Math 112
Exam 3 Big Ideas
Section 3.9 – Related Rates
 This section is about finding a rate at which something is occurring and this
is done by finding a relationship between rates.
 Some examples of related rates are:
o Sphere: How fast is the radius changing given the volume is
changing at a certain rate.
o Cone: How fast is the volume changing given the height is
changing at a certain rate.
o Distance: If one car is traveling east from a given point at one
rate and another car is traveling north from the same point at a
different rate, how fast is the distance between the cars
changing at a certain time.
 Key approach
o Create an equation that describes a relationship between the
two variables under consideration.
o If a certain dimension is always fixed, it can be included in the
above equation. If it is not fixed, it must be left as a variable.
o Differentiate the above equation implicitly and plug in the
known quantities. Solve for the unknown rate.
Sections 3.10 – Linear Approximation
 Since a tangent line at a given point is really close to the function at the point
of tangency, the equation of the tangent line can be used to approximate the
value near the point of tangency.
 The equation below can be used to approximate the value of the function f at
a point a near the point of tangency.
f (x) » f (a)+ f '(a)(x- a)

Note: This section does talk about “differentials” but we did not talk about
them and they will not be on the exam.
Section 3.11 – Hyperbolic functions
 In this section we defined two new functions which are related to the shape
that a wire creates when hung between two poles. The two functions are:
ex - e- x
ex + e- x
and cosh(x) =
sinh(x) =
2
2
 The functions tanh, csch, sech, and coth are all defined similar to the regular
sinh(x)
trig functions. For example tanh(x) =
etc.
cosh(x)
 Three properties that are helpful in working with these functions are:
sinh(-x) = -sinh(x), cosh(-x) = cosh(x), and cosh2 (x) - sinh2 (x) =1. All of these
can be derived quite easily from the original definitions of sinh and cosh.


The derivatives of all of these functions can all be computed using the
knowledge of the derivatives of exponentials.
Inverse hyperbolic functions are also defined in this section but will not be
assessed on this exam.
Section 4.1—Maximum and Minimum Values
 Critical numbers of a function are values in the domain for which the
derivative of a function is equal to 0 or does not exist.
 Critical numbers are x values that are candidates for minima and maxima to
occur but not all critical numbers will yield a max or min.
 If a max or min occurs at some number c then c is a critical number of f.
 To find the absolute max or min on a closed interval:
o Find the value of the function at the critical numbers
o Find the value of the function at the endpoints
o The largest value from the above two steps is the absolute max and
the smallest value from the above two steps is the absolute min.
 Don’t confuse the place at which a max or min occurs (an x-value) and the
actual maximum or minimum which is a y-value.
Section 4.2 – The Mean Value Theorem
 If f satisfies the following conditions
o f is continuous on a closed interval [a, b]
o f is differentiable on the open interval (a, b)
then there is a point, c, in the interval (a, b) for which the slope of the tangent
line at that point is the same as the slope of the segment between the points
on the graph of f at the endpoints of the interval [a, b]. In other words,
f (b) - f (a)
f '(c) =
b- a
Section 4.3—What does the derivative tell us about the shape of the graph?
 A function f is increasing when f '(x) > 0 and decreasing when f '(x) < 0
 If f ' changes from positive to negative at a point c, then there is a max at c.
 If f ' changes from negative to positive at a point c, then there is a min at c.
 If f '' is positive on an interval, then the function is concave up on that
interval.
 If f '' is negative on an interval, then the function is concave down on that
interval.
 If f '' changes concavity at a point then that point is an inflection point.
 If f is continuous at c, f '(c) = 0 and f ''(c) < 0 then there is a local max at c.
 If f is continuous at c, f '(c) = 0 and f ''(c) > 0 then there is a local min at c.
Section 4.4—L’Hospital’s Rule



f (x)
has an
x®a g(x)
f (x)
f '(x)
0
¥
indeterminate form of or
then lim
= lim
x®a
x®a
g(x)
g'(x)
0
¥
0
The other indeterminate forms are 0 ×¥ , ¥-¥, 0 , ¥0 , and 1¥ . Each of these
forms would need to be manipulated to get them in one of the two primary
0
¥
indeterminate forms of or
before L’Hospital’s rule can be used.
0
¥
L’Hospital’s rule is placed at this point in the chapter to allow us to better
determine vertical and horizontal asymptotes which will assist in our
graphing.
L’Hospital’s rule can be used to help find limits. If lim
Section 4.5—Using derivatives to sketch graphs
 The guidelines for sketching curves are:
o Domain
o Intercepts
o Symmetry (odd, even, periodic)
o Asymptotes
o Intervals of increasing and decreasing
o Max and mins (these following directly from determining increasing
and decreasing)
o Concavity and points of inflection
o Put it all together
 When sketching these graphs, make sure that all important features
(intercepts, asymptotes, maximums, minimums, points of inflection) are
clearly labeled!
Section 4.7—Optimization problems
 Often when finding maximums and minimums you are optimizing a situation
like minimizing cost or maximizing profit or minimizing surface area or
maximizing volume etc.
 The techniques used in finding maximums and minimums for graphing can
be used in the optimization situations.
 The first derivative test or second derivative test can be used to verify a
maximum or minimum.
Section 4.8—Newton’s Method
 Newton’s method is an iterative method which relies on the derivative that is
used to find the roots of a function.
 The basic form of Newton’s method is
f (xn )
xn+1 = xn f '(xn )

An initial approximation is made and that is x1. This is then the input to
determine the second approximation x2 . The second approximation is the
input to find the third approximation, x3 , and so forth.
Section 4.9—Antiderivatives
 Up until this point, we have been given the function and determined the rate
of change using the derivative. If, however, we are given the rate of change,
we can find the original function. To do this we would need the derivative
function and a point on the graph of the function (usually an initial condition).
 A common context for these types of problems is position, velocity, and
acceleration. Velocity is the derivative of position and acceleration is the
derivative of velocity. We also know that acceleration due to gravity is -32
feet/sec2.