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OEF Derivatives

OEF Derivatives
--- Introduction ---

This module actually contains 35 exercises on derivatives of real
functions of one variable.

Arc and Arg

Establish the correspondence between the fucntions and their derivatives in the following table.

Circle

We have a circle whose radius increases at a constant speed of centimeters per second. At the moment when the radius equals centimeters, what is the speed at which its area increases (in
/s)?

Circle II

We have a circle whose radius increases at a constant speed of centimeters per second. At the moment when its area equals square centimeters, what is the speed at which the area increases (in
/s)?

Circle III

We have a circle whose area increases at a constant speed of square centimeters per second. At the moment when the area equals cm^{2}, what is the speed at which its radius increases (in cm/s)?

Circle IV

We have a circle whose area increases at a constant speed of square centimeters per second. At the moment when its radius equals cm, what is the speed at which the radius increases (in cm/s)?

Composition I

We have two differentiable functions
and
, with values and derivatives shown in the following table.

x

-3

-2

-1

0

1

2

3

Let
be defined by
. Compute the derivative
.

Composition II *

We have 3 differentiable functions
,
and
, with values and derivatives shown in the following table.

x

-3

-2

-1

0

1

2

3

Let
the function defined by
. Compute the derivative
.

Mixed composition

We have a differentiable function
, with values and derivatives shown in the following table.

x

-2

-1

0

1

2

Let
, and let
defined by
. Compute the derivative
.

Virtual chain Ia

Let
be a differentiable function, with derivative
. Compute the derivative of
.

Virtual chain Ib

Let
be a differentiable function, with derivative
. Compute the derivative of
.

Division I

We have two differentiable functions
and
, with values and derivatives shown in the following table.

x

-2

-1

0

1

2

Let
defined by
. Compute the derivative
.

Mixed division

We have a differentiable function
, with values and derivatives shown in the following table.

x

-2

-1

0

1

2

Let
defined by
. Compute the derivative
.

Hyperbolic functions I

Compute the derivative of the function
defined by
.

Hyperbolic functions II

Compute the derivative of the function
defined by
.

Multiplication I

We have two differentiable functions
and
, with values and derivatives shown in the following table.

x

-2

-1

0

1

2

Let
. Compute the derivative
.

Multiplication II

We have two differentiable functions
and
, with values and derivatives shown in the following table.

x

-2

-1

0

1

2

Let
. Compute the second derivative
.

Mixed multiplication

We have a differentiable function
, with values and derivatives shown in the following table.

-2

-1

0

1

2

Let
defined by
. Compute the derivative
.

Virtual multiplication I

Let
be a differentiable function, with derivative
. Compute the derivative of
.

Polynomial I

Compute the derivative of the function
defined by
, for
.

Polynomial II

Compute the derivative of the function
defined by
.

Rational functions I

Compute the derivative of the function

Rational functions II

Compute the derivative of the function

Inverse derivative

Let
be the function defined by

.

Verify that
is bijective, therefore we have an inverse function
. Calculate the value of its derivative
at
.

You must reply with a precision of at least 4 significant digits.

Rectangle I

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle II

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle III

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle IV

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle V

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Rectangle VI

We have a rectangle whose at a constant speed of centimeters per second, but whose stays constant at . At the moment when equals , what is the speed (in ) at which changes?

Right triangle

We have a right triangle as follows, where AB= , and AC at a constant speed of /s. At the moment when AC= , what is the speed at which BC changes (in /s)?

Sign of a number

Construct a study of the sign of
by choosing four of the sentences given below.

,

,

,

,

Tower

Somebody walks towards a tower at a constant speed of meters per second. If the height of the tower is meters, at what speed (in m/s) does the distance between the man and the top of the tower decrease, when the distance between him and the foot of the tower is meters?

Trigonometric functions I

Compute the derivative of the function
defined by
.

Trigonometric functions II

Compute the derivative of the function
.

Trigonometric functions III

Compute the derivative of the function
defined by
at the point
.
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