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MTH 211 CALCULUS II
Students will be expected to demonstrate
an understanding of Calculus beyond the manipulation of symbols, apply
Calculus to practical problems and use current technology throughout the
course. They will demonstrate their understanding of Calculus using four
approaches--geometric, numerical, algebraic and verbal.
A comprehensive departmental final exam
testing the degree of mastery of the following course objectives is required.
1. General Objectives
1.1 Recognize and manipulate functions given
in numerical, graphical, and analytical forms.
1.2 Give reasonable approximations for values
of functions, their limits, derivatives and integrals and express the
error involved.
1.3 Use graphing calculator technology to explore
the behavior of functions, limits, derivatives, integrals and series;
to find numerical approximations for limits, derivatives, integrals
and intervals of convergence for power series; and to aid in solving
problems and verifying solutions.
1.4 Express Calculus concepts, and explain
and interpret results in well-written sentences.
1.5 Interpret the derivative as the limit of
a difference quotient that gives the slope of a linear approximation
to a graph at a point, and as instantaneous rate of change.
1.6 Interpret a definite integral both as a
limit of a Riemann Sum that is the difference in areas above and below
the x-axis, and as the net amount of change in the function whose rate
of change is the integrand.
1.7 Explain the relationship between the derivative
and the definite integral as it is expressed in both parts of the Fundamental
Theorem of Calculus.
1.8 Use derivatives and integrals to model
and solve applied problems.
1.9 Use the sign, magnitude, and units of measurement
of a solution to an applied problem to assess its reasonableness
2. Calculus I Review
2.1 Review finding derivatives of functions that are sums, differences,
products, quotients, or compositions of powers of x, sines and cosines,
logarithmic and exponential functions.
2.2 Review finding indefinite integrals of linear combinations of
powers of x, sine, cosine, and the natural exponential function and
check the results by taking the derivative.
3. Integration Techniques
3.1 Review finding indefinite and definite integrals of sums and differences
of constant multiples of powers of x, sines, cosines, and exponential
functions.
3.2 Review finding derivatives of inverse sine and tangent.
3.3 Find antiderivatives for derivatives of inverse sine and tangent.
3.4 Use the following integration techniques:
3.4.1 Integration by substitution (Review)
3.4.2 Integration by parts
3.4.3 Integration by partial fractions
3.4.4 Integration using integration tables. Problems involving tables
may need to be manipulated to fit the form given in the table.
3.4.5 Integrate functions involving products of powers of sine and
powers of cosine.
Recommended if time allows: integrate functions involving products
of powers of secant and powers of tangent.
3.4.6 Integrate using trigonometric substitutions of the form u
= asinӨ and u = atanӨ
.
Recommended if time allows: integrate using trigonometric substitutions
of the form u = asecӨ .
3.5 Use the following numerical techniques of integration to approximate
an integral for a function given algebraically, graphically, or as a
table of values; sketch the geometric representation of the approximation,
and discuss the error involved:
3.5.1 LEFT-HAND and RIGHT-HAND rules (Review).
3.5.2 MIDPOINT rule
3.5.3 TRAPEZOID rule
3.5.4 Use concavity to determine whether the Midpoint and Trapezoid
rules are over or under estimates.
3.5.5 SIMPSON'S rule.
4. Applications of Integration
4.1 Set up an approximating Riemann sum and represent its limit as
a definite integral in a variety of applications, then evaluate or approximate
the integral. The following applications are required; however, students
should be able to adapt this technique to application problems that
are similar to, but not exactly the same as, the specific applications
listed
4.1.1 Find or approximate the area between two curves. Sketch
and label the ith rectangle with its width and height,
express the area of the ith rectangle, express the total
area as the limit of a Riemann sum of these areas, write the Riemann
sum as a definite integral and find or approximate its value.
4.1.2 Find or approximate volumes by slicing.
Sketch and label the ith volume element , express the volume
of the ith element, express the total volume as the limit
of a Riemann sum of these volumes, write the Riemann sum as a definite
integral and find or approximate its value.
4.1.3 Find or approximate work done in
moving an object along a straight line by a force in the direction
of the motion. Identify the ith force element, ith
distance element and ith work element; express the total
work as the limit of a Riemann sum; write the limit of the Riemann
sum as a definite integral and evaluate or estimate it. Express the
total work done using correct units.
4.2 Find or approximate arc length using the definite integral formula
for arc length
5. First order differential equations
5.1 Translate verbal descriptions of rates of change into differential
equations.
5.2 Solve differential equations by:
5.2.1 the separation of variables technique
5.2.2 using slope fields
5.2.3 using Euler's method
5.3 Model exponential and logistic growth and decay by formulating
and solving differential equations with initial conditions. Use the
models to answer questions about amounts and times at which they occur.
6. Limits with Indeterminate Forms
6.1 Use L 'Hopital's Rule to evaluate ∞/∞,
0/0, ∞ · ∞
forms
6.2 Use logarithmic techniques to reduce the indeterminate forms 00and
∞0 to
forms to which L'Hopital's Rule applies.
6.3 Use graphical and numerical techniques to approximate limits with
indeterminate forms.
7. Improper Integrals
7.1 Evaluate improper integrals with infinite limits of integration.
7.2 Evaluate improper integrals with integrands containing infinite
discontinuities.
7.3 Use a comparison test to determine convergence or divergence of
an improper integral. If the integral converges, find its value, give
an upper bound on its value, or give an approximation of its value.
8. Infinite Series
8.1 Use the definition of an infinite series as the sum of an infinite
sequence to construct the series.
8.2 Calculate and graph a sequence of partial sums. Construct an expression
for the nth partial sum when possible.
8.3 Determine convergence or divergence of an infinite series by evaluating
the limit of the nth partial sum for geometric and telescoping series.
8.4 Use the nth term test for divergence.
8.5 Recognize geometric series, determine their convergence or divergence
and find the sum if convergent.
8.6 Recognize the harmonic series and know that it diverges.
8.7 Recognize an alternating series. Use the first n terms of an alternating
series whose terms decrease in magnitude and whose nth term goes to
0 to approximate the sum of the series. Give the magnitude of the first
neglected term as a bound in the error of the approximation.
8.8 Interpret the terms of a series as areas of rectangles of width
one and height equal to the value of the term. Interpret the Integral
Test as a comparison of an infinite integral to its upper or lower sum.
Use the integral test to determine the convergence or divergence of
p-series. Use an appropriate integral to give an upper bound for the
value of the series.
8.9 Recommended if time allows: Use the Ratio Test to determine the
convergence or divergence of series.
8.10 Construct Taylor Series expansions of functions and examine their
intervals of convergence graphically and/or numerically.
8.11 Calculate Taylor polynomial approximations. Discuss the accuracy
of the approximations by comparing graphs and by constructing the Lagrange
form of the Taylor Remainder.
8.12 Perform substitution, algebraic manipulation, and term by term
differentiation and integration of known power series to form new series.
8.13 Construct and use the first several terms of a series to approximate
an integral.
(revised 08/20/04)
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