Pure: DifferentiationEdexcel A-Level Maths: Topic test

1. 1
   A curve has equation $y=5\sin3x+2\mathrm{e}^{-x}$, where $x$ is in radians.

   (a)

   What is $\dfrac{\mathrm{d}y}{\mathrm{d}x}$?

   [1 mark]

   • A$5\cos3x-2\mathrm{e}^{-x}$
   • B$15\cos3x+2\mathrm{e}^{-x}$
   • C$15\cos3x-2\mathrm{e}^{-x}$
   • D$-15\cos3x-2\mathrm{e}^{-x}$

   (b)

   What is the gradient of the curve where $x=0$?

   [1 mark]

   • A$13$
   • B$17$
   • C$15$
   • D$-2$

   (c)

   Find the equation of the tangent to the curve at the point where $x=0$, giving your answer in the form $y=mx+c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(x)=3x^2-x$.

   (a)

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

   [1 mark]

   • A$6x+3h-1$
   • B$6xh+3h^2-h$
   • C$6xh+3h^2+h$
   • D$3h^2-h$

   (b)

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

   [1 mark]

   • A$6x+3h-1$
   • B$3x-1$
   • C$6x$
   • D$6x-1$

   (c)

   Find $f''(x)$. Hence state, with a reason, whether the curve $y=f(x)$ is convex or concave for all $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A farmer encloses a rectangular field next to a straight river using $240$ m of fencing. There is no fence along the river. An extra fence divides the field into two parts, so there are three fences perpendicular to the river, each of length $x$ m, and one fence of length $y$ m parallel to the river.

   (a)

   Show that the area of the field, $A$ m$^2$, is given by $A=240x-3x^2$.

   [3 marks]

   (b)

   Use calculus to find the maximum possible area, and justify that it is a maximum.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=x\mathrm{e}^{-2x}$ for $x\ge0$.

   (a)

   Use the product rule to find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. Hence find the exact coordinates of the stationary point of $C$ and determine its nature.

   [6 marks]

   (b)

   Show that $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=4(x-1)\mathrm{e}^{-2x}$. Hence find the $x$-coordinate of the point of inflection of $C$, justify that it is a point of inflection, and state the values of $x$ for which $C$ is convex.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The curve $C$ has equation $x^3+y^3=6xy$.

   (a)

   Which equation is obtained by differentiating both sides with respect to $x$?

   [1 mark]

   • A$3x^2+3y^2=6y+6x$
   • B$3x^2+3y^2\dfrac{\mathrm{d}y}{\mathrm{d}x}=6y$
   • C$3x^2+3y^2=6y+6x\dfrac{\mathrm{d}y}{\mathrm{d}x}$
   • D$3x^2+3y^2\dfrac{\mathrm{d}y}{\mathrm{d}x}=6y+6x\dfrac{\mathrm{d}y}{\mathrm{d}x}$

   (b)

   The point $(3,3)$ lies on $C$. What is the gradient of $C$ at this point?

   [1 mark]

   • A$1$
   • B$-1$
   • C$0$
   • D$-3$

   (c)

   Find the equation of the tangent to $C$ at $(3,3)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A rumour spreads through a school of $800$ students. At time $t$ days after it starts, $N$ students have heard it. The rate of increase of $N$ is proportional to the product of the number of students who have heard it and the number who have not.

   (a)

   Which differential equation models this, where $k$ is a positive constant?

   [1 mark]

   • A$\dfrac{\mathrm{d}N}{\mathrm{d}t}=kN(800-N)$
   • B$\dfrac{\mathrm{d}N}{\mathrm{d}t}=k(800-N)$
   • C$\dfrac{\mathrm{d}N}{\mathrm{d}t}=-kN(800-N)$
   • D$\dfrac{\mathrm{d}N}{\mathrm{d}t}=\dfrac{kN}{800-N}$

   (b)

   When $N=20$, the rate of increase of $N$ is $15.6$ students per day. What is the value of $k$?

   [1 mark]

   • A$0.78$
   • B$0.02$
   • C$0.001$
   • D$0.0195$

   (c)

   Find the number of students who have heard the rumour when the rate of increase is greatest.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Sand is poured onto level ground at a constant rate of $12$ cm$^3$ per second. It forms a cone whose height is always equal to its base radius, $r$ cm, at time $t$ seconds. The volume of a cone is $\frac13\pi r^2h$ and its curved surface area is $\pi rl$, where $l$ cm is the slant height.

   (a)

   Show that the volume of the cone is $V=\frac13\pi r^3$, and find $\dfrac{\mathrm{d}r}{\mathrm{d}t}$ in terms of $r$.

   [3 marks]

   (b)

   Find the rate at which the curved surface area of the cone is increasing when $r=4$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   In this question $x$ and $t$ are real numbers and all angles are measured in radians.

   (a)

   Prove from first principles that the derivative of $\sin x$ is $\cos x$. You may use $\sin(x+h)=\sin x\cos h+\cos x\sin h$ and the small angle approximations $\sin h\approx h$ and $\cos h\approx1-\frac{h^2}{2}$ for small $h$.

   [6 marks]

   (b)

   A curve has parametric equations $x=t^2+2t$, $y=t^3-12t$. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$, and find the coordinates of the points on the curve at which the tangent is horizontal.

   [6 marks]

   Total for question 8: 12 marks

End of questions