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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