Nagoya University, G30 program Fall 2015

Calculus I Instructor : Serge Richard

Midterm

Exercise 1 Compute the following limits:

1. limx→1−(x−1)|x−1| e

(x−1)2,

2. limx→0

√1+x−1x ,

3. limx→0(ex−1)2

x2 .

Exercise 2 Compute the derivative of the following functions:

1. f : R → R, f(x) = 3x4 + x sin(x),

2. g : R → R, g(x) = 1cos2(3x)+1

,

3. h : R∗ → R, h(x) = |x|ex,

Exercise 3 The function hyperbolic tangent tanh is defined for any x ∈ R by

tanh(x) =sinh(x)

cosh(x)=

ex − e−x

ex + e−x.

1. Compute the derivative of tanh and find the critical point(s) of tanh,

2. Determine the maximal domain on which an inverse for this function can be defined, and call this

inverse arctanh,

3. Compute the derivative of the function arctanh.

Exercise 4 Sketch the following curve as precisely as possible:

f : R \ {1} ∋ x 7→ x2 + x

x− 1∈ R.

Exercise 5 State as precisely as possible the Mean Value Theorem. Can you illustrate its content with

a drawing ?

Exercise 6 Consider the curve in R2 defined by the relation

F (x, y) = x3 + y3 − 6xy = 0.

1. Find the equation of the tangent line at the point (3, 3),

2. For x, y > 0, at what point (x, y) is the tangent to the curve a horizontal line ?

1