honors analysis. mr. manker is traveling to california. he drives at a constant pace: 100 miles the...
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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
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 CHANGE
The 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!)