Nagoya University, G30 program Fall 2016
Calculus I Instructor : Serge Richard
Final examination
Exercise 1 Consider a function f : R → R sufficiently differentiable, and let x0 ∈ R.
1. Write the Taylor’s expansion up to order n with an expression for the remainder,
2. If f(x) = ex and x0 = 1, write the polynomial of degree n that you obtain in the Taylor’s expansion,
3. For the same function and the same x0, provide a simple estimate for the remainder term.
Exercise 2 Compute the following integrals:
a)
∫cos3(x)dx,
∫xα ln(x)dx for any α ≥ 0,
∫ 1
−1
√1− x2dx.
For the last integral you can use the equality cos(x)2 = 1+cos(2x)2 .
Exercise 3 Compute the derivatives of the following functions (and simplify the results, if possible):
a) x 7→ sin((x2 + 1)2
), b) x 7→ ex − 1
ex + 1, c) x 7→ xx.
Exercise 4 Compute the following limits:
a) limx→0
ex − 1− sin(x)
x2 + x3, b) lim
x→0
cos2(x)− 1
x4.
Exercise 5 Consider the sequence of numbers (aj)j∈N with aj =(−1)j
j .
1. Is the corresponding series∑
j∈N aj convergent ?
2. Is the corresponding series absolutely convergent ?
3. Is the power series∑
j∈N |aj |xj convergent for x = 12 ?
All answers must be explained.
Exercise 6 Let f : R → R be a function sufficiently many times differentiable and with f(0) = 1.
Consider now the function x 7→ 1xf(x) which is not continuous at x = 0. We would like to give a
meaning to the integral∫ 1−1
1xf(x)dx. For that purpose, we consider for any ε > 0 the expression
Iε :=
∫ −ε
−1
1
xf(x)dx+
∫ 1
ε
1
xf(x)dx.
1. Justify why Iε is well-defined for any ε > 0,
2. By considering a Taylor’s expansion of f around 0, show that limε→0 Iε exists.
For information, the above limit is called a principal value integral.
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