Average Rate of Change
Polynomials
Rational Functions
Word Problems
Misc
100
Mike owns an orchard and uses several polynomials to model the growth of his trees. Given x is time, and f(x) is height--write an expression that will help mike compute the average rate of change for a tree from month 2 to month 5.
\frac{f(5)-f(2)}{3}
100
Explain in detail why a point of inflection cannot also be a critical point (minimum or maximum)
A critical point occurs when the slope is zero. Critical points occur inside intervals that are concave up or concave down. A point of inflection is where a graph will change concavity, meaning it cannot be where slope is 0. The key difference is a critical point is a change in direction, and a point of inflection is a change in concavity.
100
Explain, in detail, the difference between a gap and a vertical asymptote. Give an example!
Answers vary
100
The parent function for rationals is 1/x.
what are all the properties of this function? Holes, asymptotes, intercepts, etc...
Vertical asymptote x=0
Horizontal asymptote y=0
No: gaps, intercepts
100
A 3rd degree polynomial has 1 real root. How many complex roots will it have?
2 because the total number of roots should add up to the degree of the polynomial.
200
Which kind of polynomial has a constant second difference?
-Linear
-Quadratic
-Cubic
-Exponential
Quadratic
Generally, given
ax^2+bx+c
then
\Delta^2f(x)=2a\cdot s^2
where s is the step size
200
A wedding caterer charges a 200 service fee and 24 per plate (cost of food for one guest). Johanna has gotten RSVPs accepting the invitation from 72 guests, but is still waiting to hear back from the other 18. She also has to pay for her own food and the groom's, who are not in the guest count. Let C(p) represent Johanna's catering cost for "p" plates of food. What is the range of C(x) ?
1976<=C(p)<=2408
200
Create a rational function that has:
A gap at x=4, a horizontal asymptote at y=6 , and a vertical asymptote at x=3
\frac (6(x-4)(x+1))((x-4)(x-3)
200
An insect was discovered in the rainforest of Costa Rica. Environmentalists found that a function can describe the population change over time, t months.
P(t)=\frac(45(1+0.6t))(3+0.02t)
According to the model, what size will the group of insects grow to, but not exceed? What factors do you think leads to this projection?
y=1350. Territory size, food availability, competition with other insects/animals, incomplete data.
200
What concavity must a graph have on an interval in order for the function to have a local minimum? Explain!
Concave up!
300
Find the second difference of:
5x^2-6x-8
with a step size of 1
\Delta^2f(x)=2a, 2(5)=10
300
Sketch this graph. Then highlight where the critical points are, which section is concave up or down, and the point of inflection.
concave down:
-\infty<x<4
concave up:
4<x<\infty
critical points: (2,5), (6,1) Point of inflection: (4,3)
300
What is the lim as
\lim _{x\to -3^-}f(x)
\frac(2x^2+12x+16)(x^2+7x+12)
\infty
300
The average cost per unit to produce x units of plywood is given by
C(x)=\frac(300)(x+10)
If the cost per unit is $1.50, how many units have been produced?
x=190 units
300
The slope of f(x) can be modeled by the function g(x)=x^2-9 . Where are the two critical points of f(x)?
x=\pm3
400
For some function f(x) , the average rate of change between x=3 and x=4 is 12, then from x=4 and x=5 the average rate of change is 18.
How would you describe the interval
3<=x<=5
The interval is increasing concave up
400
The graph of f is shown, if f(a)=f(b)=4 and a \ne b then what is a \cdot b?
a\cdot b=10
400
When does a rational function have an x-intercept? Be detailed.
A factor in the numerator must become zero with an x-value that doesn't cause the denominator to be zero.
400
The concentration C of a certain antibiotic in a patient's bloodstream is given by
C(t)=\frac(50t)(t^2+25)
In order for the antibiotic to be effective, 4 or more mg/dl of the antibiotic must be present in the bloodstream. Assuming you took the medication at t=0, when does the drug become effective and for how long?
At t=2.5 you hit a concentration of 4. After t=10.5 the concentration will drop below 4.
400
A cubic polynomial's slope can be modeled with which kind of function?
quadratic!
500
Given h(x) is a quadratic, find the value of j+k 
j+k=-22
500
The cost of renting a bowling alley for an event is a $200 flat fee and $25 per person that attends. Let C(x) represent the cost per person when x people attend the event. Write C(x)
C(x)=\frac{25x+200}{x}
500
Write the equation of a rational function 'g' that satisfies the following:
1) x-intercept at x=5
2) vertical asymptote at x=-2
3) horizontal asymptote at y=2/7
4) has a gap at x=-3
\frac(2(x-5)(x+3))(7(x+2)(x+3)
500
Can a vertical asymptote and a gap be at the same x-value?
Yes technically, but you won't see the gap!
500
Explain why odd degree polynomials will always have at least 1 guaranteed real root.
if you look at the end behavior, the polynomial MUST pass through y=0