth College
Instructor’s overview - 1
These slides are not meant to be prescriptive, but serve as a skeletal outline of the
course syllabus using active learning methods throughout the class.
In these slides we assume that most classes have the same general format:
1. (5-10 minutes) A brief introductory mini-lecture, often recapitulating and
extending the end of the previous class, leading into the first group activity.
2. (10-15 minutes) Group activity meant to explore a new topic or extend and old
one.
3. (5-10 minutes) A short follow up on the results and lead-in to a second piece
which is sometimes lecture/discussion and sometimes another period for
group work.
4. (10-15 minutes) Whatever the second piece is.
5. (5-10 minutes) Follow up on second piece
6. (5-10 minutes) A short introduction to the homework, preparation, and next
piece of material for the next class.
Instructor’s overview – 2
The problems used in the slides below are indicative what level
we expect throughout the course. Instructors will want to
supplement these with both easier and harder examples as
class progress dictates.
One goal in providing these templates is to help ensure a
uniform level of instruction to which we can test.
For each set of group work, there are a sequence of questions,
ordered by difficulty. Each one is aimed to build up to the next
with the goal of completing the last one, which is the most
abstract. Those that are circled are candidates for write-up and
grading by group.
Average final mastery by topic (F15)
Midterm 1
Midterm 2
Anti-differentiation and indefinite
integrals
𝑑
𝐹
𝑑𝑥
𝑥 = 𝐹′ 𝑥 = 𝑓 𝑥
∫ 𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶
Examples:
Function
Antiderivative
sin(𝑥)
𝑥2
+𝐶
2
− cos 𝑥 + 𝐶
cos(𝑥)
sin 𝑥 + 𝐶
𝑒𝑥
𝑒𝑥 + 𝐶
𝑥
Rectilinear motion
If 𝑝(𝑡) gives the position of an object in motion at time
𝑡, then
′
we know that 𝑣 𝑡 = 𝑝′(𝑡) is its velocity and 𝑎 𝑡 = 𝑣 𝑡 = 𝑝′′(𝑡) is
its acceleration.
Using anti-differentiation, we can start with the acceleration
due to gravity and work our way backwards.
If 𝑎 𝑡 = −9.8 𝑚/𝑠 2 then 𝑣 𝑡 = −9.8𝑡 + 𝐶.
What is the constant? If we know 𝑣 0 = 𝑣0 then 𝐶 = 𝑣0 .
Continuing, we anti-differentiate again yielding
9.8 2
𝑝 𝑡 =− −
𝑡 + 𝑣0 𝑡 + 𝐷.
2
Again, if we know 𝑝 0 = 𝑝0 then 𝐷 = 𝑝0 .
𝑣0
𝑝0
Group Work:
The Siege of Syracuse
1.
If the trajectory of a boulder is given by
(𝑥′′ 𝑡 , 𝑦 𝑡 ) with 𝑥2 0 = 𝑦 0 = 0 and we set
𝑦 0 = −9.8 𝑚/𝑠 and 𝑥 0 = 0, what are the
functions 𝑥(𝑡) and 𝑦 𝑡 ?
2.
If, as in the diagram, the catapult launches
with a fixed speed
but variable angle 𝜃, what
′
are 𝑥′(0) and 𝑦 0 ?
𝑠
3.
4.
Given all of this information, where does the
boulder hit the x-axis?
𝜃
𝑥′ 𝑡
Suppose a ship sits at (100 𝑚, 0) and you
have a catapult that launches with an initial
speed of 𝑠 = 25. How do we set up the
catapult to hit the ship?
(𝑥 ′ 𝑡 , 𝑦 ′ 𝑡 )
𝑦′ 𝑡
Challenge problems
1. In the catapult problem, we fixed the trajectory on a
plane. Can you generalize to all of three-dimensional
space?
2. What is the maximum range of a catapult with speed 𝑠?
How is that range achieved?
3. What are the benefits and drawbacks of placing catapult
higher or lower than the sea level?
Finding areas
What is the area of an object in
the plane?
Simplification: What is the area
between a curve 𝑦 = 𝑓(𝑥) and the
𝑥-axis?
A question we can answer: What
is the area of a rectangle?
Riemann sums
𝑦 = 𝑒 −𝑥
𝐴 = ℎ𝑤 = 𝑒 −2 ⋅ 1
With four rectangles (as in the figure)
the left endpoint Riemann sum is
𝑒 −0 ⋅ 1 + 𝑒 −1 ⋅ 1 + 𝑒 −2 ⋅ 2 + 𝑒 −3 ⋅ 1
ℎ
=𝑓 2
= 𝑒 −2
4
𝑒−
=
𝑖=1
𝑖−1
Δ𝑥
w = Δ𝑥 = 1
Riemann sums
For 𝑛 rectangles, Δ𝑥 =
𝑏−𝑎
𝑛
.
Sample points are given by:
{𝑎, 𝑎 + Δ𝑥, 𝑎 + 2Δ𝑥, … , 𝑎 + 𝑛Δ𝑥 = 𝑏}
Left endpoints: 𝑎, 𝑎 + Δ𝑥, 𝑎 + 2Δ𝑥, … , 𝑎 + 𝑛 − 1 Δ𝑥
Right endpoints: {𝑎 + Δ𝑥, 𝑎 + 2Δ𝑥, … , 𝑎 + 𝑛Δ𝑥 = 𝑏}
Area of 𝒌𝒕𝒉 box:
𝑓
Left endpoints: 𝑓 𝑎 + 𝑘 − 1 Δx Δ𝑥 =
Right endpoints: 𝑓 𝑎 + kΔx Δ𝑥 =
𝑓
𝑎+
𝑘 𝑏−𝑎
𝑎+
𝑛
𝑛
Riemann sums:
Left endpoints:
Right endpoints:
𝑛
𝑘=1 𝑓
𝑘−1 𝑏−𝑎 𝑏−𝑎
𝑛
𝑛
𝑘 𝑏−𝑎 𝑏−𝑎
+
𝑛
𝑛
𝑎+
𝑛
𝑘=1 𝑓
𝑎
𝑘−1 𝑏−𝑎
𝑛
𝑛
𝑏−𝑎
𝑏−𝑎
Riemann sums
Group work
Let 𝑓 𝑥 = 𝑥 2 − 𝑥 + 1.
3
1.
Estimate ∫1 𝑓 𝑥 𝑑𝑥 using Riemann sums with right endpoints with 3 rectangles.
2.
Write down the Riemann sum with left endpoints for ∫1 𝑓 𝑥 𝑑𝑥 with 𝑛
rectangles.
3.
Write down the Riemann sum with right endpoints for ∫𝑎 𝑓 𝑥 𝑑𝑥 with 6
rectangles.
4.
Write down the Riemann sum with left endpoints for ∫𝑎 𝑓 𝑥 𝑑𝑥 with 𝑛
rectangles.
3
𝑏
𝑏
Evaluating definite integrals
Find I =
4
∫1
𝑥 3 − 4𝑥 𝑑𝑥.
Left hand endpoints:
𝐼 = lim
𝑛→∞
𝑛
𝑖=1
1 + (𝑖 −
3 3
1)
𝑛
− 4 1 + (𝑖 −
Right hand endpoints:
𝐼 = lim
𝑛→∞
𝑛
𝑖=1
1+
3 3
𝑖
𝑛
−4 1+
3
𝑖
𝑛
3
𝑛
3
1)
𝑛
3
𝑛
Definite integrals
Group work
We’ll assign each group one of three simple
functions: 𝑓 𝑥 = 𝑥, 𝑥 2 , or 𝑥 3 . For your
function, complete the following problems:
𝑏
1.
Write down the definition of ∫𝑎 𝑓 𝑥 𝑑𝑥 using Riemann sums.
2.
Find Δ𝑥 for the sum using 𝑛 rectangles and compute the heights of the
rectangles using right or left-hand endpoints (your choice).
3.
Simplify the summands using algebra and the formulae above.
4.
Compute the resulting limits.
5.
Compute ∫𝑎 𝑓 𝑥 𝑑𝑥 using Riemann sums.
𝑏
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus
Group Work
Pick on of the graphs to the
right to 𝑥be 𝑓(𝑥) and let
𝑔 𝑥 = ∫0 𝑓 𝑡 𝑑𝑡.
1. At what values of 𝑥 do local maxima and
minima of 𝑔(𝑥) occur?
2. Where does 𝑔(𝑥) obtain its global maximum on
this interval?
3. On which intervals is 𝑔 𝑥 concave up and
concave down?
4. Sketch a graph of 𝑔 𝑥 .
Calculating areas with Riemann sums
Consider the Gaussian function
2
𝑥−𝜇
1
𝑓 𝑥 =
𝑒 2𝜎2
2𝜎 2 𝜋
1.
Integration techniques:
𝑢-substitution
FTC applied to the chain rule:
′
𝑓 𝑔 𝑥
= 𝑓 ′ 𝑔 𝑥 𝑔′ 𝑥
𝑏
𝑓 ′ 𝑔 𝑥 𝑔′ 𝑥 𝑑𝑥 = 𝑓 𝑔 𝑏
⟹
𝑎
− 𝑓(𝑔 𝑎 )
Substitution
Group work
Integration techniques:
integration by parts
FTC applied to the product rule:
′
𝑓 𝑥 𝑔 𝑥 = 𝑓 ′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′ 𝑥
𝑏
𝑓 ′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′ 𝑥 𝑑𝑥 = 𝑓 𝑏 𝑔 𝑏 − 𝑓 𝑎 𝑔(𝑎)
⟹
𝑎
Rewritten:
𝑏
∫𝑎 𝑓
′
𝑥 𝑔 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑔 𝑥
|𝑏𝑎
𝑏 ′
− ∫𝑎 𝑓
𝑥 𝑔 𝑥 𝑑𝑥
Integration by parts
Group work