Units and Rates
Function Notation
Functions and Graphs
Rates of Change
Calculus
100
How many crab apples, each of which weighs 1.5 ounces, are needed to balance three 2-pound weights? (16 ounces = 1 pound)
64 crab apples
100
Your happiness, h, depends on the amount of support that you give your friends, s. Write a statement to express this relationship using function notation.
h=f(s)
100
Sketch the graph of
f(x)=x^2
Then sketch the graph of
f(x+3)-4
Correctly label the vertex, x-intercepts, and y-intercept of the new graph.
Vertex is (-3,-4) ; x-intercepts are x=-5,-1 ; y-intercept is y=5.
100
What is the average rate of change of the function below between x=-2 and x=0?
ARC_[-2,0] =3/2
100
What is the derivative of
f(x)= 7x^5 - 3x^3 + x - 4?
f'(x) = 35x^4 - 9x^2 + 1
200
A space shuttle leaves the earth’s atmosphere at a speed of 25,667 feet per second (TRUE!).
What is this speed in kilometers per hour?
(1 m=3.281 ft; 1 km=1000m)
28,163 km/hr
200
g(x)=2^x
Choose a value for a to show that the following equation is true:
g(-a)=1/g(a)
Answers vary.
Example: a=3
200
Sketch the graph of
y=|x|
Then sketch the graph of
y=a|x|+b
if a and b are both some number less than -1.
Answers vary, but should include a narrowing, a “flip” over the x-axis, and a shift down.
200
Calculate the average rate of change of the function
f(x)=1/x
between x=2 and x=5
ARC_[2,5]=(f(5)-f(2))/(5-2)
ARC_[2,5]=(1/5-1/2)/3
ARC_[2,5]=-1/10
200
Calculate the limit of the function as x approaches 3:
lim_{x->3} (x^2 - 9)/(x - 3)
Answer: 6
Simplify first:
(x^2 - 9)/(x - 3) = ((x+3)(x-3))/(x-3) = x + 3
Then direct substitution:
lim_{x->3} (x + 3) = (3) + 3 = 6
300
The average adult needs about 800 000 liters of oxygen per year. How much oxygen, in liters, does the average adult need per minute?
1.5 liters of oxygen per minute.
300
h(x)=(x+3)(x-3)
What is the value of
h(x+3)-h(x-3)
?
12x
300
Sketch a graph with the following characteristics:
Domain: [-10,10] and Range: [-3,8]
y-intercept is y=6 ; x-intercepts are x=-5,7
Function increasing over the interval (-5,0) and decreasing over all other intervals
Rate of change increasing over the interval (-5,0) and increasing over all other intervals
Answers vary
300
Create a function whose average rate of change between x=2 and x=4 is equal to 5.
Answers vary
300
Determine the following limits:
1. lim_{x->0} 1/x
2. lim_{x->\infty} 1/x
3. lim_{x->\infty} x
1) 1/0 -> DNE
2) 1/\infty -> 0
3) \infty -> DNE