Math 1210-007
Midterm Review
Fall 2010
Applications of the Derivative
Related Rates:
1.
Determine quantities two interest (e.g. r = radius, A = area)
2.
Determine the relation between the quantities (e.g.
3.
Differentiate the relation with respect to time in order to determine how the rates
of the two quantities are related (e.g.
)
4.
Plug in the values that are given for the quantities, and the rate of change of one
of the quantities, and solve the equation in step 3 for the unknown rate.
Sometimes only one of the two quantity's value is given, in which case you use
the original relation (before differentiating) to determine the value of the other
quantity.
Critical Points:
)
There are three types of critical points...
1.
Stationary points are the values of x where
2.
Singular points are values of x where
3.
End points are the largest and smallest values over which the function
being considered. (i.e. on the interval
the end points are
and
Extreme Values:
(i.e. numerator is 0).
is undefined (i.e. denominator is 0).
is
)
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 1210-007
Midterm Review
Fall 2010
Relative Extrema & Inflection Points
1.
2.
3.
and
means
has a relative maximum at x.
and
means
has a relative minimum at x.
and
changes sign at x (i.e. goes from negative to positive or
from positive to negative) means
an inflection point at x.
Optimization Problems:
1.
Determine all quantities of interest (e.g. h = height, w = width, A = area)
2.
Determine relationship between quantities (e.g.
3.
Identify the objective function (e.g.
4.
Use the relations 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.
Integration
Indefinite Integrals (Anti-Differentiation):
1.
2.
3.
4.
5.
6.
7.
,
)
)
Math 1210-007
Midterm Review
Fall 2010
Fundamental Theorems of Calculus:
1.
1st Fundamental Theorem:
2.
2nd Fundamental Theorem:
Method of Substitution (Indefinite Integrals):
1.
2.
Determine the correct definition for u in terms of x.
(e.g. for
the correct definition of u is
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.
)
)
Method of Substitution (Definite Integrals):
1.
2.
3.
Determine the correct definition for u in terms of x.
(e.g. for
the correct definition of u is
Determine du by using the formula
(e.g. if
then
so
).
)
Determine the new limits of integration given the definition of u.
(e.g. when
,
and when
,
limits of integration are from
to
)
. So the new
4.
Substitute u and du into the integral so that it is completely in terms of u. Also,
substitute the new limits of integration with respect to u.
(e.g.
)
5.
Evaluate the definite integral in terms of u. Note that there it is not necessary to
substitute back in for x when evaluating a definite integral.
(e.g.
)
Math 1210-007
Midterm Review
Fall 2010
Calculating Exact Anti-derivatives:
1.
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
and find
including the " + C ".
b.
Use the fact that
to determine C.
2.
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
and find
including the " + C ".
b.
Use the fact that
to determine C.
c.
Use the techniques of integration to anti-differentiate of
and find
including the " + D ".
d.
Use the fact that
to determine D
Physics
Two-Dimensional Motion:
= horizontal acceleration
= horizontal velocity
= horizontal position
= vertical acceleration
= vertical velocity
= vertical position
= horizontal acceleration
= initial horizontal velocity
= initial horizontal position
= vertical acceleration
= initial vertical velocity
= initial vertical position
When acceleration is constant you can use the following formulas:
Horizontal Motion:
Vertical Motion:
Right is the positive direction, Left is the negative direction.
Up is the positive direction, Down is the negative direction.
Math 1210-007
Midterm Review
Fall 2010
Vectors:
1.
2.
3.
4.
magnitude of the vector
direction of the vector
horizontal component of the vector
vertical component of the vector
Physics:
1.
Newton's Second Law:
2.
(i.e. the net force is equal to the sum of all forces on the object)
3.
When gravity is the only force acting on an object then you may assume
a.
b.
4.
Make sure that all of the units in your problem are consistent.
Practice Problems:
Section 2.8: #1 - 9
Section 3.1: #5 - 26
Section 3.2: #1 - 10
Section 3.2: #11 - 18
Section 3.3: #6 - 10
Section 3.4: #10 - 16
Section 3.8: #1 - 36
Section 4.4: #1 - 14
Section 4.4: #15 - 26
Section 4.4: #35 - 52
(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)
(finding relative maxima and minima)
(optimization problems)
(finding anti-derivatives)
(evaluating definite integrals)
(finding anti-derivatives using the substitution method)
(evaluating definite integrals using the substitution method)
Physics: (See practice problems and examples in the following files available online)
- Forces:
- Vectors:
- 2D Motion:
- 1D Practice:
- 2D Practice:
http://www.math.utah.edu/~camacho/files/Forces.pdf
http://www.math.utah.edu/~camacho/files/Vectors.pdf
http://www.math.utah.edu/~camacho/files/Planar_Motion.pdf
http://www.math.utah.edu/~camacho/files/pp_11_8.pdf
http://www.math.utah.edu/~camacho/files/pp_11_9.pdf