### Unit 4 Powerpoints

```Honors Analysis
 Mr.
Manker is traveling to California.
He drives at a constant pace:
100 miles the first two hours
210 miles the next three hours
240 miles the next four hours
Create a graph
showing his
distance and time
 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?
 Write
the domain and range using interval
notation:
 Give
the domain and range using interval
notation:
 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.
 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?

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.
 Graph
piecewise function f shown below:
2 − 2 − 6 ≤  < −2
=
−4 −2≤ <2
2 − 6
2≤<4
2 + 5   < 1
=
3 + 2   ≥ 1
Solve for k so that function f is continuous:
2 + 6   ≤ 2
=
+ 4   > 2
 Solve
for k so that f is continuous:
−2 + 4   < 3
f(x)=
+
≥ 3
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.
 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.
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?
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?
The distance of a ball in feet above the
ground, d, after t seconds is modeled by the
equation   = − 2 + 16 + 5.
How high is the ball off the ground when it is
first thrown?
How might you estimate the 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!
 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
 Evaluate
values using the Greatest Integer
Function
 Simple modular arithmetic problems
Area under the curve (above the x-axis):
Definite Integral
In symbols:
6

0

 Estimate
the area under the curve between
x = 0 and x = 3 using n = 3 trapezoids.
=− +1
2
+4
3
− −1
0
2
+ 4
 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
= 2 + 4
 Determine values that make piecewise
functions continuous
 Evaluate Greatest Integer Function values
 Modular Arithmetic








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!)
```