CALCU-solve question d,ei,eii,eiii,eiv,evc. Explai

solve question d,ei,eii,eiii,eiv,ev

Generated Graph

Solution by Steps

step 1

To explain the shape of the graph of

c(t) = t e^{-t}

, consider the component functions

y = t

and

y = e^{-t}

step 2

The graph of

y = t

is a straight line passing through the origin with a slope of 1

step 3

The graph of

y = e^{-t}

is an exponential decay function, with a y-intercept at

(0, 1)

and approaching the t-axis as an asymptote as

t

goes to infinity

step 4

The graph of

c(t)

will have a y-intercept at

(0, 0)

since

c(0) = 0 \cdot e^{0} = 0

step 5

The graph of

c(t)

will have no x-intercepts other than the origin, as

c(t)

is never zero for t > 0

step 6

The graph of

c(t)

will approach the t-axis as an asymptote as

t

goes to infinity, similar to

y = e^{-t}

step 7

The maximum value of

c(t)

occurs at

t = 1

, which can be found by setting the derivative equal to zero and solving for

t

[question d] Answer

The function

c(t) = t e^{-t}

would not be suitable for modelling the concentration of a safe and effective drug because it approaches zero as

t

approaches infinity, implying that the drug would eventually have no effect. Additionally, the concentration increases initially and then decreases, which may not be desirable for maintaining a therapeutic level.

Key Concept

Model suitability based on function behavior

Explanation

A suitable drug concentration model should maintain a therapeutic level over time, not decrease to zero.

step 1

To find the new rule after dilation by a factor of 3 from the t-axis, multiply the original function

c(t)

by 3

step 2

The new function is

c_3(t) = 3t e^{-t}

[question ei] Answer

The new rule after dilation is

c_3(t) = 3t e^{-t}

.

Key Concept

Dilation of functions

Explanation

Multiplying a function by a constant factor dilates the graph vertically by that factor.

step 1

To determine if

c_3(t) = 3t e^{-t}

is suitable for modelling the concentration of an effective drug, consider the behavior of the function

step 2

The function

c_3(t)

has a maximum value greater than that of

c(t)

, which may allow for a higher initial concentration

step 3

However, like

c(t)

,

c_3(t)

also approaches zero as

t

approaches infinity, which may not be ideal for maintaining a therapeutic level over time

[question eii] Answer

The function

c_3(t) = 3t e^{-t}

may not be suitable for modelling the concentration of an effective drug as it still approaches zero over time.

Key Concept

Long-term behavior of drug concentration models

Explanation

A model that approaches zero may not maintain the necessary therapeutic level over time.

step 1

To determine the range of values for

k

where

k \in \mathbb{Z}^+

such that

c_k(t) = k t e^{-t}

is suitable for modelling drug concentration, consider the maximum value and long-term behavior

step 2

The function must have a sufficient maximum value to reach therapeutic levels but should not approach zero too quickly

step 3

Since the maximum value of

c(t)

is

1/e

and

c_k(t)

is

k/e

,

k

must be chosen to ensure the maximum value is within the desired therapeutic range

[question eiii] Answer

The range of values for

k

must be chosen such that the maximum value of

c_k(t)

is within the therapeutic range and does not approach zero too quickly.

Key Concept

Therapeutic range and maximum value

Explanation

The maximum value of the drug concentration model should be within the therapeutic range for effectiveness.

step 1

Choosing the minimum suitable value of

k

is advisable to avoid excessive concentration, which could lead to toxicity

[question eiv] Answer

It is advisable to choose the minimum suitable value of

k

to avoid potential toxicity.

Key Concept

Minimizing risk of toxicity

Explanation

A lower value of

k

reduces the risk of the drug concentration becoming too high.

step 1

To choose a suitable value of

k

, consider the therapeutic range and the maximum value of

c_k(t)

step 2

Sketch the graph of

c_k(t)

based on the chosen value of

k

and compare the maximum value to that of

c(t)

step 3

Ensure that the graph of

c_k(t)

has a maximum value within the therapeutic range and does not approach zero too quickly

[question ev] Answer

A suitable value of

k

should be chosen based on the therapeutic range, and the graph of

c_k(t)

should be sketched to compare the maximum values.

Key Concept

Choosing

k

based on therapeutic range

Explanation

The value of

k

determines the maximum concentration and should be chosen to ensure effectiveness without risking toxicity.

Solve question fi,fii,g,hi,hii,

Solution by Steps

step 1

To construct a suitable rule

c_m(t)

for the dilation factor

m

from the

y

-axis, we need to multiply the function

f(t)

by

m

step 2

The rule for

c_m(t)

is

c_m(t) = m \cdot f(t)

step 3

To find the minimum suitable integer value of

m

, we need to ensure that the maximum concentration is less than

3 \mathrm{mg} / \mathrm{litre}

and the concentration is greater than

1 \mathrm{mg} / \mathrm{litre}

for at least 6 hours

step 4

Without the specific function

f(t)

, we cannot calculate the exact value of

m

. However, we can say that

m

must be chosen such that the conditions are met

[fi] Answer

The rule is

c_m(t) = m \cdot f(t)

, and the minimum suitable integer value of

m

depends on the specific function

f(t)

.

Key Concept

Dilation from the y-axis

Explanation

The rule

c_m(t)

represents the concentration of the drug after dilation, and

m

must be chosen to satisfy the given conditions for drug effectiveness.

Solution by Steps

step 1

Choose an appropriate integer value of

m

based on the conditions given for the drug to be effective

step 2

Sketch the graph of

c_m(t)

on a suitable domain, ensuring that the maximum concentration occurs within

1-6

hours and does not exceed

3 \mathrm{mg} / \mathrm{litre}

step 3

Identify all key features of the graph, including the maximum point, the duration where the concentration is above

1 \mathrm{mg} / \mathrm{litre}

, and any intercepts with the axes

[fii] Answer

The appropriate integer value of

m

and the sketch of

c_m(t)

depend on the specific function

f(t)

and must meet the conditions for drug effectiveness.

Key Concept

Graphing and Conditions for Effectiveness

Explanation

The graph of

c_m(t)

must be sketched to visually confirm that the chosen value of

m

results in a model that satisfies all the required conditions for an effective drug.

Solution by Steps

step 1

To determine suitable values for

m

and

k

, we need to consider the conditions for the drug to be effective

step 2

The value of

m

affects the y-axis dilation, and

k

affects the x-axis dilation

step 3

Choose

m

and

k

such that the maximum concentration is less than

3 \mathrm{mg} / \mathrm{litre}

, the concentration is greater than

1 \mathrm{mg} / \mathrm{litre}

for at least 6 hours, and the maximum concentration occurs within

1-6

hours

step 4

Without the specific function

f(t)

, we cannot provide exact values for

m

and

k

. However, we can state that they must be chosen to satisfy the given conditions

[g] Answer

Suitable values for

m

and

k

depend on the specific function

f(t)

and must be chosen to satisfy the conditions for drug effectiveness.

Key Concept

Choosing Dilation Factors

Explanation

The values of

m

and

k

must be chosen to ensure that the model of the drug concentration meets the effectiveness criteria.

Solution by Steps

step 1

State the rule for

c_m(t)

using the chosen values of

m

and

k

step 2

The rule is

c_m(t) = m \cdot f(kt)

[hi] Answer

The rule for the chosen values of

m

and

k

is

c_m(t) = m \cdot f(kt)

.

Key Concept

Rule for Dilation

Explanation

The rule

c_m(t)

incorporates both the dilation from the y-axis by

m

and the dilation from the x-axis by

k

.

Solution by Steps

step 1

Provide graphical and/or algebraic evidence that the model satisfies the conditions for an effective drug

step 2

Graphically, ensure that the graph of

c_m(t)

shows the concentration above

1 \mathrm{mg} / \mathrm{litre}

for at least 6 hours, below

3 \mathrm{mg} / \mathrm{litre}

, and the maximum within

1-6

hours

step 3

Algebraically, demonstrate that for the domain of

t

where the drug is effective, the inequalities 1 < c_m(t) < 3 hold true

step 4

Without the specific function

f(t)

, we cannot provide the exact graphical or algebraic evidence. However, the evidence must show that all conditions are met

[hii] Answer

Graphical and algebraic evidence must confirm that the model meets the conditions for an effective drug, but specific evidence depends on the function

f(t)

.

Key Concept

Evidence of Effectiveness

Explanation

The model must be supported by evidence, either graphical or algebraic, that demonstrates the drug concentration remains within the effective range for the required duration.

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