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Math 125: II - Dr. Loveless Essential Course Info My Course Website: math.washington.edu/~aloveles/ Homework Log-In (use UWNetID): webassign.net/washington/login.html Directions for Webassign code purchase: math.washington.edu/webassign Math Department 125 Course Page: math.washington.edu/~m125/

First week to do list 1. Read 4.9, 5.1, 5.2, and 5.3 of the book. Start attempting HW. 2. Print off the “worksheets” and bring them to quiz sections.

Today Week 1 assignments  Syllabus/Intro Closing time is always 11pm.  Section 4.9 - HW 1A,1B closes Wed - - HW 1C closes Fri (covers 4.9, 5.1, and 5.2) Expect 6-8 hrs of work, start today!

What we will do in this course: 5. Practicing , Trig and Precalc We learn calculus which Students often say: The hardest part of provides the essential tools for calculus is you have to know all your engineering, science and economics. precalculus, and they are right.

1. Ch. 5 – Defining the Integral Improving your algebra, trig and - Def’n and basic techniques precalculus skills will be one of the best benefits you will gain from this course. 2. Ch. 6 – Basic Integral Applications (Arguably as valuable as the course - Areas, Volumes content itself). - Average Value - Measuring Work You will use these skills often in your other courses at UW, so embrace, 3. Ch. 7 – Integration Techniques practice, and hone your precalculus - by parts, trig, trig sub, partial frac. skills.

4. Ch. 8-9 – More Applications - , Center of Mass - Differential Equations

Entry Task: Differentiate 7 1. 퐹(푥) = − 5√푥3 + 4ln (푥) 푥10 6푥 2. 퐺(푥) = 푒 + 5 tan(푥) + 휋 3. 퐻(푥) = 2 tan−1(푥) − 푒

4. 퐽(푥) = 푥3 cos(4푥) + ln(2)

4.9 Antiderivatives Def’n: If g(x) = f’(x), then we say g(x) = “the of f(x)”, and f(x) = “an of g(x)” Example: Give an antiderivative of 푔(푥) = 푥2.

Examples (you do): Find the general antiderivative of 1. 푓(푥) = 푥6 1 1 2. 푔(푥) = cos(푥) + + 푒푥 + 푥 1+푥2 5 3 3. ℎ(푥) = + √푥2 √푥 푥−3푥2 4. 푟(푥) = 푥3

Initial Conditions There is no way to know what “C” is unless we are given additional information. Such information is called an initial condition.

Example: 푓’(푥) = 푒푥 + 4푥 and 푓(0) = 5 Find 푓(푥).

Example: 푓′′(푥) = 15√푥, and 푓(1) = 0, 푓(4) = 1 푓(푥). Find

Example: Ron steps off the 10 meter high dive at his local pool. Find a formula for his height above the water. (Assume his acceleration is a constant 9.8 m/s2 downward)

5.1 Defining Area Calculus is based on limiting processes that “approach” the Calc. I exact answer to a rate question. Visual:

In Calculus I, you defined f’(x) = ` of the at x’ 푓(푥 + ℎ) − 푓(푥) = lim ℎ→0 ℎ

Calc. II In Calculus II, we will see that Visual: antiderivatives are related to the area `under’ a graph 푛

∗ = lim ∑ 푓(푥𝑖 )∆푥 푛→∞ 𝑖=1

Riemann sums set up: We build a procedure to get better and better approximations of the area “under” 푓(푥).

1. Break into n subintervals. 푏−푎 ∆푥 = and 푥𝑖 = 푎 + 𝑖∆푥 푛

2. Draw n rectangles.

Area of each rectangle =

(height)(width) = 푓(푥∗)∆푥 𝑖

3. Add up rectangle areas.

Example: Approx. the area under 푓(푥) = 푥3 from x = 0 to x = 1 using n = 3 subdivisions and right-endpoints to find the heights.

You do: Approx. the area under 푓(푥) = 푥3 from x = 0 to x = 1 using n = 4 subdivisions and left-endpoints to find the heights.

I did this again with 100 subdivisions, then 1000, then 10000.

Here is a summary of my findings:

General Pattern: (right-endpoint) Adding up the rectangle areas For 푓(푥) = 푥3 on x = 0 to x = 1. 푛 푛 𝑖 3 1 3 Sum = ∑ 푥𝑖 ∆푥 = ∑ ( ) 1−0 1 푛 푛 ∆푥 = = 𝑖=1 𝑖=1 푛 푛 푛 3 1 𝑖 𝑖 1 푥 = 0 + 𝑖 = Exact Area = lim ∑ ( ) 𝑖 푛 푛 푛→∞ 푛 푛 3 𝑖=1 3 𝑖 Height = 푓(푥𝑖) = 푥𝑖 = ( ) 푛 3 3 𝑖 1 Area = 푓(푥𝑖)∆푥 = 푥𝑖 ∆푥 = ( ) 푛 푛

Definition of the Definite Integral We define the exact area “under” f(x) from x = a to x = b curve to be 푛 lim ∑ 푓(푥𝑖)∆푥, 푛→∞ 𝑖=1 푏−푎 where ∆푥 = and 푛 푥𝑖 = 푎 + 𝑖∆푥.

We call this the definite integral of f(x) from x = a to x = b, and we write 푏 푛 ∫ 푓(푥)푑푥 = lim ∑ 푓(푥𝑖)∆푥 푛→∞ 푎 𝑖=1

Example: Write down this definition 푓(푥) = 푥 for the √ on the interval 푥 = 5 to 푥 = 7.