IB Questionbank

User interface language: English | Español

Date	May 2014	Marks available	2	Reference code	14M.2.sl.TZ1.5
Level	SL only	Paper	2	Time zone	TZ1
Command term	Find	Question number	5	Adapted from	N/A

Question

The population of deer in an enclosed game reserve is modelled by the function \(P(t) = 210\sin (0.5t - 2.6) + 990\), where \(t\) is in months, and \(t = 1\) corresponds to 1 January 2014.

Find the number of deer in the reserve on 1 May 2014.

[3]

Find the rate of change of the deer population on 1 May 2014.

[2]

b(i).

Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.

[1]

b(ii).

Markscheme

\(t = 5\) (A1)

correct substitution into formula (A1)

eg \(210\sin (0.5 \times 5 - 2.6) + 990,{\text{ }}P(5)\)

\(969.034982 \ldots \)

969 (deer) (must be an integer) A1 N3

[3 marks]

evidence of considering derivative (M1)

eg \(P'\)

\(104.475\)

\(104\) (deer per month) A1 N2

[2 marks]

b(i).

(the deer population size is) increasing A1 N1

[1 mark]

b(ii).

Examiners report

[N/A]

b(i).

[N/A]

b(ii).

Syllabus sections

Topic 6 - Calculus » 6.1 » Derivative interpreted as gradient function and as rate of change.

Show 56 related questions

12N.1.sl.TZ0.4a: Find \(f'(x)\) .
12N.1.sl.TZ0.4b: The graph of \(f\) has a gradient of \(3\) at the point P. Find the value of \(a\) .
12M.1.sl.TZ1.3a: Write down \(f'(x)\) .
12M.1.sl.TZ1.3b(i) and (ii): The tangent to the graph of f at the point \({\text{P}}(0{\text{, }}b)\) has gradient m...
12M.1.sl.TZ1.3c: Hence, write down the equation of this tangent.
10N.1.sl.TZ0.2a: Find \(g'(x)\) .
10N.1.sl.TZ0.2b: Find the gradient of the graph of g at \(x = \pi \) .
10N.1.sl.TZ0.10a(i), (ii) and (iii): (i) Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) . (ii) ...
10N.1.sl.TZ0.10b: Given that the area of T is \(2k + 4\) , show that k satisfies the equation...
09N.2.sl.TZ0.5: Consider the curve with equation \(f(x) = p{x^2} + qx\) , where p and q are constants. The...
09M.1.sl.TZ1.3: Let \(f(x) = {{\rm{e}}^x}\cos x\) . Find the gradient of the normal to the curve of f at...
09M.2.sl.TZ1.10c: The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} - 2)\) is parallel to...
09M.2.sl.TZ1.10d: The graph of f is decreasing for \(p < x < q\) . Find the value of p and of q.
09M.2.sl.TZ2.6a: Write down the gradient of the curve at P.
10M.2.sl.TZ1.9a: Show that \(A = 10\) .
10M.2.sl.TZ1.9b: Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of k.
10M.2.sl.TZ1.9c(i), (ii) and (iii): (i) Using your value of k , find \(f'(x)\) . (ii) Hence, explain why f is a...
10M.2.sl.TZ1.9d: Let \(g(x) = - {x^2} + 12x - 24\) . Find the area enclosed by the graphs of f and g .
SPNone.2.sl.TZ0.2c: Write down the gradient of the graph of f at \(x = 3\) .
11N.1.sl.TZ0.9c: Find \(f'(x)\) .
11N.1.sl.TZ0.9a(i), (ii) and (iii): Use the graph to write down the value of (i) a ; (ii) c ; (iii) d .
11N.1.sl.TZ0.9b: Show that \(b = \frac{\pi }{4}\) .
11N.1.sl.TZ0.9d: At a point R, the gradient is \( - 2\pi \) . Find the x-coordinate of R.
11N.2.sl.TZ0.10a: Sketch the graph of f .
11N.2.sl.TZ0.10c: Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .
11N.2.sl.TZ0.10b(i) and (ii): (i) Write down the x-coordinate of the maximum point on the graph of f . (ii) Write...
11N.2.sl.TZ0.10d: Find the interval where the rate of change of f is increasing.
11M.2.sl.TZ1.8d: In the first rotation, there are two values of t when the bucket is descending at a rate of...
11M.2.sl.TZ1.8a: Show that \(a = 4\) .
11M.2.sl.TZ1.8b: The wheel turns at a rate of one rotation every 30 seconds. Show that...
11M.2.sl.TZ1.8c: In the first rotation, there are two values of t when the bucket is descending at a rate of...
13M.1.sl.TZ1.3b: Find the gradient of the curve of \(f\) at \(x = \frac{\pi }{2}\) .
13M.1.sl.TZ1.10b: Find the set of values of \(x\) for which \(f\) is increasing.
13M.2.sl.TZ1.9e: Find the maximum rate of change of \(f\) .
13M.2.sl.TZ2.10b: Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the...
14M.2.sl.TZ1.5b(ii): Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.
13N.2.sl.TZ0.7b(i): Find the rate of change of area when \(x = 2\).
13N.2.sl.TZ0.7b(ii): The area is decreasing for \(a < x < b\). Find the value of \(a\) and of \(b\).
15M.2.sl.TZ1.6: Let \(f(x) = \frac{{\ln (4x)}}{x}\) for \(0 < x \le 5\). Points...
15N.1.sl.TZ0.10b: Find the set of values of \(x\) for which the graph of \(f\) is concave down.
16M.2.sl.TZ2.9a: Write down the equation of the horizontal asymptote of the graph of \(f\).
16N.2.sl.TZ0.10b: (i) Write down the value of \(k\). (ii) Find \(g(x)\).
16M.2.sl.TZ2.9b: Find \(f'(x)\).
16N.2.sl.TZ0.10a: (i) Find the value of \(c\). (ii) Show that \(b = \frac{\pi }{6}\). (iii) Find...
16N.2.sl.TZ0.10c: (i) Find \(w\). (ii) Hence or otherwise, find the maximum positive rate of change of...
16M.2.sl.TZ2.9d: Given that \(g'(1) = - e\), find the value of \(a\).
16M.2.sl.TZ2.9c: Write down the value of \(b\).
16M.2.sl.TZ2.9e: There is a value of \(x\), for \(1 < x < 4\), for which the graphs of \(f\) and \(g\)...
17M.2.sl.TZ2.8a: Find the value of \(p\).
17M.2.sl.TZ2.8b.i: Write down the coordinates of A.
17M.2.sl.TZ2.8b.ii: Write down the rate of change of \(f\) at A.
17M.2.sl.TZ2.8c.i: Find the coordinates of B.
17M.2.sl.TZ2.8c.ii: Find the the rate of change of \(f\) at B.
17M.2.sl.TZ2.8d: Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\)...
17N.1.sl.TZ0.5a: Find \((g \circ f)(x)\).
17N.1.sl.TZ0.5b: Given that \(\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3\), find the...

Hide related questions

Question

Markscheme

Examiners report

Syllabus sections

View options