Pregunta
- The basics Consider the function
a. What is its rate of change when
is 0 ?
b. What is its rate of change when
is 1 ?
c. What is its second derivative?
d. What is its fifth derivative?
a. What is its rate of change when
b. What is its rate of change when
c. What is its second derivative?
d. What is its fifth derivative?
Ask by Sherman Summers. in Canada
Mar 14,2025
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Respuesta verificada por el tutor
Responder
Simplified Answers:
(a) Rate of change when
:
0
0
(b) Rate of change when
:
20
20
© Second derivative:
(d) Fifth derivative:
Solución
Step 1. Write down the function
We have
Step 2. Compute the first derivative
Differentiate each term:
- Derivative of
is . - Derivative of
is . - Derivative of
is . - Derivative of
is . - Derivative of
is . - Derivative of
is .
Thus,
(a) Rate of change when
Evaluate
at
:
(b) Rate of change when
Evaluate
at
:
Step 3. Compute the second derivative
Differentiate
:
- Derivative of
is . - Derivative of
is . - Derivative of
is . - Derivative of
is . - Derivative of
is .
Thus,
© The second derivative is
Step 4. Compute the fifth derivative
We calculate the derivatives term by term from the original
:
-
For
: All derivatives are (for orders ). -
For
: -
-
- Higher derivatives become
.
-
-
For
: -
-
-
-
- Subsequent derivatives are
.
-
-
For
: -
-
-
-
-
.
-
-
For
: -
-
-
-
-
.
-
-
For
: -
-
-
-
-
-
.
-
Now, summing the contributions only from terms that survive to the fifth derivative:
- From
and and : Their fifth derivatives are . - From
: The fifth derivative is . - From
: The fifth derivative is .
Thus,
(d) The fifth derivative is
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Bonus Knowledge
To tackle the function
, we’ll dive into each part step by step.
a. The rate of change of a function at a certain point is found using the derivative. The first derivative
of our function is calculated as
. Evaluating this at
gives
.
b. Evaluating
gives
. So, the rate of change when
is
.
c. The second derivative
is the derivative of
:
.
d. The fifth derivative
can be computed from the original function; since the highest degree is
, the derivatives reduce to zero after the sixth, leading us to conclude
at
(as it comes from the
term).
So the results are
,
,
, and
.
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