Math 1210-003
Camacho
Spring 2012
Midterm #1 Review
Limits
Rules
1.
2.
3.
4.
5.
6.
7.
The limit does not exist at a point where the function has a...
1.
Jump (e.g.
2.
Asymptote
at
(e.g.
)
at
)
One sided limits
1.
(limit approaching from the right side)
2.
(limit approaching from the left side)
Continuity: A function,
1.
2.
3.
, is continuous at
exists.
exists.
.
if...
Math 1210-003
Camacho
Spring 2012
Midterm #1 Review
Trigonometric Limits
1.
2.
Derivatives
Definition:
provided that the limit exists. Otherwise the function is not differentiable at that value of .
Rules:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Math 1210-003
Camacho
Spring 2012
Midterm #1 Review
Higher Order Derivatives:
1.
2.
3.
4.
Implicit Differentiation:
1.
Differentiate both sides with respect to , keeping in mind the chain rule when
differentiating an expression involving (since is itself a function of ).
2.
Move all terms having a
3.
Factor
to one side (it may be necessary to distribute first)
and solve (the expression will likely depend both on
and ).
Applications of the Derivative
Related Rates:
1.
Draw a diagram representing the situation.
2.
Identify the two quantities of interest (e.g. r = radius, A = area). In most cases, the rate
of one variable will be given, and the rate of the other variable is being asked for.
3.
Identify all information given about quantities in the problem (e.g.
4.
Determine the relation between the quantities (e.g.
5.
Differentiate the relation with respect to time and algebraically solve for the unknown
derivative (e.g.
).
6.
Substitute the known values of each variable to find the numerical value of the
unknown rate. Sometimes the quantity of only one variable is given while both are
needed. To find the value of the other quantity use the relation between the two
quantities, substitute the known quantity and solve for the unknown.
,
).
)
Math 1210-003
Camacho
Spring 2012
Midterm #1 Review
Critical Points: There are three types of critical points...
1.
Stationary points are the values of x where
(e.g. numerator is 0).
2.
Singular points are values of x where
3.
End points are the boundary points of the domain on which
(e.g. on the interval
the end points are
and
)
is undefined (e.g. denominator is 0).
is being considered.
Extreme Values: Maximum and Minimum values of a function
1.
The extreme values of a function MUST occur at a critical point
2.
To find the extreme values of a function first find the critical points, then evaluate
the function at all known critical points to determine which is the maximum value
and which is the minimum value.
Monotonicity & Concavity:
1.
2.
3.
4.
5.
means
means
means
means
means
is increasing
is decreasing
has a horizontal tangent line
is concave up
is concave down
Math 1210-003
Camacho
Spring 2012
Midterm #1 Review
Practice Problems
Limits:
Section 1.3: #1 - 12
Section 1.3: #25 - 34
Section 1.3: #41 - 44
Section 1.6: #1 - 16
(evaluating limits)
(using limit theorems)
(left and right side limits)
(identifying continuous functions)
Derivatives:
Section 2.3: #1 - 20
Section 2.4: #1 - 18
Section 2.5: #1 - 20
Section 2.6: #1 - 14
Section 2.7: #1 - 12
(finding derivatives using rules of differentiation)
(differentiating trigonometric functions)
(differentiating using the chain rule)
(higher order derivatives)
(implicit differentiation)
Applications of Differentiation:
Section 2.8: #1 - 9
Section 3.1: #5 - 26
Section 3.2: #1 - 10
Section 3.2: #11 - 18
(related rates problems)
(finding critical points and extreme values)
(finding where a function increases or decreases)
(finding where a function concave up or concave down)