Calculus I March 30-April 3

Time Allotment: 40 minutes per day

Student Name: ________________________________

Teacher Name: ________________________________

Calculus March 23-27

1

Packet Overview Date Objective(s) Page Number

Monday, March 30 Average Value of a Function 2-4

Tuesday, March 31 Connecting Position, Velocity, and Acceleration Functions Using Integrals

5-9

Wednesday, April 1 Connecting Derivatives and Integrals to Motion

10-14

Thursday, April 2 Rate Curves 15-19

Friday, April 3 1. More Rate Curve Practice

2. Cumulative Quiz covering everything in this week’s packet

20-22

Additional Notes: This week’s packet is mostly hand-written. You should treat each lesson as a guided worksheet. Be on the lookout for graphs that need to be graphed by you, and for questions you need to answer. Most places for you to answer questions will be inside boxes I drew, but there may be a few questions you need to answer without a box. Be sure as always to work all assigned example problems and exercises. For the exercises and quizzes, do your best to work the problem with a pencil, then check the solutions on the last pages of your packet with a red pen.

Though not required to complete these assignments, Khan Academy’s AP Calculus AB series of videos are a helpful resource for supplemental learning. We’ll be going over the chapter on Applications of Integrals

I hope you enjoy this packet. It was a joy to make for you all, especially as these application lessons of integrals are some of my favorite topics in our calculus curriculum. Email if you have questions!

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___________________________

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the GHNO Academy Honor Code.

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Calculus March 23-27

2

Monday, March 30 Calculus Unit: Integral Applications Lesson 1: Average Value of a Function Unit Overview: Integral Applications All of our hard work learning the theory and mechanics behind integrals is about to pay off! Our next unit will explore applications of integrals. Some of these applications include applying integrals to quantities in physics (position, velocity, acceleration, forces, work, energy, etc.) and calculating areas and volumes of geometric shapes, including deriving the volume formulas of cylinders, cones, and spheres. By the end of this unit, you will understand more deeply why the physics equations we use in physics class work. You will also never have to memorize another physics equation for the rest of physics I, nor will you have to remember area and volume equations because you will be able to use calculus to derive them! Objective: Show how the average value of a function can be used to calculate the area under the curve. 1. Find the average value of a function, given two points. 2. Use the average value formula to find the area under the curve. Introduction to Lesson 1 Today we’ll learn what the average value of a function means. In its essence, it is the average of two endpoints on an interval, and is located halfway between the two endpoints (see the first figure at the top of the next page). What’s really interesting about this average value is that if you multiply it by the interval, you’ll geometrically get the area of a rectangle that is equal to the area under the curve (see the graph at the bottom of the page). The question to think about today is to ponder why that is the case. Why is the area of that rectangle equal to the area under the curve?

Calculus March 23-27

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Tuesday, March 31 Calculus Unit: Applications of Integrals Lesson 2: Connecting Position, Velocity, and Acceleration Functions Using Integrals Objective: Be able to do this by the end of this lesson. 1. Distinguish distance and displacement. 2. Distinguish speed and velocity. 3. Apply knowledge that the area under a velocity versus time graph is displacement. 4. Use integrals to find the area under the curve to find displacement, or positions of objects. Introduction to Lesson 2 Read this lesson carefully and fill in any blanks or graphs!

Calculus March 23-27

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Wednesday, April 1 Calculus Unit: Applications of Integrals Lesson 3: Connecting Derivatives and Integrals to Motion Objective: Be able to do this by the end of this lesson. 1. Use integrals, or antiderivatives to work backwards to get position, velocity, and acceleration functions. 2. Use integrals to solve one dimensional motion problems. Mr. Bailey escapes a 6th Period Calculus April Fool’s prank this year! Introduction to Lesson 3 This section introduces some of the most beautiful and fundamental calculus applications in physics. You’ll start by reviewing what happened when you take the derivative of position and velocity functions, and then you’ll end by seeing what happens when you take the antiderivative of acceleration and velocity functions!

Calculus March 23-27

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Thursday, April 2 Calculus Unit: Applications of Integrals Lesson 4: Rate Curves Objective: Be able to do this by the end of this lesson. 1. Compare rate of change functions to their graphs. 2. Draw graphs of functions to find the area under the curve. Introduction to Lesson 4 Remember how we learned that the definite integral gives you the area under the curve of a function between two boundaries? In this lesson, we’re going to review that concept, as well as learn how to solve word problems requiring us to set up integrals.

Calculus March 23-27

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Friday, April 3 Calculus Unit: Applications of Integrals Lesson 4: Evaluating Integrals of Even and Odd Functions Objective: Be able to do this by the end of this lesson. 1. Practice setting up and solving rate curve problems. 2. Take a quiz on the week’s work! Introduction to Lesson 4 This is a practice day! Warmup by solving some rate curve problems introduced yesterday, then take the quiz and have a great weekend!

Name:__________________________

Quiz

1. A particle moves in a straight line with velocity in meters per seconddeterminedby the functionv(t),wheret is time in seconds. Att= 3, theparticle’s distance from the starting point was 9 meters in the positivedirection.Whatexpressionshouldweusetodeterminetheparticle’spositionatt=4seconds?

a) ! 4 − !(3)

b) 9 + !! 4

c) 9 + ! ! !"!!

d) ! ! !"!

!

2. Thevelocityofaparticlemovingalongthex-axisisv(t)=t2+t.Att=1,itspositionis1.Whatisthepositionoftheparticle,s(t),atanytimet?

3. The cumulative cost of purchasing and maintaining Julia’s computer is

increasingatarateofr(t)dollarsperyear(wheretisthetimeinyears).Att=1,Juliahadspentatotalof$420onhercomputer.Whatdoes420 + ! ! !" = 570!

! mean?

4. Asaparticlemovesalong thenumber line, itspositionat timet iss(t), itsvelocityisv(t),anditsaccelerationisa(t)=3t2.Ifv(0)=3ands(0)=1,whatiss(2)?

5. Whatistheaveragevalueof6x2+8ontheinterval[3,5]?

Quiz

1. A particle moves in a straight line with velocity in meters per seconddeterminedby the functionv(t),wheret is time in seconds. Att= 3, theparticle’s distance from the starting point was 9 meters in the positive

direction.Whatexpressionshouldweusetodeterminetheparticle’sposition

att=4seconds?

a) ! 4 − !(3)b) 9+ !′(4) c) 9+ ! ! !"!

!

d) ! ! !"!!

Answer:9+ ! ! !"!!

Sincewe know that at t=3, the particle has traveled 9meters. To find the

particlespositionafter4seconds,wemustaddthedistancethattheparticle

has then traveled (the displacement) in the time interval [3, 4] to 9, the

distancebetweenbeingfoundby

! ! !"!!

2. Thevelocityofaparticlemovingalongthex-axisisv(t)=t2+t.Att=1,itspositionis1.Whatisthepositionoftheparticle,s(t),atanytimet?

Answer: ! ! = !! !! + !

! !! + !!

Finding the indefinite integral for v(t), we arrive at

! ! = !! !! + !

! !! + !

which tells us the position of the particle at some time t. Since we know its position at t = 1, we can solve for c

1 = !! 1! + !

! 1! + !

! = 1− !! + !! = 1− !

!

! = !!

We can then plug in c to the formula above to get our answer.

3. The cumulative cost of purchasing and maintaining Julia’s computer isincreasingatarateofr(t)dollarsperyear(wheretisthetimeinyears).Att=1,Juliahadspentatotalof$420onhercomputer.

Whatdoes420 + ! ! !" = 570!! mean?

Answer:Juliahadspentatotalof$570maintainingandpurchasingmaterial

forhercomputerbytheendofthe5thyear.Shespent$420bytheendofher

firstyear,and ! ! !" !! tellsustheamountshespentfromyears1through

5.

4. Asaparticlemovesalong thenumber line, itspositionat timet iss(t), itsvelocityisv(t),anditsaccelerationisa(t)=3t2.

Ifv(0)=3ands(0)=1,whatiss(2)?

Answer:s(2)=11

v(t)= ! ! !" = !! ∗ 3!! + !! 0 = 3 = 0! + !.! = 3.! ! = !! + 3 = !

! !! + 3! + !

! 0 = 1 = !! 0! + 3 ∗ 0+ !. ! = 1

! 2 = !! 2! + 3 ∗ 2+ 1 = 4+ 6+ 1 = 11

5. Whatistheaveragevalueof6x2+8ontheinterval[3,5]?

Answer:106

!!"# = !!!! 6!! + 8 !" !

!

6!! + 8 !" = !! !

! + 8! + ! 6!! + 8 !" !

! = 2 5! + 8 5 + ! − [2 3! + 8 3 + !] 290−78=212(thec’scancel) !!"# = !! 212 = 106