Algebra Review
Limits - Solving
Limits - Graphs
Continuity
Derivatives
100
Factor:
3x^2 +6 +2x +x^3
(x^2 +2)(x+3)
100
List the properties of limits
Product, quotient, sum, x, constant, power
100
What is the limit as x approaches 1.
Does not exist.
100
What are the 3 criteria for a function to be continuous at x=a?
1. f(a) exists
2. Lim x--->a exists
3. Lim = f(a)
100
What is the definition of a derivative without looking at your notes?
Lim h--->0 f(x+h)-f(x)/h
200
Graph: 3x^2 + 5
Desmos
200
What is the lim as x goes to 3 of (2x+1)
7
200
What is the lim as x approaches 4?
5
200
Is this function continuous at x=3? If not, what is the type of discontinuity. Be able to explain what criteria fails.
Jump discontinuity
200
Using the definition of derivative, what is the derivative of 5x+6
5
300
Describe the domain and range of: Y = 1/x +5
x cannot equal zero
y cannot equal 5
300
What is the limit as x goes to 4 of (x^2 - 6x +8)/(x-4)
2
300
Graph the piecewise function and determine limits as x approaches 0 and 2
f(x) =
1/2x +1 if x<0
x-1 if 0<=x<2
x if x>=2
300
Without graphing, what kind of discontinuity will occur at x=3?
f(x) =( x^2 -7x +12)/(x-3)
What is the value of f(3)?
Removable/hole at x=3
-1
300
Explain in your own words what a derivative is and how the definition of a derivative illustrates this.
The lim as the distance of x values of secant lines goes towards 0- approximates the slope of the tangent line.
400
Create a line through (3,5) and (2,-1) using point slope form
y-5 = 6(x-3)
400
What is the lim as x goes to 0 of (x+6)1/2-61/2/x
1/2(61/2)
400
Does this lim as x approaches 2 exist? Be able to prove why or why not.
No, approaches 3 from the right and 8 from the left.
400
At what points is the graph discontinuous? What are the limits at those points?
x=0: Lim DNE
x=1: Lim = 1
400
Using the definition of derivative, what is the derivative of 3x^3. Graph the derivative
9x^2
500
f(x) = (3x2 + 5x)1/2
Find f(x+h)
(3x2+6hx+3h2+5x+5h)1/2
500
What is the lim as x goes to 0 of sin(1/x)
DNE!
Lecture example
500
Describe all the limits (at issue points)
-2: DNE
1: 3
3: DNE
500
Removable or Nonremovable discontinuity?
f(x) {-9x if x<= 2, x^2 -2x+3 if x>2
Nonremovable, Jump
500
What is the derivative of (3x+5)^1/2
3/2(3x+5)^1/2