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Basics: Inflection
points, concave down and concave up, Regular partition of intervals, general
Riemann sums, special Riemann sums: Left Point Sum, Right Point Sum, Midpoi=
nt
Sum, Trapezoid Sum( =3D [Left+Right] / 2), Simp=
son
Sum(=3DT/3+2*M/3). Definition of definite integral. Geometric interpretation of definite integral in terms of are=
a.
Average value of a function. Simple rules for de=
finite
integrals: Theorems 4.2 and 4.3. Definition of
anti-derivatives.
Techniques: Determining
the signs of the second derivatives of functions, determining the interval =
of concave
down and concave up and points of inflection, 2nd derivative test
for local extrema. Curve sketching: horizontal asympt=
otes,
vertical asymptotes, intervals of increasing and decreasing, intervals of
concave up and concave down, local maximum and local minimum points, inflec=
tion
points. Story problems in optimization: objective functions and constraints.
Formulate optimization problems and solve them. Hand calculation for left p=
oint
sum, right point sum, midpoint sum, trapezoid sum, and Simpson sum. Memoriz=
e 2
special formulas: 1+2+…+n=3Dn(n+1)/2 and
1^2+2^2+…+n^2=3Dn(n+1)(2n+1)/6. Use definition to find definite integ=
rals:
i.e. taking limit of Riemann sums as the partition tends to infinity. Find
anti-derivatives for all elementary functions: power functions, exponential funct=
ions,
logarithmic functions, trigonometric functions. Application to projectile
problems: determining the motion of a projectile or falling body in terms of
its position, velocity, acceleration; time of impact, maximum height, impact
velocity.
Elementary Functions: Properties of elementary functions: power functions, exponential
functions, logarithmic functions, trigonometric functions.
Calculator Skills: Approximating definite integrals by left, right, mid, trapezoid, and
Simpson rules, sketch graphs, tracing intersection points and roots, finding
numerical limits.
Miscellaneous Techniques: Finding critical points, finding absolute extrem=
a
on closed intervals, 1st derivative test for local extrema, 2nd de=
rivative
test for local extrema. Det=
ermining
the signs of derivative functions over intervals segmented by critical poin=
ts,
determining increasing and decreasing intervals of functions. 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 functions featuring local extrema, intervals of increasing and decreasing, inte=
rvals
of concave down and concave up, and points of inflection. L’Hopital’ rule=
for 0/0
type, ∞/∞ type, 0*∞ type. Finding limits of ration=
al
functions as x → ∞. Distance, area, volume formulas for element=
ary
geometric objects: rectangle, triangle, circle, ellipse, rectangular solid,
cylinders.
Quadratic fo=
rmula
for roots, factorization of a^2-b^2, a^3-b^3, long division, multiply and
divide to maintain and transform quantities. Basic forms and shapes of
polynomials: linear function, parabola, cubic polynomials, definitions of
trigonometric functions, exponential functions, and logarithmic functions,
basic identities of trigonometric functions including double angle and half
angle formulas, basic rules, identities, and limiting properties as x ͛=
4; ∞
for exponential and logarithmic functions, special values of trigonometric
functions and exponential and logarithmic functions
Partial Review Problems: All homework, quiz, lecture example problems, and sample exams