Hayfield Secondary AP Summer “Continuity of Learning” Cover Sheet Course

Teacher Names & Email Addresses Jack Mika: jjmika@fcps.edu Mr. Glynn: TEGlynn@fcps.edu Ms. Dash: madash@fcps.edu

Title AB/BC Calculus Summer 2020: Continuity of Learning BC has ONE section that is BC only. Clearly labeled Otherwise, the opportunities are the same. Calculus is Calculus! AND: This opportunity is based on Algebra and Pre-Calculus, you do not need to learn or use calculus to complete this activity!!

Date Available June 2020

Date Due: NONE Optional work to help you prepare: Not collected

Objective/Purpose of Opportunity Review material from Previous Math classes that will be useful for success in AB/BC CALCULUS

Description of Worksheet No grade

Grade Value of Worksheet Non graded optional work

Tools/Resources Needed to Complete Opportunity

Old Notes from previous classes (precalculus and Algebra 2 material ON LINE SITES . PatrickJMT is great!) http://patrickjmt.com/

Estimated Length 3-6 hours depending on how much you remember Suggested supplies for class in the fall at home or in school: You should be organized! At least one composition book for homework. A notebook for class notes (3 ring or composition book) A 3 ring binder to keep all handouts/quizzes/test. Dry erase markers for use in class (if we are face to face)

Hayfield SS: Summer practice

NOTES:

• Section IV is for BC students only. Sections X and XI are for students that have seen limits and

derivatives: BC Students should have seen these. ALL other sections are helpful for both classes.

• Section XIV is for studying/memorizing only. There is no written work.

• Suggestion: If you work on these activities, begin to organize this and all your work throughout

the year. Having your papers organized is a key study skill in all your classes.

I For each of the following functions

➢ Make a table of values for the function that clearly shows the main points.

➢ Sketch the graph for the function. When you sketch the graph of the function, you should be

able to label all the points from your table. That is: if the point is in the table, know where the

point is located on the graph

➢ ELIMINATE THE WORD “IT” from your vocabulary! We will talk about this during the year.

You should be aware of the following types of points: These important points should include any key

features such as:

• The roots (x-intercepts)

• Y-intercepts

• Asymptotes: vertical and horizontal

• Local maximum and minimum values.

• Period if applicable

Tables should be big enough to illustrate these features

Also, know how to label the following:

• Label by name any local maximum or minimum values eg. This is a local max and use an arrow to

point to the dot on the graph.

• Sketch asymptotes with a “broken line”.

• Show in the table the roots and y-intercept

Here are your functions:

a) 2

2

6 5

xy

x x

+=

− +

b) 3 1xy e= −

c)

2xy e−=

Be sure to do this problem in RADIANS: Leave points in terms or pi

II For each of the following, find the area described.

• Be sure to sketch the picture described. Draw the functions for the given domain

• Shade the region enclosed by the functions described.

• Show how you calculated the total area described.

a) The area enclosed by the following lines:

• 2y x=

• 3y =

• 2 8y x= − +

• The horizontal x-axis ( 0y = )

b) The area enclosed by the functions

• 29y x= −

• The horizontal x-axis ( 0y = )

c) Use 2 rectangles to estimate the area enclosed by the graphs of:

• 2xy =

• 0x =

• 3x =

• The x-axis: 0y =

Give an explanation for your estimate. Do you think the estimate is TOO HIGH or TOO

LOW. (Hint: draw rectangles inside or outside of your picture and use a calculator to help

you find the areas of the rectangles)

III Find the “inverse function” for each of the following:

Show all work for FINDING the inverse function.

Graph the inverse function and the original function on the same axis (x-y plane)

You should also draw the line y x= as a broken line. Recall that a function and its inverse are

symmetrical about the line y x= .

a) 22 1; 0y x x= −

b) 2xy e=

c) cos ; 0y x x =

IV THIS SECTION IS BC ONLY: AB should continue on to section V

Summation: Find the following sums and answer the related questions:

a) 32

1

1

n n=

=

What do you think will happen to the sum as n gets larger?

Hint: Consider the sum of the first 64 terms!

b) 16

21

1

n n=

=

What do you think will happen to the sum as n gets larger?

c) 1

(1 )n

n+ =

What do you think will happen to the terms of this sequence as n gets larger?

Do this for the following values to try to get a “feel” for what is happening: Use these values

of n: 1, 10, 100, 1000, 10 000, 100 000 This should give you an estimate for the number.

Do you recognize the number you see?

What number is this approaching as n approaches infinity?

NOTE ABOUT c) above:

Recall:

• As n gets very large, 1

0n→

• As n gets very large, (1.00001)n → That means the expression get REALLY big for big values of

“n”

• (1) 1n = for any value of n.

• Notice the above answer shows that some of the obvious rules may not apply when they are in

conflict in the same problem.

V Find the following composition of functions:

Given the following:

( ) sinf x x= , 2( ) 2 3 1g x x x= − + , 5

( )1

h xx

=+

Find:

a) ( ( ))f h x =

b) ( ( ))g h x =

c) ( ( ))h g x =

VI Write the following function as the composition of TWO functions: (decompose into two

functions)

Ex: 2

3( )

5h x

x x=

+ is the composition:

( ) ( ( ))h x f g x= where: 3

( )f xx

= is the “big picture” and 2( ) 5g x x x= + is the detail in the function.

Let f(x) be the “big picture) and g(x) be the details within

a) 2 4( ) ( 5 6)h x x x= + +

b) 2( ) sin( 6 1)h x x x= + +

c) 2

6( )

(3 5)h x

x=

+

VII Apply Log rules to rewrite the following as sums / differences and coefficients for logs.

Ex: 3 2ln(5 ) ln 5 3ln 2lnx y x y = + +

1. 3ln(7 )x =

2. 3

2ln( )

x

y=

3. 3ln(2 )x =

4. 5ln(( 7 3)(2 ))x x+ =

VIII Solve for y:

a) ln 2y x= +

b) 3 5 2 7xy y x+ = +

c) 2 23 18 2 5x y xy x− = +

IX Review from Pre-Calculus Honors

Find the following limits: Given the graphs below of ( )f x on the left and ( )g x on the right. Use these graphs to

answer the following questions:

( )f x ( )g x

1. 1

lim ( )x

f x+→

=

2. 1

lim ( )x

g x+→

=

3. 1

lim ( )x

f g x−→

=

4. 1

lim ( )x

g f x−→

=

Given the table below:

a lim ( )x a

f x+→

lim ( )x a

f x−→

( )f a

-1 4 6 4

0 -3 -3 5

1 2 2 2

Find the following:

5. 0

lim ( )x

f x→

=

6. 1

lim ( )x

f x→−

=

7. 1

lim ( )x

f x→

=

Find the following Limits:

8. 3

2 3lim

3x

x

x−→−

− +=

+

9. 2

3

5 6lim

3x

x x

x→

− +=

−

10.

2

sinlim

2x

x

x→

=

11. 3 2

3

2 5 2020lim

2021 7 8x

x x x

x x→

− + −=

+ −

12. 2

3 20

2 5lim

8 2 7x

x x

x x x→

+=

+ +

X From Honors Pre-Calculus: Find the following derivatives: Given ( )f x , find dy

dx

1. 2( ) 3 2 5f x x x= + +

2. 2 3( ) ( 2 5)(7 4 )f x x x x x = + + + +

3. 23 2 5

( )2 7

x xf x

x

+ +=

+

4. ( ) 3f x =

5. 7( ) (2 3)f x x= +

XI For each of the following:

Given 2( ) 3 2 4f x x x= + − , find the following:

1. ( )f a

2. (5)f

3. ( ) (5)f x f−

4. ( ) (5)

5

f x f

x

−

− (simplify your answer!)

5. ( ) ( )f x h f x

h

+ − (simplify your answer)

XII. Graphs/Properties of Functions: You may want to organize the following in a way you can save and use for a reference for the year. BECOME FAMILIAR WITH THE FOLLOWING FUNCTIONS, GRAPHS AND THE GIVEN PROPERTIES OF THESE FUNCTIONS. .

The Function Domain Range Type of Symmetry:

x-axis, y-axis, origin

Odd, Even,

Neither Intervals of

Increasing/Decreasing

a) y = 1

b) y = x

c) y = x2

d) y = x3

e) y = x4

f) y =

g)

h)

i)

j) y = |x|

k)

l) y = lnx

m) y = ex

n) y = sinx

o) y = cosx

XIII Try to imagine what a sketch would look like with the following properties: Try to make up a scenario that would

a) Sketch a piece of a graph that is increasing and concave up. *Make up a scenario/situation that may be reasonably represented by this graph! BE CREATIVE!

b) Sketch a piece of a graph that is increasing and concave down. Make up a scenario/situation that may be reasonably represented by this graph!

c) Sketch a piece of a graph that is decreasing and concave up. Make up a scenario/situation that may be reasonably represented by this graph

d) Sketch a piece of a graph that is decreasing and concave down. Make up a scenario/situation that may be reasonably represented by this graph

e) For all values of the domain, x, the function ( ) 0,f x ( )f x is increasing for all values of x

and ( )f x is concave up for all values of x.

f) (0) 2f = , ( )f x is increasing for all x >0, ( )f x is concave down for all x >0.

g) (3) 5f = , ( )f x is increasing for all x <3, ( )f x is decreasing for all x >3. ( )f x is a

continuous (no breaks in the graph).

h) (3) 5f = , ( )f x is increasing and concave up for all x <3, ( )f x is decreasing and concave

down for all x >3. ( )f x is a continuous (no breaks in the graph).

i) The population is growing at a rate that is directly proportional to the current population.

That is: as the population increases, so does the rate of growth for the population. THIS IS EXPONENTIAL GROWTH

XIV The following should be familiar. Know the Pythagorean and Reciprocal Identities

Reciprocal Identities Quotient Identities Pythagorean Identities

xx

csc

1sin =

xx

sin

1csc =

xx

sec

1cos =

xx

cos

1sec =

xx

cot

1tan =

xx

tan

1cot =

sintan

cos

coscot

sin

xx

x

xx

x

=

=

2 2

2 2

2 2

sin cos 1

tan 1 sec

1 cot csc

x x

x x

x x

+ =

+ =

+ =

Co-Function Identities Odd/Even Identities

sin cos2

− =

cos sin

2

− =

csc sec2

− =

sec csc

2

− =

tan cot2

− =

cot tan

2

− =

Odd Even

( )sin sin − = − ( )cos cos − =

( )csc csc − = − ( )sec sec − =

( )tan tan − = −

( )cot cot − = −

Double Angle Identities Half Angle Identities

2 2

2

2

sin 2 2sin cos

cos 2 cos sin

cos 2 2cos 1

cos 2 1 2sin

x x x

x x x

x x

x x

=

= −

= −

= −

2

2

1 cos 2sin

2

1 cos 2cos

2

xx

xx

−=

+=

The Radian Measures and Coordinates MUST be memorized

Remember: coordinateyr

y−==sin , coordinatex

r

x−==cos , and

coordinatex

coordinatey

x

y

−

−==tan

KNOW AND UNDERSTAND ALL THE FOLLOWING FORMULAS and their uses.

Equation of a line:

CByAx

xxmyy

bmxy

=+

−=−

+=

)( 11

Slope of a line:

12

12

xx

yym

−

−=

Parallel lines have the same slope

Perpendicular lines have slopes that are “negative reciprocals”

Area of a rectangle is length x width

MATERIALS FOR CLASS.

You should have

• Composition book for homework

• Notebook for class notes as well as a ring binder or large trapper to keep test, quizzes, handouts, etc.

• Dry erase markers so that we can use individual white boards for class practice.

• A Graphing Calculator

THE NEXT PAGES ARE FOR YOUR READING AND UNDERSTANDING. This will be a great

resource throughout the year but also very helpful to understand as soon as possible to know WHY

we are doing the things we do.

CHANGE: Please read through this to try to understand the most important concepts that you will see

regularly throughout the year. These concepts show up as a MAJOR PART of the exam every year.

Understanding these concepts is one of the MOST IMPORTANT FOUNDATIONS FOR THIS COURSE!

• Rate of change

Average rate of change

Instantaneous rate of change

• Total Change

• Average Value of a function

SLOPE: Gives a rate of change

Given a function ( )f x :

The Average rate of change of the function, ( )f x over the interval from “a” to “b” is the secant line:

Average rate of change of ( ) ( )

( )f b f a

f xb a

−=

− which is the SLOPE of the secant line.

The instantaneous rate of change of ( )f x at the point ( , ( ))c f c is the derivative of the function evaluated

when x c= : '( )f c

The Total Change of a function over the interval (a,b) is the INTEGRAL of the RATE OF CHANGE of the

function.

Total change of ( )f x from time “a” to time “b” = '( )

b

a

f x dx

The AVERAGE VALUE of a function is the “sum” (the integral) of the function divided by the “width”

(the distance from “a” to “b”) of the function.

Average value of ( )f x over the interval (a,b) : 1

( )

b

ave

a

f f x dxb a

=−

NB:

Given a function: the “AVERAGE RATE OF CHANGE” is the slope of the secant.

Given a rate of change of a function, the AVERAGE rate of change is the AVERAGE VALUE of the rate of

change (average value formula above applied to '( )f x ).

Let’s consider some of the above with reference to a common application.

Position of a function: ( )s t

Velocity of a function: ( ) '( )v t s t= This the slope of position at any given time.

Acceleration of a function: ( ) '( ) ''( )a t v t s t= = This is the slope of velocity at any given time.

SLOPE: Gives a rate of change

Given a function ( )s t :

The Average rate of change of the function, ( )s t over the interval from “a” to “b” is the secant line:

Average rate of change or AVERAGE VELOCITY* over the interval (a,b) is given by the SLOPE

( ) ( )s b s a

b a

−

− which is the SLOPE of the secant line from “a” to “b”

The AVERAGE ACCELERATION over the interval (a,b) would be the Average rate of change of the

velocity. ( ) ( )v b v a

b a

−

−

The instantaneous rate of change, or VELOCITY AT TIME: t c= is the derivative of the position

function evaluated when t c= : so ( ) '( )v c s c=

The acceleration at t c= : ( ) '( )a c v c= , the rate of change of the velocity at time “c”

The Total Change of a function over the interval (a,b) is the INTEGRAL of the RATE OF CHANGE of the

function.

Since Total Change of position is displacement and rate of change of position is velocity

Total displacement of ( )s t from time “a” to time “b” = ( ) '( ) ( ) ( )

b b

a a

v t dt s t dt s b s a= = −

The AVERAGE VELOCITY of a function is the “sum” (the integral) of the velocity divided by the “width”

(the time from “a” to “b”) of the velocity function.

Average velocity over the interval (a,b) : 1 ( ) ( )

( )

b

ave

a

s b s av v t dt

b a b a

−= =

− −*

*Notice average velocity formula above by using position or velocity

Examples to practice the above concepts: The problems are made to be set up. There is insufficient

information to solve the problem.

1. From 1:00 pm to 5:00 pm, people are entering a stadium

at a rate of ( ) .....r t =

1000 people are already in the stadium at 1:00 pm.

a. How many people are in the stadium at 3:00 pm?

b. What is the average rate that people are entering the

stadium between 1 pm and 5 pm?

2. On July 4th the temperature outside is given by the

function T(t); where midnight is t-0 and time, t, is in hours.

a. What is the average rate of change in temperature from 9

AM until 1:30 PM?

b. How is the temperature changing at 6:00 PM

c. What was the average outside temperature for the 4th of

July.

The following problems are from the 2018 FRQ

3. People enter a line to get onto an escalator at a rate of

( )r t which is people per second and where t is in seconds. As people are entering the line for the

escalator, others are exiting the line and getting onto the escalator at constant rate 0.7 people per

second. There are 20 people in line for the escalator at time t=0.

a. How many people enter the line in the first 2 minutes?

b. How many people are in line at the end of 2 minutes?

c. For the first 5 minutes, someone is always in line for the

escalator. After 5 minutes, when is the first time, k, that there is no-one in line for the

escalator.

4. A particle moves along the x-axis with a velocity given

by ( )v t for 0 10t . At t = 0 , the particles position is 8− . ( (0) 8s = − )

a. Find the particles acceleration at t = 3.

b. Find the position of the particle at t = 3.

c. Interpret the meaning of 3

0

( )v t dt

5. The height of a tree is given by the function ( )H t where

( )H t is measured in meters and t is measured in years.

a. Using correct units, interpret '(6)H in the context of this

problem.

b. Find an expression for the average height of the tree over

the interval 2 10t

c. Write an expression for the average rate of growth of the

tree over the time period from 3 10t .

Let’s try one that has a defined function: (you can answer part a. and simply represent the answer to part

b.

6. Let ( )f x be the function defined by : ( ) cosxf x e x=

a. Find the average rate of change of ( )f x on the interval

0 x

b. What is the slope of the function at3

2x

= ?

A very challenging problem (for now)

7. Researchers on a boat are measuring plankton cells in a

sea. At a depth of h, meters, the density of the plankton cells, in millions of cells per cubic meter, is

modeled by ( )p h

a. Using correct units, interpret '( )p h

b. Consider a vertical column of water in the sea with

horizontal cross section of a constant area of 3 square meters. How many plankton are in the

water between a height (depth) of h = 0 and h = 30 meters.

Summary Notes for AP Exam: Different kinds of CHANGE

• Average rate of change of ( ) ( )

( )f b f a

f xb a

−=

− which is

the SLOPE of the secant line for ,x a b

• '( )f c is The instantaneous rate of change of ( )f x at

the point ( , ( ))c f c

OR Derivative at a point ( , ( ))c f c is ( ) ( )

'( ) limx c

f x f cf c

x c→

−=

− OR

0

( ) ( )'( ) lim

h

f c h f cf c

h→

+ −=

• Displacement/Total CHANGE of ANY rate of change

Total change of ( )f x from time “a” to time “b” = '( )

b

a

f x dx where '( )f x is the rate of

change

Examples: temperature, crowds, mail sorted, plant growing, cars passing, gravel cleaned etc.

• AVERAGE: Average value of a continuous ( )f x over

the interval [a,b] : 1

( )

b

ave

a

f f x dxb a

=−