Math 1310-001
Camacho
Spring 2013
Midterm #2 Review
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.
11.
Math 1310-001
Camacho
Spring 2013
Midterm #2 Review
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 ).
Implicit Differentiation can be used for…

Differentiating Inverse Trig Functions

Differentiating Functions in which the base and exponents are both variable.
Linearization of a Function
and
near
Applications of the Derivative
Related Rates:
0.
Understand the problem and draw a diagram the situation.
1.
Identify and name the quantities of interest. In most cases, the rate of one quantity will
be given, and the rate of another quantity is being asked for.
2.
Use information in problem to write a relation between the quantities.
5.
Differentiate the relation with respect to time to get a relation between the rates of the
quantities.
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 1310-001
Camacho
Spring 2013
Midterm #2 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.
Absolute 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
Local Extreme Values
1st Derivative Test: If


Local Max if
Local Min if
or
when
when
2nd Derivative Test: If If


Local Max if
Local Min If
, then
AND
AND
, then
when
when
Math 1310-001
Camacho
Spring 2013
Midterm #2 Review
Optimization Problems:
0.
Understand the problem and draw a diagram.
1.
Identify the quantities of interest
2.
Use information in problem to determine the objective function and constraints.
3.
Use constraints to write the objective function in terms of only one variable.
4.
Maximize/Minimize the objective function within the appropriate domain.
5.
Explain the results in a way that answers the original question
Practice Problems
Derivatives:
Section 3.2: #3 – 24, 25 – 27
Section 3.3: #1 – 22
Section 3.4: #7 – 36, 37 – 40, 41 – 44
Section 3.5: #3 – 16, 17 – 26
Section 3.6: #17 – 29, 17 – 26
Section 3.7: #2 – 20, 21 – 27
Section 3.9: #5 – 8, 15 – 19
(product and quotient rules)
(trigonometric derivatives)
(chain rule)
(implicit differentiation)
(inverse trig derivatives)
(derivatives of logarithms)
(linearization of functions)
Application of Derivatives:
Section 4.1: #1 – 30
Section 4.2: #15 – 22, 23 – 38, 41 – 54
Section 4.3: #7 – 16, 21 – 32
Section 4.5: #5 – 46, 27 – 32
Section 4.6: #2 – 30
(related rates)
(absolute maximum and minimum values)
(increasing, decreasing, concavity, asymptotes)
(l’Hopital’s rule)
(optimization problems)