Math 1210 – 003
Camacho
Spring 2012
Final Exam 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
Final Exam 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.
Higher Order Derivatives:
1.
2.
3.
4.
Math 1210 – 003
Camacho
Spring 2012
Final Exam 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 ).
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.
,
).
)
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.
Math 1210 – 003
Camacho
Spring 2012
Final Exam Review
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.
Optimization Problems:
1.
Determine all quantities of interest (e.g. h = height, w = width, A = area)
2.
Determine the relationships between quantities (e.g.
3.
Identify the objective function (e.g.
4.
Use the relationships to write the objective function in terms of only one
variable (e.g.
, therefore
).
5.
Determine the maximum/minimum value of the objective function by first
finding all critical points and then plugging the critical points into the objective
function to determine which yields the optimal value. In some problems you are
looking for the maximum value and in others you will be looking for the
minimum value.
,
)
)
Math 1210 – 003
Camacho
Spring 2012
Final Exam Review
Integration
Indefinite Integrals (Anti-Differentiation):
1.
2.
3.
4.
5.
6.
7.
8.
Fundamental Theorems of Calculus:
Method of Substitution (Indefinite Integrals):
1.
2.
Identify inner function and set equal to u.
(e.g. for
the inner function
Determine du by using the formula
(e.g. if
then
so
).
)
3.
Substitute u and du into the integral so that it is completely in terms of u.
(e.g.
)
4.
Anti-differentiate in terms of u. (e.g.
5.
Substitute x into the solution using the definition of u.
(e.g.
)
)
Math 1210 – 003
Camacho
Spring 2012
Final Exam Review
Method of Substitution (Definite Integrals):
1.
Find the anti-derivative,
, using the u-substitution.
2.
Evaluate the integral by calculating
Calculating Exact Anti-derivatives:
1.
2.
Given a function's derivative,
, and some known value for the function,
, you should be able to use this information to determine
.
a.
Use the techniques of integration to anti-differentiate of
including the " + C ".
b.
Use the fact that
and find
to determine C.
Given a function's second derivative,
, some known value for the function's
first derivative
, and some known value for the function
, you
should be able to use this information to determine
.
a.
Use the techniques of integration to anti-differentiate of
including the " + C ".
b.
Use the fact that
c.
Use the techniques of integration to anti-differentiate of
including the " + D ".
d.
Use the fact that
and find
to determine C.
to determine D
and find
Math 1210 – 003
Camacho
Spring 2012
Final Exam Review
Applications of the Integral
Mean Value of a Function
Area of Region
1.
Between two function,
and
:
Area =
Solve
to find intersection points.
2.
Absolute Area of f(x):
a.
Find all zeros of f(x) (e.g.
b.
Evaluate integrals separately between each of the zeros
(e.g.
,
,
,
)
c.
Add absolute values of all integrals
(e.g. A =
3.
Horizontal Slicing when functions are in terms of .
(e.g. A =
)
Volume of a Solid
Vertical Slicing:
Horizontal Slicing:
Disc Method:
Washer Method:
Shell Method:
or
or
or
[Note: If slices and axis are perpendicular, use disc or washer method. If perpendicular,
use shells method.]
Math 1210 – 003
Camacho
Spring 2012
Final Exam Review
Arc Length
Standard Function:
Parametric Function:
Work
(General work)
(Work on a spring)
(Pumping fluid out of a tank)
Center of Mass
Math 1210 – 003
Camacho
Spring 2012
Final Exam Review
Practice Problems
Limits:
Section 1.3: #1 - 12 (evaluating limits)
Section 1.3: #25 - 34 (using limit theorems)
Section 1.3: #41 - 44 (left and right side limits)
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 3.3: #6 - 10
Section 3.4: #10 - 16
Section 3.8: #1 - 36
(related rates problems)
(finding critical points and extreme values)
(finding relative maxima and minima)
(optimization problems)
(finding anti-derivatives)
Integrals:
Section 4.3: #9 – 16
Section 4.4: #1 - 14
Section 4.4: #15 - 26
Section 4.4: #35 - 52
Section 4.5: #1 – 14
(properties of definite integrals)
(evaluating definite integrals)
(finding anti-derivatives using the substitution method)
(evaluating definite integrals using the substitution method)
(finding average value of a function)
Applications of Integration:
Section 5.1: #11 – 28
Section 5.2: #1 – 15
Section 5.3: #1 – 12
Section 5.4: #1 – 12
Section 5.5: #1 – 8
Section 5.5: #19 – 24
Section 5.6: #1 – 5
(calculating area of a region)
(find volumes using disc method and washer method)
(finding volumes using shell method)
(finding arc length of standard and parametric functions)
(work done on a spring)
(work against gravity)
(center of mass)