01:640:135 Chapter Notes - Chapter 1-5: Antiderivative, Riemann Sum
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Suggestions for Studying for the Final Exam in Math 135
The ﬁnal exam in Math 135 covers the entire course. However, students can expect that there
will be a slight emphasis on the material in Chapter 5, since that material has not been covered on
earlier hour exams.
Students should carefully review the study suggestions for the hour exams. In reviewing the
material in Chapter 5, students should be sure that they
•Understand the deﬁnition of an indeﬁnite integral or antiderivative.
•Are able to check whether a given function Fis an antiderivative of another function f.
•Know antiderivatives for polynomials, sin x, cos x, sec2x, sec xtan x, and ex, and can compute
antiderivatives of functions related to these by the method of substitution.
•Understand Σ notation for sums.
•Understand the deﬁnition of a Riemann sum and can compute the value of a Riemann sum
given the function, the interval, the partition, and the choice of representative points.
•Understand that deﬁnite integrals are limits in an appropriate sense of Riemann sums.
•Understand the interpretation of deﬁnite integrals as the “net” area under the graph of a
function, where area above the x-axis is counted positively and area below the x-axis is
•Can evaluate deﬁnite integrals using antiderivatives.
•Can diﬀerentiate functions deﬁned as deﬁnite integrals with varying upper or lower limits.
Chapter 5 contains The Fundamental Theorem of Calculus, in fact two versions of that theorem.
Certainly students should pay attention to a theorem that is described as the fundamental result
of the subject.
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