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Math 172 Exam 1. Question and answers verified to pass 2024/2025
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Math 172 Exam 1. Question and answers verified to pass 2024/2025 Math 172 Exam 1 The definite integral (def 1) - correct answer A sequence of real numbers (xn)∞n=1 is said to converge to a real number l ∈ R, if for every ε > 0 there exists a natural number N := N(ε) ∈ N such that ...

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  • October 15, 2024
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Math 172 Exam 1
The definite integral (def 1) - correct answer ✔A sequence of real numbers
(xn)∞n=1 is said to converge to a real number l ∈ R, if for every ε > 0 there
exists a natural number N := N(ε) ∈ N such that for all n ≥ N, |xn − l| < ε.
If (xn)∞n=1 converges to l, we write
lim xn = l
n→∞


The definite integral (def 2) - correct answer ✔Let f be a function defined on
a closed interval [a, b] and let (xi)n
i=0
be the regular partition of [a, b],
i.e. xi = a + i ((b−a)/n) for all i ∈ {0, 1, . . . , n}. If lim Σ exists and gives the
n→∞
same value for all possible choices of points x*i ∈ [xi−1, xi], then we say
that f is integrable on [a, b].
If f is integrable on [a, b] we denote the definite integral of f on [a, b] by
∫b f(x)dx := lim Σi+1 f(x*i) (b-a)/n
a n→∞


Fundamental Theorem of Calculus I - correct answer ✔Let f : [a, b] → R be
continuous
on [a, b]. Then, F(x) = ∫x f(t)dt is
a
well-defined for all x ∈ [a, b]. More-

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