5.3 Differentiating polynomialsIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A function is defined by $f(x)=4x^3-6x^2+5x-9$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$12x^3-12x^2+5x$
   • B$12x^2-12x+5$
   • C$4x^4-6x^3+5x^2-9x$
   • D$12x^2-6x+5$

   (b)

   Find the gradient of the graph of $f$ at the point where $x=3$.

   [1 mark]

   • A$60$
   • B$113$
   • C$149$
   • D$77$

   (c)

   Find the values of $x$ at which the gradient of the graph of $f$ is $5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function is defined by $g(x)=3x^2+\frac{4}{x}-\frac{2}{x^2}$, for $x\neq0$.

   (a)

   Find $g'(x)$.

   [1 mark]

   • A$6x+\frac{4}{x^{2}}-\frac{4}{x^{3}}$
   • B$6x-4+\frac{4}{x}$
   • C$6x-\frac{4}{x^{2}}+\frac{4}{x^{3}}$
   • D$6x-\frac{4}{x^{2}}-\frac{4}{x^{3}}$

   (b)

   Find the gradient of the graph of $g$ at the point where $x=2$.

   [1 mark]

   • A$11.5$
   • B$12.5$
   • C$10$
   • D$10.5$

   (c)

   Find $g'(-1)$ and state whether $g$ is increasing or decreasing at $x=-1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The cost, $C$ AED, of producing $x$ items in a day is modelled by $C(x)=0.5x^3-6x^2+40x+200$, for $0<x\le20$. The rate of change of cost, in AED per item, is given by $C'(x)$.

   (a)

   (i) Find $C'(x)$.
   (ii) Find the rate of change of cost when $10$ items are produced.

   [3 marks]

   (b)

   The rate of change of cost is $40$ AED per item. Find the number of items produced, justifying your choice of solution.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A closed cylindrical can has volume $500\text{ cm}^3$ and radius $r$ cm. Its total surface area, $A\text{ cm}^2$, is given by $A(r)=2\pi r^2+\frac{1000}{r}$, for $r>0$.

   (a)

   (i) Write $A(r)$ with $r^{-1}$ in place of $\frac{1000}{r}$ and hence find $A'(r)$.
   (ii) Find $A'(5)$.
   (iii) Interpret your answer to (ii).

   [6 marks]

   (b)

   (i) Use your GDC to solve $A'(r)=0$.
   (ii) Find the value of $A$ at this value of $r$.
   (iii) Find $A'(3)$, and comment on how the surface area changes either side of your value of $r$ from (i), using $A'(5)=22.8$.

   [6 marks]

   Total for question 4: 12 marks

End of questions