ch. 4 notes

CH. 4 NOTESHonors Analysis

ROAD TRIP! Mr. Manker is traveling to California.

He drives at a constant pace:100 miles the first two hours210 miles the next three hours240 miles the next four hours

Create a graph showing his distance and time

PIECEWISE FUNCTIONS What is the domain of the function? The

range? Write the equation of the function for

distance (after time t hours) using piecewise notation

What is Mr. Manker’s rate of change (speed) on the interval (0, 2)?

What is his speed on the interval (2, 5)? What is his speed on the interval (5, 9)? What aspects of this scenario aren’t

realistic? Why?

DOMAIN/RANGE Write the domain and range using

interval notation:

DOMAIN/RANGE Give the domain and range using

interval notation:

STEP FUNCTIONS The post office offers flat-rate mailing of

packages: $1.50 for a package weighing less than 4 oz, $2.50 for a package weighing 4 oz to less than 8 oz, and $3.50 for a package weighing 8 oz to 12 oz.

Graph this function. Is it continuous? Explain.

GREATEST INTEGER FUNCTION The greatest integer function returns

the greatest integer LESS THAN OR EQUAL TO the input value.

Greatest integer notation: f(x) = or (or occasionally f(x) = [x]). Sometimes called “floor function” How can it be graphed on a calculator? Is this function continuous?

GREATEST INTEGER FUNCTION

DISCRETE VS CONTINUOUS

A t-shirt company sells t-shirts for $8 apiece if five or fewer are sold, $6 apiece if 5-10 are purchased, or $5 apiece if more than 10 are purchased.

Why is this considered to be a DISCRETE function?

Sketch a graph.

GRAPHING A PIECEWISE FUNCTION Graph piecewise function f shown below:

WRITING PIECEWISE FUNCTIONS

IS THE FUNCTION CONTINUOUS?

𝑓 (𝑥 )={2 𝑥+5𝑖𝑓 𝑥<13𝑥+2 𝑖𝑓 𝑥≥1

CONTINUOUS PIECEWISE FUNCTIONSSolve for k so that function f is continuous:

𝑓 (𝑥 )={2𝑥+6 𝑖𝑓 𝑥≤2𝑘𝑥+4 𝑖𝑓 𝑥>2

CONTINUOUS PIECEWISE FUNCTIONS Solve for k so that f is continuous:

f(x)=

SOLVING RATE PROBLEMS

Jose left the airport and traveled toward the mountains. Kayla left 2.1 hours later traveling 35 mi/hr faster in an effort to catch up to him. After 1.2 hours Kayla finally caught up. Find Jose’s average speed.

SOLVING RATE PROBLEMS Two cars 276 miles apart start to travel

toward each other. They travel at rates differing by 5 mi/hr. If they meet after 6 hours, find the rate of each.

SOLVING RATE PROBLEMS

A boat traveled 336 miles downstream and back. The trip downstream took 12 hours. The trip back took 14 hours. What is the speed of the boat in still water? What is the speed of the current?

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

AVERAGE RATE OF CHANGE

INSTANTANEOUS RATE OF CHANGE

INSTANTANEOUS RATE OF CHANGE

INSTANTANEOUS RATE OF CHANGE

About how fast is the ball going after 1 sec?

About how fast is it going at 3 sec? Are there multiple possible speeds?

About how fast is it going at 8 seconds?

INSTANTANEOUS RATE OF CHANGEThe distance of a ball in feet above the ground, d, after t seconds is modeled by the equation .

How high is the ball off the ground when it is first thrown?

How might you estimate the instantaneous rate of change?

INSTANTANEOUS RATE OF CHANGE The rate of change of a function at an

INSTANT in time.

Example: The speed on the speedometer of your car at a certain moment in time.

Instantaneous rate of change is also the slope of the tangent line drawn to a point on a curve. In calculus, it is known as the DERIVATIVE!

QUIZ TOPICS Evaluate, analyze, and graph piecewise

functions Determine domain and range of a

function using the graph Determine values that make piecewise

functions continuous Solve distance = rate * time word

problems Calculate average rate of change of a

function from a table or function Estimate instantaneous rate of change

of a function

ADDITIONAL TOPICS Evaluate values using the Greatest

Integer Function Simple modular arithmetic problems

Area under the curve (above the x-axis):Definite Integral

In symbols:

THE TRAPEZOIDAL RULE Estimate the area under the curve

between x = 0 and x = 3 using n = 3 trapezoids.𝑦=− (𝑥+1 )2+4

∫0

3

(− (𝑥−1 )2+4 )𝑑𝑥

TEST TOPICS – DAY ONE Evaluate, analyze, and graph piecewise

functions Write the equation of piecewise functions Determine domain and range of a function

using the graph (or given a function such as

Determine values that make piecewise functions continuous

Evaluate Greatest Integer Function values Modular Arithmetic

WRITE THE EQUATION OF THE PIECEWISE FUNCTION:

TEST TOPICS – DAY TWO Solve distance = rate * time word problems (use chart

setup!) Calculate average rate of change of a function from a

table or function Estimate instantaneous rate of change of a function Estimate definite integrals by counting blocks on a

graph (WATCH OUT FOR GRAPH SCALE!!) Calculate definite integrals by calculating areas

(constant functions, linear functions, etc.) Estimate definite integrals (area under the curve)

using the Trapezoidal Rule (may be given function OR a table of values – always best to draw a graph first!!)

Determine units for rate problems (y unit divided by x unit!)

Determine units for integral/area problems (x unit times y unit!)