IB DP Maths Topic 2.2 Use of technology to graph a variety of functions, including ones not specifically mentioned SL Paper 2

 

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Question

Let \(f(x) = 4{\tan ^2}x – 4\sin x\) , \( – \frac{\pi }{3} \le x \le \frac{\pi }{3}\) .

On the grid below, sketch the graph of \(y = f(x)\) .


[3]
a.

Solve the equation \(f(x) = 1\) .

[3]
b.
Answer/Explanation

Markscheme

     A1A1A1     N3

Note: Award A1 for passing through \((0{\text{, }}0)\), A1 for correct shape, A1 for a range of approximately \( – 1\) to 15.

[3 marks]

a.

evidence of attempt to solve \(f(x) = 1\)     (M1)

e.g. line on sketch, using \(\tan x = \frac{{\sin x}}{{\cos x}}\)

\(x = – 0.207\) , \(x = 0.772\)     A1A1     N3

[3 marks]

b.

Question

Let \(f(x) = x\cos x\) , for \(0 \le x \le 6\) .

Find \(f'(x)\) .

[3]
a.

On the grid below, sketch the graph of \(y = f'(x)\) .


[4]
b.
Answer/Explanation

Markscheme

evidence of choosing the product rule     (M1)

e.g. \(x \times ( – \sin x) + 1 \times \cos x\)

\(f'(x) = \cos x – x\sin x\)     A1A1     N3

[3 marks]

a.


     A1A1A1A1     N4

Note: Award A1 for correct domain, \(0 \le x \le 6\) with endpoints in circles, A1 for approximately correct shape, A1 for local minimum in circle, A1 for local maximum in circle.

[4 marks]

b.

Question

Consider \(f(x) = 2 – {x^2}\) , for \( – 2 \le x \le 2\) and \(g(x) = \sin {{\rm{e}}^x}\) , for \( – 2 \le x \le 2\) . The graph of f is given below.


On the diagram above, sketch the graph of g.

[3]
a.

Solve \(f(x) = g(x)\) .

[2]
b.

Write down the set of values of x such that \(f(x) > g(x)\) .

[2]
c.
Answer/Explanation

Markscheme


     A1A1A1     N3

[3 marks]

a.

\(x = – 1.32\) , \(x = 1.68\) (accept \(x = – 1.41\) , \(x = 1.39\) if working in degrees)     A1A1     N2

[2 marks]

b.

\( – 1.32 < x < 1.68\) (accept \( – 1.41 < x < 1.39\) if working in degrees)     A2     N2

[2 marks]

c.

Question

The velocity v ms−1 of an object after t seconds is given by \(v(t) = 15\sqrt t  – 3t\) , for \(0 \le t \le 25\) .

On the grid below, sketch the graph of v , clearly indicating the maximum point.


[3]
a.

(i)     Write down an expression for d .

(ii)    Hence, write down the value of d .

[4]
b(i) and (ii).
Answer/Explanation

Markscheme


     A1A1A1     N3

Note: Award A1 for approximately correct shape, A1 for right endpoint at \((25{\text{, }}0)\) and A1 for maximum point in circle.

[3 marks]

a.

(i) recognizing that d is the area under the curve     (M1)

e.g. \(\int {v(t)} \)

correct expression in terms of t, with correct limits     A2     N3

e.g. \(d = \int_0^9 {(15\sqrt t } – 3t){\rm{d}}t\) , \(d = \int_0^9 v {\rm{d}}t\)

(ii) \(d = 148.5\) (m) (accept 149 to 3 sf)     A1     N1

[4 marks]

b(i) and (ii).

Question

Let \(g(x) = \frac{1}{2}x\sin x\) , for \(0 \le x \le 4\) .

Sketch the graph of g on the following set of axes.


[4]
a.

Hence find the value of x for which \(g(x) = – 1\) .

[2]
b.
Answer/Explanation

Markscheme


     A1A1A1A1     N4

Note: Award A1 for approximately correct shape, A1 for left end point in circle, A1 for local maximum in circle, A1 for right end point in circle.

[4 marks]

a.

attempting to solve \(g(x) = – 1\)     (M1)

e.g. marking coordinate on graph, \(\frac{1}{2}x\sin x + 1 = 0\)

\(x = 3.71\)     A1     N2

[2 marks]

b.

Question

Consider the expansion of \({(x + 2)^{11}}\) .

Write down the number of terms in this expansion.

[1]
a.

Find the term containing \({x^2}\) .

[4]
b.
Answer/Explanation

Markscheme

12 terms     A1     N1

[1 mark]

a.

evidence of binomial expansion     (M1)

e.g. \(\left( \begin{array}{l}
n\\
r
\end{array} \right){a^{n – r}}{b^r}\) , an attempt to expand, Pascal’s triangle

evidence of choosing correct term     (A1)

e.g. 10th term , \(r = 9\) , \(\left( {\begin{array}{*{20}{c}}
{11}\\
9
\end{array}} \right)\) , \({(x)^2}{(2)^9}\)

correct working     A1

e.g. \(\left( {\begin{array}{*{20}{c}}
{11}\\
9
\end{array}} \right){(x)^2}{(2)^9}\) , \(55 \times {2^9}\)

\(28160{x^2}\)     A1     N2

[4 marks]

b.
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