Practice section 7 of 8

The diagram shows $y=f^{\prime}(x)$ meeting the x-axis at $x=-3$ and $x=1$. Given $f(-3)=4$ and $f(1)=1$:

(i) State the coordinates and natures of the stationary points of $y=f(x)$. [3]

(ii) Find the range of $x$ for which the curve is strictly decreasing. [2]

(iii) Sketch $y=f(x)$. [2]

The curve $C$ is $y=f(x)=\dfrac{ax^2+b}{x+c}$, where $a,b,c$ are constants, and $y=-x+2$ is an asymptote.

(i) Show that $a=-1$, $c=2$. [2]

(ii) Given $b=1$, sketch $C$, showing all asymptotes, stationary points and axial intercepts, if any. [4]

(iii) Sketch $y=f^{\prime}(x)$, showing all asymptotes, axial intercepts and stationary points, if any. [3]

The line $y=mx+3/m$ is tangent to the parabola $y^2=kx$ for every $m\ne0$. Find $k$ and the coordinates of contact in terms of $m$. [4]

Sand from a chute forms a conical pile whose altitude is always $4/3$ of its base radius.

(i) How fast is its volume increasing when the radius is $1$ m and increases at $1/8\text{ cm s}^{-1}$? [4]

(ii) How fast is its radius increasing when it is $2$ m and the volume increases at $1000\text{ cm}^3\text{ s}^{-1}$? [3]

Point $Q(x,y)$ lies on $4x^2+xy+y^2=36$, with $y\ge0$. The curve meets the x-axis at $P,R$ as drawn.

(i) Show that triangle $PQR$ has area $A=3y$. [3]

(ii) Find $dy/dx$ in terms of $x,y$. [2]

(iii) Hence find the x-value where $A$ is stationary and use the second derivative test to determine its nature. [6]

If $x$ increases at $8$ units/s, find the rate of change of $A$ when $x=0$. [3]

A flower-bed for a large garden consists of an $x$ m by $y$ m rectangle topped by a semicircle of diameter $x$ m. A low wall takes $3$ hours per metre along the straight parts and $9$ hours per metre around the semicircular arc. A total of $180$ hours is available.

Use differentiation to find $x,y$ giving maximum area. [10]