Exam III Study Guide
408C Calculus
4.4.13
Questions on the second exam will be mainly drawn from the list of topics below but my also
include topics not listed. You should know how to do the following:
4.2 The Mean Value Theorem
• Understand the hypotheses and conclusions of Rolle’s Theorem.
• Decide when Rolle’s Theorem is valid and determine which numbers satisfy its conclusion.
• Understand the hypotheses and conclusions of the Mean Value Theorem.
• Decide when the Mean Value Theorem is valid and determine which numbers satisfy its
conclusion.
4.3 How Derivatives Affect the Shape of a Graph
• Find the intervals on which a function is increasing, decreasing, concave up, and concave
down.
• Locate inflection points of a graph.
• Apply the First and Second Derivative Tests to determine if a curve has a maximum or
minimum value at a critical point.
4.4 Indetermine Forms and l’Hospital’s Rule
• Recognize an indeterminate form.
• Use l’Hospital’s rule to evaluate limits of the form limx→a
f (x)
.
g(x)
• Rewrite limits such as limx→0+ x ln x or limx→0+ xx so that l’Hospital’s rule can be applied.
4.5 Summary of Curve Sketching
• Find the domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, concavity, minima, maxima, and inflection points of a function.
• Use the above information to make an accurate sketch of a given function.
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Exam III Study Guide
408C Calculus
4.4.13
4.7 Optimization Problems
• Follow the protocol discussed in class to solve optimization problems.
4.9 Antiderivatives
• Understand the relationship between an antiderivative and a derivative.
• Observe why a given function f (x) has a family of antiderivatives F (x) + C instead of a
unique antiderivative.
• Find antiderivatives of a standard collection of functions (given in a table on page 345 of the
text).
• Determine antiderivatives of sums and differences of functions.
5.1 Areas and Distances
• Estimate the area under a given curve using rectangles with right endpoints and left endpoints
lying on the curve.
• Recognize area under a curve as a limit of the rectangle sums as the number of rectangles
approaches infinity.
• Use velocity data for an object in motion to estimate the position of that object at a given
time.
5.2 The Definite Integral
• Understand the definition of the definite integral.
• Be able to write an integral as a limit using the formula in Theorem 4 on page 374.
• Realize that integrating a curve which lies below the x-axis contributes a negative amount to
the net area under the curve.
• Utilize properties of integrals to rewrite them in different forms.
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Exam III Study Guide
408C Calculus
4.4.13
5.3 The Fundamental Theorem of Calculus
• Know the Fundamental Theorem of Calculus, Part 1.
• Apply the Fundamental Theorem of Calculus, Part 2 to evaluate definite integrals.
• Using the FTC, take derivatives of functions like
Z
x3
f (x) =
2
e−x dx.
x2
5.4 Indefinite Integrals and the Net Change Theorem
• Understand the relationship between an indefinite integral and an antiderivative.
• Evaluate indefinite integrals for a standard collection of functions.
• Realize a definite integral of a rate of change as the net change of a quantity over a time
period or an interval.
• Distinguish between displacement and total distance traveled, and calculate each quantity.
5.5 The Substitution Rule
• Use u-substitution to evaluate definite and indefinite integrals.
• Recognize even and odd functions, and use this property to calculate their integrals.
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