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Test 2
Popper 07 today
1
Test 2 Review
Problem type one
Find the zeros
(x axis intercepts; roots)
Command: roots
f ( x)  0.005 x5  23x3  .01x  12
Ans (−.81,0)
Problem type two
Evaluate the function/finding and evaluating derivatives
Using GGB to compute
f ( x)  8 x3 x 2  3
f (1) 
f '(3) 
f ''(9) 
Command: derivative
Let’s do this one real time together
Popper 07
Questions 1 and 2
2
What is the second derivative of
f ( x) 
e x ( x  1)3
e2
f ''(3.12) 
Problem type 3
Limit at a point
lim
x 3
x3  2 x 2  3x
x 3
Command limit[f]
Ans: 12
Popper 07
Question 3
3
Limit at infinity
Review Horizontal Asymptotes:
Ax n  junk
Bx d  junk
5 x 3  junk
x  junk  3 x d
lim
d 2
d 3
d 6
Popper 07
Questions 4 and 5
4
Just the facts:
14 questions:
12 multiple choice
02 free response
06 points each
14 points each
Formulas:
Technology:
DQ, Cost, Revenue, and Profit
GGB and a small online calculator
72 points
28 points
Topics
Multiple choice:
Find the zeros
ROOTS
Calculate the LIMIT
h approaches a, h approaches infinity
both rational functions
Characterize the discontinuity
It is continuous
It is discontinuous because
f(2) DNE
lim DNE
defined, exists…do not match
Find the derivative in functional form
Evaluate the derivative at a point; evaluate the second derivative at a point
AVERAGE rate of change
P(a) – the value at a point; P’(a) – the rate of change at a point
Break-even quantity and price
Equilibrium point (quantity, price)
demand and supply
Free response:
Given a table of values, find the regression model, Rsquared, predict value
Given f(x), find the equation of the tangent line at a given x = −3
TIMING:
2 minutes each multiple choice
8 minutes each free response
10 minutes checking
24
16
10
50 minutes THEN press “submit”
5
Problem type 4
Continuity – the definition requires:
Where is the function discontinuous and WHY is it discontinuous there?
A
because the function is undefined there
B
because the limit is not defined there
C
even though the function is defined there, the limit isn’t the function value
6
Problem type 5
Given the following function modeling GNP
G( x)  0.024 x3  0.12 x 2  0.25x  57
where G is in billions of dollars and x is years since 2003
Find the GNP for 2005, be sure to report your units.
Find the average rate of change of GNP from 2001 to 2003.
At what rate is GNP changing at the beginning of 2004?
Popper 07
Questions 6 and 7
7
Problem type 6
Application problems:
A
A student of a foreign language can learn new words according to the
following formula
N (t ) 
9.27t 2  13t  7
30.3 x 5
Where N is the number of words and t is the time in hours.
How many new words will the student learn after 5 hours of study?
What is the rate of learning new words after 30 minutes of study?
8
B
Break-even points
The Googone company produces gallons of their product. Find the break-even
point if the production cost is $11 per gallon, the selling price is $23.50 per gallon,
and the company’s fixed costs are $2137.09.
9
C
Market equilibrium
Given
Demand:
d ( x)  8 x  49
Supply:
s( x)  4 x  425
Find the equilibrium quantity and price. Round to the nearest unit.
10
Free Response problems:
A
Given the following table, find the cubic regression model:
x
y
0
11
1
13
2
15
3
14
4
16
5
19
6
20
What is the value of R2 for your model? Is it a good one? How do you know?
Use your model to predict the value for x = 9
11
B
Given:
f ( x)  3x 2  5x  4
Find the derivative of the function:
Find the slope of the tangent line at x = 1
Find the function value at x = 1
Find the equation of the tangent line to the graph at x = 1
Popper 07
Questions 8, 9, and 10
12