17M.2.SL.TZ1.T_3

pestleMathematicsAISLPaper 217M· sl-2-5-modelling-functionssource ↗

Consider the function $f(x) = 0.3{x^3} + \frac{{10}}{x} + {2^{ - x}}$ .

Consider a second function, $g(x) = 2x - 3$ .

Calculate $f(1)$ .

[2]

Sketch the graph of $y = f(x)$ for $- 7 \leqslant x \leqslant 4$ and $- 30 \leqslant y \leqslant 30$ .

[4]

Write down the equation of the vertical asymptote.

[2]

Write down the coordinates of the $x$ -intercept.

[2]

Write down the possible values of $x$ for which $x < 0$ and $f’(x) > 0$ .

[2]

Find the solution of $f(x) = g(x)$ .

[2]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$0.3{(1)^3} + \frac{{10}}{1} + {2^{ - 1}}$ (M1)

Note: Award (M1) for correct substitution into function.

$= 10.8$ (A1)(G2)

[2 marks]

M17/5/MATSD/SP2/ENG/TZ1/03.b/M (A1)(A1)(A1)(A1)

Note: Award (A1) for indication of correct window and labelled axes.

Award (A1) for correct shape and position for $x < 0$ (with the local maximum, local minimum and $x$ -intercept in relative approximate location in ${{\text{3}}^{{\text{rd}}}}$ quadrant).

Award (A1) for correct shape and position for $x > 0$ (with the local minimum in relative approximate location in ${{\text{1}}^{{\text{st}}}}$ quadrant).

Award (A1) for smooth curve with indication of asymptote (graph should not touch $y$ -axis and should not curve away from the $y$ -axis). The asymptote is only assessed in this mark.

[4 marks]

$x = 0$ (A2)

Note: Award (A1) for “ $x = {\text{(a constant)}}$ ” and (A1) for “ ${\text{(a constant)}} = 0$ ”.

The answer must be an equation.

[2 marks]

$( - 6.18,{\text{ }}0){\text{ }}( - 6.17516 \ldots ,{\text{ }}0)$ (A1)(A1)

Note: Award (A1) for each correct coordinate. Award (A0)(A1) if parentheses are missing.

[2 marks]

$- 4.99 < x < - 2.47{\text{ }}( - 4.98688 \ldots < x < - 2.46635 \ldots )$ (A1)(A1)

Note: Award (A1) for both correct end points, (A1) for strict inequalities used with 2 endpoints.

[2 marks]

$0.3{x^3} + \frac{{10}}{x} + {2^{ - x}} = 2x - 3$ (M1)

Note: Award (M1) for equating the expressions for $f$ and $g$ or for the line $y = 2x - 3$ sketched (positive gradient, negative $y$ -intercept) on their graph from part (a).

$(x = ){\text{ }} - 1.34{\text{ }}( - 1.33650 \ldots )$ (A1)(G2)

Note: Award a maximum of (M1)(A0) or (G1) for coordinate pair seen as final answer.

[2 marks]

Examiners’ report

[N/A]