**Exam 2 Review Topics**

**(Note: L’Hopital Rule will not
be covered by Exam 2)**

**Topics: **Derivatives,
chain rule, implicit differentiation, related rate, equation of tangent line, tangent
line approximation, linear approximation, critical points (2 types), inflection
points, local extrema, absolute extrema in closed intervals, absolute extrema
in unbounded intervals, intervals of increasing and decreasing, intervals of concave
down and concave up, optimization, story problem of optimization, mean value
theorem, families of functions.

**Techniques: **All
differentiation rules: summation rule, scalar product rule, product rule,
quotient rule, chain rule, implicit differentiation, related rate problem.
Derivatives of inverse functions, derivatives of inverse trigonometric functions.
Linear approximation, equation of tangent line. Finding critical points,
finding absolute extrema on intervals, 1^{st} derivative test for local
extrema, 2^{nd} derivative test for local
extrema. Determining the signs of derivative functions over intervals segmented
by critical points, determining increasing and decreasing intervals of
functions, technique of one-point testing. Determining the signs of the second
derivatives of functions, determining the interval of concave down and concave
up and points of inflection. Sketch graphs of inverse functions. Sketch graphs
of functions featuring x- and y-intercepts, vertical and horizontal asymptotes,
local extrema, intervals of increasing and decreasing,
intervals of concave down and concave up, and points of inflection. Mean Value
Function Theorem. L’Hopital’ rule for 0/0 type, ∞/∞
type, 0*∞ type, 1^∞ type. Story problems of optimization.
Determining the signs of the second derivatives of functions using one-point
testing method, determining the interval of concave down and concave up and
points of inflection, 2^{nd} derivative test for local extrema. Curve
sketching: horizontal asymptotes, vertical asymptotes, intervals of increasing
and decreasing, intervals of concave up and concave down, local maximum and
local minimum points, inflection points. Story problems in optimization: decision
variable, objective function, and constraint. Formulate optimization problems
and solve them.

**More Techniques: **Quadratic formula for roots, factorization of a^2-b^2, a^3-b^3, multiply
and divide to maintain and transform quantities. Basic forms and shapes of trig
and inverse trig functions, exponential functions, logarithmic functions, and
polynomials: linear function, parabola, cubic polynomials. Definitions of
trigonometric functions. Definitions of inverse trigonometric functions and
exponential functions (logarithmic functions). Basic rules, identities, and
limiting properties as x → ∞ for exponential and logarithmic
functions. Special values of trigonometric functions and exponential and
logarithmic functions. Derivative of power functions, exponential functions,
logarithmic functions, trigonometric functions and their inverses.
Antiderivatives of power functions, exponential functions, trigonometric
functions and their inverses.

**Calculator Skills:** Sketch graphs, tracing intersection points and roots, finding numerical
limits

**Partial Review Problems**: All homework, quiz, lecture
example problems, and sample exams. __Repeat
the problems until you can do them without any help. __