CalculusIB Maths: Applications and Interpretation SL: Topic test

1. 1
   The number of bacteria in a culture, $N$ thousand, $t$ hours after the start of an experiment is modelled by $N(t)=0.5t^2+2t$. A student uses a GDC to find the average rate of change of $N$ between $t=3$ and $t=3+h$. The results are $5.05$ for $h=0.1$, $5.005$ for $h=0.01$ and $5.0005$ for $h=0.001$.

   (a)

   Write down the best estimate of the instantaneous rate of change of $N$ at $t=3$, in thousand bacteria per hour.

   [1 mark]

   • A$5.05$
   • B$5$
   • C$10.5$
   • D$2$

   (b)

   Which statement about the value found in part (a) is correct?

   [1 mark]

   • AIt is the value of $N$ when $t=3$.
   • BIt is the average rate of change of $N$ over the first 3 hours.
   • CIt is the gradient of the tangent to the graph of $N$ at $t=3$.
   • DIt is the area under the graph of $N$ between $t=0$ and $t=3$.

   (c)

   Differentiate $N(t)$ and hence show that the instantaneous rate of change at $t=3$ agrees with your answer to part (a).

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A function $p$ is defined for $-5\le x\le5$ and has derivative $p'(x)=8+2x-x^2$.

   (a)

   Find the interval on which $p$ is increasing.

   [1 mark]

   • A$-2<x<4$
   • B$x<-2$ or $x>4$
   • C$-4<x<2$
   • D$x<4$

   (b)

   Which statement about the graph of $p$ at $x=4$ is correct?

   [1 mark]

   • AIt has a local minimum.
   • BIt has a point of inflection, because the gradient is zero.
   • CThe function is decreasing at $x=4$.
   • DIt has a local maximum.

   (c)

   Find the greatest value of $p'(x)$ and the value of $x$ at which it occurs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=2x^3-5x^2+x+7$. The point $A$ on $C$ has $x$-coordinate $2$.

   (a)

   Find the $y$-coordinate of $A$ and the gradient of $C$ at $A$.

   [3 marks]

   (b)

   Find the equation of the normal to $C$ at $A$. Give your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A solar farm's power output, $P$ kW, $t$ hours after 06:00 is modelled by $P(t)=6t-0.5t^2$ for $0\le t\le12$.

   (a)

   (i) Find $P'(t)$ and hence find the time at which the power output is greatest. [3]
   (ii) Find the greatest power output. [1]
   (iii) The total energy produced, in kWh, is the area under the graph of $P$ for $0\le t\le12$. Write down an integral for this area and use your GDC to evaluate it. [2]

   [6 marks]

   (b)

   (i) Use the trapezoidal rule with 4 intervals of equal width to estimate the total energy produced between 06:00 and 18:00. [3]
   (ii) Using your answer to part (a)(iii), find the percentage error in this estimate, and state, with a reason, whether it is an underestimate or an overestimate. [3]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $k$ is defined by $k(x)=3x^4-\frac{5}{x^2}+x$ for $x\neq0$.

   (a)

   Find $k'(x)$.

   [1 mark]

   • A$12x^3-10x^{-3}+1$
   • B$12x^3+10x^{-1}+1$
   • C$12x^3+10x^{-3}+1$
   • D$12x^3-5x^{-3}+1$

   (b)

   Find the value of $k'(1)$.

   [1 mark]

   • A$23$
   • B$3$
   • C$-1$
   • D$22$

   (c)

   Find the equation of the tangent to the graph of $y=k(x)$ at the point where $x=1$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $f(x)=0.5x^2+1$. The region $R$ is bounded by the graph of $y=f(x)$, the $x$-axis and the lines $x=1$ and $x=4$. The curve $C$ has gradient function $f(x)$ and passes through the point $(2,5)$.

   (a)

   Which expression gives the area of $R$?

   [1 mark]

   • A$\int_0^4(0.5x^2+1)\,dx$
   • B$\int_1^4 x\,dx$
   • C$f(4)-f(1)$
   • D$\int_1^4(0.5x^2+1)\,dx$

   (b)

   Find the area of $R$.

   [1 mark]

   • A$7.5$
   • B$13.5$
   • C$24$
   • D$15.75$

   (c)

   Find the equation of $C$ in the form $y=g(x)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The rate at which a household uses electricity, $P$ kW, is recorded every two hours, starting at midnight. The five readings, taken at $t=0,2,4,6,8$ hours after midnight, are $1.2,\ 0.8,\ 2.4,\ 3.1$ and $1.9$ kW respectively.

   (a)

   Use the trapezoidal rule with all five readings to estimate the total energy used, in kWh, between $t=0$ and $t=8$.

   [3 marks]

   (b)

   The company charges 0.40 AED per kWh.
   (i) Find the cost of the energy estimated in part (a). [2]
   (ii) A meter later shows the true energy used was 16.4 kWh. Find the percentage error in the estimate, correct to 3 significant figures. [2]

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A bakery produces $x$ hundred loaves per day, where $x>0$. The average cost per loaf, $A$ AED, is modelled by $A(x)=0.02x+\frac{18}{x}$.

   (a)

   (i) Find $A'(x)$. [2]
   (ii) Find the number of loaves per day that minimises the average cost. [2]
   (iii) Find the minimum average cost per loaf. [2]

   [6 marks]

   (b)

   (i) Find the equation of the tangent to the graph of $y=A(x)$ at $x=20$. [4]
   (ii) Find the values of $x$ for which the average cost is decreasing. [2]

   [6 marks]

   Total for question 8: 12 marks

End of questions