Applications of Differentiation
Question 1
2017 CHIJ AM SA1 P1
$PQR$ is an isosceles triangle with $PQ=PR=50$cm and $QR=28$cm as shown in the diagram above. A rectangle $ABCD$ is drawn inside the triangle with $AB$ on $QR$, and $D$ and $C$on $PQ$ and $PR$ respectively.
(i)
If $AB=2x$cm, show that the area $A$ cm$^{2}$ of the rectangle is given by $A=\frac{48x\left( 14-x \right)}{7}$.
[3]
(ii)
Obtain an expression for $\frac{\text{d}A}{\text{d}x}$.
[1]
(iii)
Given that $x$ can wary, find the value of $x$ for which the area of the rectangle is stationary.
[2]
(iv)
Explain why this value of $x$ gives the largest area possible.
[1]
(i) 
(ii) 
(iii) 
(iv) 
Question 2
2017 PLMGS P1
A farmer uses $160$m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a rectangle, with the dimensions shown.
(i)
Show that the area of the plot is $\frac{320\sqrt{3}x-\left( 3\sqrt{3}+6 \right){{x}^{2}}}{4}$m$^{2}$.
(ii)
Given that $x$ can vary, find the value of $x$ for which the area of the plot is stationary.
(iii)
Explain why this value of $x$ gives the farmer the largest possible area for the plot. Find this area and give your answer correct to the nearest square metre.
(i) 
(ii) 
(iii) 
Question 3
2022 CCHMS AM S4 P1 Q4
A curve has the equation $y=\frac{6x-4}{x+3}$ for $x>-3$.
(a)
Find an expression for $\frac{\text{d}y}{\text{d}x}$.
[2]
(b)
Explain whether $y$ is an increasing or decreasing function.
[1]
(c)
Showing full working, determine whether the gradient of the curve is an increasing or decreasing function.
[2]
(a)
(b)
(c)
Question 4
2022 CCHMS AM S4 P1 Q7
The equation of a curve is $y=\frac{{{\left( ax+1 \right)}^{3}}}{6}-7$, where $a$ is a positive constant.
(a)
Explain why the curve only has one stationary point.
[3]
(b)
In the case where $a=3$, explain why this stationary point is a point of inflexion.
[2]
(a)
(b)
Question 5
2022 CHIJ SNGS AM S4 P1 Q2
The function $\text{f}$ is defined by $\text{f}\left( x \right)=\frac{x+1}{4x+1}$, where $x\ne -\frac{1}{4}$.
Determine, with working, whether $\text{f}$ is an increasing or decreasing function.
[4]
Question 6
2022 CHIJ SNGS AM S4 P1 Q6
Express $1+10x-2{{x}^{2}}$ in the form $p{{\left( x+q \right)}^{2}}+r$, where $p$, $q$ and $r$ are constants.
[3]
Hence, state
(i)
the range of values of $x$ for which $y=1+10x-2{{x}^{2}}$ is a decreasing function,
[1]
(ii)
the range of values of $k$ for which the line $y=k$ does not intersect the curve $y=1+10x-2{{x}^{2}}$,
[1]
(iii)
the maximum value of $y=2+20x-4{{x}^{2}}$.
[1]
(i)
(ii)
(iii)
Question 7
2022 CCHMS AM S4 P2 Q6
A spherical ball is completely filled with air. Air is then being released at a constant rate of $k$cm$^{3}$ per second.
(a)
Given that the rate of change of the radius of the ball is $-1.5$cm/s, when the radius is $4$cm, find the value of the constant $k$, leaving your answer in terms of $\pi $.
[3]
(b)
Find the rate of change of the surface area of the ball when the rate of change of its radius is one-third the rate of change of its volume.
[5]
(a)
(b)
Question 8
2022 CHS AM S4 P1 Q12
(b)
The function $\text{f}$ is defined by $y=\frac{7x-2}{2x+3}$, where $x\ne -1.5$.
It is known that $x$ and $y$ vary with time $t$, and that $x$ decreases at a rate of $0.2$ units per second. Find the rate at which $y$ is changing at the instant when $x=3$ units.
[4]
Question 9
2022 SCGS AM S4 P2 Q2
[The volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi {{r}^{2}}h$.]
An empty, inverted cone has a height of $60$cm and base radius $20$cm. The circular base is held horizontal and uppermost. Water is poured into the cone at a constant rate.
(a)
When the depth of the water in the cone is $h$ cm, show that the volume of the water in the cone is $\frac{\pi {{h}^{3}}}{27}$.
[2]
The water level is rising at a rate of $3$ cm per minute when the depth of the water is $12$ cm.
(b)
Find the rate at which water is being poured into the cone, leaving your answer in terms of $\pi $.
[3]
(a)
(b)
Question 10
2022 NTSS AM S4 P1 Q4
(a)
Given that $y=\frac{{{\text{e}}^{x}}}{{{x}^{2}}+1}$, $x\ne 1$, explain, with working, whether $~y$ is an increasing or decreasing function.
[3]
(b)
Air is escaping from a hole in a spherical balloon of radius $r$ cm in such a way that the total volume, $V$ cm$^{3}$, is decreasing at a constant rate of $25\pi $cm$^{3}$/s. Assuming that the balloon retains its shape, calculate the rate of change of $r$ when $r=5$.
[3]
(a)
(b)
