Applications of Differentiation (AQA GCSE Further Maths): Exam Questions

Point $A$lies on the curve $y=x^2+5x+8$

The $x$-coordinate of $A$ is – 4.

Show that the equation of the normal to the curve at $A$ is $3y=x+16$.

The normal at $A$ also intersects the curve at $B$.

Work out the $x$-coordinate of$B$.

$P$ is the point on the curve $y=a{x}^3+10x^2$ where $x=2$.

The gradient of the normal to the curve at $P$ is $-\frac{1}{4}$.

Work out the value of $a$.

$y=12x+\frac{3}{x}$

Show that $y$ has a minimum value when   $x=0.5$.

$P$is a point on a curve.

The curve has gradient function $\frac{x^5-17}{10}$.

The tangent to the curve at $P$ is parallel to the line $3x-2y=9$. Work out the $x$-coordinate of $P$.

The curve $y=\mathrm{f}\left(x\right)$ has $\frac{dy}{dx}={(x+2)}^5+{(x+2)}^3$.

The curve has exactly one stationary point at $P$ where $x=–2$.

Use the expression for $\frac{dy}{dx}$ to show that $P$ is a minimum point.

A curve has equation $y=2x^2+3x–9$.

At a point $P$ on the curve, the tangent is parallel to the line $y=4-5x$.

Work out the coordinates of $P$.

You must show your working.

$y=\mathrm{f}\left(x\right)$ is the graph of a cubic function.

$y<0$ for $x<5$

$y⩾0$ for $x⩾5$

The function is

increasing for $x<–1$

decreasing for $–1<x<2$

increasing for $x>2$

Draw a possible sketch of $y=\mathrm{f}\left(x\right)$ for values of $x$ from $–2$ to $6$

The continuous curve $y=$g($x$)  has exactly two stationary points.

The stationary points are

• a maximum point at $P\left(3a,b\right)$ where $a>0$ and $b<0$

•  a minimum point at $Q\left(-a,3b\right)$

On the axes below, sketch the curve.

Label points $P$ and $Q$ on your sketch.

$\mathrm{f}\left(x\right)=2x^3–12x^2+25x–11$

Use differentiation to show that $\mathrm{f}\left(x\right)$ is an increasing function for all values of $x$.

The diagram shows a sketch of the cubic curve   $y=\frac{1}{3}{x}^3-x^2-3x+k$

where $k$ is a constant.

The $x$-axis is a tangent to the curve at its minimum point.

Work out the value of $k.$

$k=...................$

The curve $y=2x^3-5$ intersects the $y$-axis at $C$.

The tangent to the curve at $P\left(2,11\right)$ intersects the $y$-axis at $D$.

Work out the length $CD$.

......................units