Math 1210-003
Camacho
Spring 2012
Midterm Review
Applications of the Derivative
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
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
).
,
)
)
Math 1210-003
Camacho
5.
Spring 2012
Midterm Review
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.
Newton’s Method
To solve
, start with an initial guess, . Then take successive guesses
according to the following recursive formula until your approximation is reasonably
close to the true solution:
Integration
Sigma Notation
Rules
1.
2.
3.
Special Formulas
1.
2.
3.
Math 1210-003
Camacho
Spring 2012
Midterm Review
Indefinite Integrals (Anti-Differentiation):
1.
2.
3.
4.
5.
6.
7.
8.
Fundamental Theorems of Calculus:
1.
1st Fundamental Theorem:
2.
2nd Fundamental Theorem:
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
Midterm 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
Midterm Review
Practice Problems
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 3.7: #1 – 16
Section 3.8: #1 - 36
(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)
(Newton’s Method)
(finding anti-derivatives)
Section 4.1: #15 – 24
Section 4.3: #9 – 16
Section 4.4: #1 - 14
Section 4.4: #15 - 26
Section 4.4: #35 - 52
(sigma notation)
(properties of definite integrals)
(evaluating definite integrals)
(finding anti-derivatives using the substitution method)
(evaluating definite integrals using the substitution method)