Part 1
First pattern has been designed for you. You are required to find the functions and the circle used for this design, based on the description below.

The following functions/relations have been used to design this pattern:
- one logarithmic function and its inverse
- two cubic functions
- one circle

Recommended function types:
$$
\begin{array}{l}
y_{1}=f(x)=a \ln (b x+c) \
y_{2}=f^{-1}(x) \
g(x)=a x\left[(x-h)^{2}+k\right] \
h(x)=b x\left[(x-h)^{2}+k\right]
\end{array}
$$

Mathematical rules used for this design:
(i) the four functions start at the origin.
(ii) logarithmic function and its exponential inverse end at $(4,4)$
(iii) logarithmic function has $a=0.5$
(iv) the point of inflection of the top cubic (green function) is located at $x=1.6$
(v) the maximum value of the top cubic (green function) aligns vertically with the stationary point of inflection of the bottom cubic (blue function)
(vi) the minimum point of the top cubic (green function) is located at $(2,2)$
(vii) the area of the circle is one third of the area between the two cubic functions and its centre aligns with the minimum point of the top cubic.
(viii) the two cubic functions always meet on the logarithmic function.

Based on the above information find:
1. The four functions and the equation of the circle.
2. Calculate the areas between each curve combination (use of technology indicated).

Refer to the cubic functions found from above rules. Assume that their respective found $\boldsymbol{h}$ and $\boldsymbol{k}$ values do not change.
Yeor 12 Mothernatical Methods
2021
By varying only their leading coefficients, $a$ and $b$ respectively, investigate:
3. The change in the area between the two cubic functions with respect to the total area (area between the logarithm and the exponential function).
4. How $\boldsymbol{a}$ and $b$ vary as the area between the two cubic functions approaches a maximum or a minimum.

Hence,
5. Find two cubic functions such that the area between them is $1 /$ of the total area(area between the logarithm and the exponential function).

Question

Part 1
First pattern has been designed for you. You are required to find the functions and the circle used for this design, based on the description below.

The following functions/relations have been used to design this pattern:
- one logarithmic function and its inverse
- two cubic functions
- one circle

Recommended function types:
$$
\begin{array}{l}
y_{1}=f(x)=a \ln (b x+c) \
y_{2}=f^{-1}(x) \
g(x)=a x\left[(x-h)^{2}+k\right] \
h(x)=b x\left[(x-h)^{2}+k\right]
\end{array}
$$

Mathematical rules used for this design:
(i) the four functions start at the origin.
(ii) logarithmic function and its exponential inverse end at $(4,4)$
(iii) logarithmic function has $a=0.5$
(iv) the point of inflection of the top cubic (green function) is located at $x=1.6$
(v) the maximum value of the top cubic (green function) aligns vertically with the stationary point of inflection of the bottom cubic (blue function)
(vi) the minimum point of the top cubic (green function) is located at $(2,2)$
(vii) the area of the circle is one third of the area between the two cubic functions and its centre aligns with the minimum point of the top cubic.
(viii) the two cubic functions always meet on the logarithmic function.

Based on the above information find:
1. The four functions and the equation of the circle.
2. Calculate the areas between each curve combination (use of technology indicated).

Refer to the cubic functions found from above rules. Assume that their respective found $\boldsymbol{h}$ and $\boldsymbol{k}$ values do not change.
Yeor 12 Mothernatical Methods
2021
By varying only their leading coefficients, $a$ and $b$ respectively, investigate:
3. The change in the area between the two cubic functions with respect to the total area (area between the logarithm and the exponential function).
4. How $\boldsymbol{a}$ and $b$ vary as the area between the two cubic functions approaches a maximum or a minimum.

Hence,
5. Find two cubic functions such that the area between them is $1 /$ of the total area(area between the logarithm and the exponential function).

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given the logarithmic function $y_1 = f(x) = a \ln(bx + c)$ and its inverse $y_2 = f^{-1}(x)$, we know from the problem that $a = 0.5$. ‖ ‖ step 2 ⋮ To find $b$ and $c$, we solve the equation $0.5 \ln(bx + c) = 4$ for $b$ and $c$. From the Asksia-LL calculator, we have $c = e^8 - bx$. ‖ ‖ step 3 ⋮ Next, we solve $e^{0.5 \ln(b \cdot 4 + c)} = 4$ for $b$ and $c$. From the Asksia-LL calculator, we have $c = -4(b - 4)$. ‖ ‖ step 4 ⋮ For the cubic functions $g(x) = a x \left[(x-h)^2 + k\right]$ and $h(x) = b x \left[(x-h)^2 + k\right]$, we use the given conditions to find $a$, $b$, $h$, and $k$. ‖ ‖ step 5 ⋮ Given that the point of inflection of the top cubic (green function) is at $x = 1.6$, and the minimum point is at $(2,2)$, we solve for $a$, $h$, and $k$. ‖ ‖ step 6 ⋮ The area of the circle is one third of the area between the two cubic functions, and its center aligns with the minimum point of the top cubic. We use this information to find the radius of the circle. ‖ ‖ step 7 ⋮ To find the total area between the logarithmic function and its inverse, we integrate the difference between the two functions from $x = 0$ to $x = 4$. ‖ ‖ step 8 ⋮ To find the area between the two cubic functions, we integrate the difference between $g(x)$ and $h(x)$ from $x = 0$ to $x = 4$. ‖ ‖ step 9 ⋮ To investigate how $a$ and $b$ vary as the area between the two cubic functions approaches a maximum or minimum, we analyze the leading coefficients of the cubic functions. ‖ ‖ step 10 ⋮ Finally, to find two cubic functions such that the area between them is $1/3$ of the total area, we adjust the leading coefficients $a$ and $b$ accordingly. ‖ ∻Answer∻ ⚹ The four functions and the equation of the circle are found using the given conditions and solving for the unknowns. The areas between each curve combination are calculated using integration. ⚹ ∻Key Concept∻ ⚹Integration and solving for unknowns in functions⚹ ∻Explanation∻ ⚹The problem involves finding specific functions and calculating areas between them using integration, based on given conditions.⚹◊1. What is the equation of the logarithmic function given in the image?⍭ Generate me a similar question◊

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