Differentiation (AS)AQA A-Level Maths: Topic test

1. 1
   The function $g$ is defined by $g(x)=x^{2}\sqrt{x}-\dfrac{4}{\sqrt{x}}$ for $x>0$.

   (a)

   Which expression is $g'(x)$?

   [1 mark]

   • A$\frac52x^{3/2}-2x^{-3/2}$
   • B$\frac52x^{3/2}+2x^{-3/2}$
   • C$\frac52x^{5/2}+2x^{-1/2}$
   • D$\frac52x^{3/2}+8x^{-3/2}$

   (b)

   What is the value of $g'(4)$?

   [1 mark]

   • A19.75
   • B40.25
   • C20.25
   • D22

   (c)

   Find $g''(x)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=x^{2}+4x$.

   (a)

   Which expression is equal to $f(x+h)-f(x)$?

   [1 mark]

   • A$2xh+4h$
   • B$h^{2}+4h$
   • C$x^{2}+h^{2}+4x+4h$
   • D$2xh+h^{2}+4h$

   (b)

   What is $\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$?

   [1 mark]

   • A$2x+4$
   • B$2x+h+4$
   • C$2x$
   • D$4$

   (c)

   Find the coordinates of the stationary point of the curve $y=f(x)$ and determine whether it is a maximum or a minimum.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^{3}-4x^{2}+x+6$ and passes through the point $P$ where $x=3$.

   (a)

   Show that $P$ has coordinates $(3,0)$ and find the equation of the tangent to $C$ at $P$.

   [3 marks]

   (b)

   Find the equation of the normal to $C$ at $P$. The tangent and the normal at $P$ meet the $y$-axis at $A$ and $B$. Find the length $AB$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company's profit $P$ thousand pounds when it sells $x$ hundred units of a product is modelled by $P=-x^{3}+9x^{2}-15x-8$ for $0\le x\le8$.

   (a)

   Find the coordinates of the stationary points of the profit curve and determine their nature.

   [6 marks]

   (b)

   Find the values of $x$ for which the profit is increasing. Show that the profit curve has a point of inflection at $x=3$ and explain what this means for the rate at which profit changes.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $C$ has equation $y=x^{4}-8x^{2}$.

   (a)

   How many stationary points does $C$ have?

   [1 mark]

   • A1
   • B2
   • C3
   • D4

   (b)

   What is the nature of the stationary point of $C$ at $x=0$?

   [1 mark]

   • ALocal minimum
   • BPoint of inflection
   • CNot a stationary point, since the gradient is not zero
   • DLocal maximum

   (c)

   Find the set of values of $x$ for which $C$ is concave.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Water flows into a tank from a tap and also leaks out through a small hole in the base. The volume $V$ cm$^3$ of water in the tank, $t$ minutes after the tap is opened, is modelled by $V=40t-t^{2}$ for $0\le t\le30$.

   (a)

   What is the rate of change of $V$, in cm$^3$ per minute, when $t=5$?

   [1 mark]

   • A30
   • B175
   • C35
   • D40

   (b)

   At what time is the volume of water in the tank greatest?

   [1 mark]

   • A$t=40$
   • B$t=20$
   • C$t=30$
   • D$t=400$

   (c)

   Find $\dfrac{\mathrm{d}^2V}{\mathrm{d}t^2}$ and interpret its value in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $f$ is defined by $f(x)=x^{3}-2x$.

   (a)

   Expand and simplify $f(x+h)-f(x)$.

   [3 marks]

   (b)

   Use your answer to differentiate $f$ from first principles. Hence find the coordinates of the points on the curve $y=f(x)$ where the tangent is parallel to the line $y=10x+1$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The curve $C$ has equation $y=x^{4}-6x^{2}+8x$.

   (a)

   Find the coordinates of the stationary points of $C$ and determine their nature.

   [6 marks]

   (b)

   Find the values of $x$ for which $C$ is concave. Find the coordinates of the other point of inflection of $C$ and the equation of the tangent there.

   [6 marks]

   Total for question 8: 12 marks

End of questions