Practice section 5 of 8

The diagram shows $y=f^{\prime}(x)$, passing through the origin with turning points $(-3,0)$ and $(-1.5,-1)$. The x-axis and $x=1$ are its two asymptotes.

(i) Find the range of $x$ for which $y=f(x)$ is strictly increasing and concave downwards.

(ii) State the x-coordinates and natures of all stationary points of $y=f(x)$.

(iii) Given $f(0)=1$, sketch $y=f(x)$ for $x<1$, clearly indicating all stationary points, asymptotes and intersections with the axes.

The equation of a closed curve is $(x+2y)^2+3(x-y)^2=27$.

(i) Show by differentiation that its gradient is $\dfrac{dy}{dx}=\dfrac{y-4x}{7y-x}$.

(ii) Find the equations of the tangents parallel to (a) the x-axis, (b) the y-axis.

The curve $C$ has equation $2^{-y}=x$. Its point $A$ has x-coordinate $a>0$. Show that $dy/dx=-1/(a\ln2)$ at $A$, and find the equation of the tangent there.

Hence find the tangent to $C$ that passes through the origin. The line $y=mx$ intersects $C$ at two distinct points. Write down the range of $m$.

The curve has equation $y^2-xy=-1$.

(i) Find all tangent equations parallel to the y-axis.

(ii) State and justify whether the curve has any stationary points.

(iii) Find the area bounded by the axes and the normal at the point where $y=2$.

The right-angled triangle $ABC$ has $AB=c$ cm, $BC=a$ cm and $AC=b$ cm, with right angle at $C$ and fixed area $100\text{ cm}^2$.

Express $b$ in terms of $a$. Given that $BC$ increases at $3\text{ cm s}^{-1}$, find the rate of change of $c$ when $a=20$.

The diagram shows rectangle $PQRS$ inscribed in a semicircle of radius $10$ cm, with $P,Q$ on its diameter and $S,R$ on its arc. Given $PQ=x$ cm, find the perimeter $p$ cm in terms of $x$.

Show that its area is $A=\dfrac{x}{2}\sqrt{400-x^2}\text{ cm}^2$. Find the exact value of $p$ when $A$ has its stationary value.