19M.1.SL.TZ1.S_5

pestleMathematicsAASLPaper 119M· sl-5-10-indefinite-integration-reverse-chain-by-substitutionsource ↗

The derivative of a function $f$ is given by $f'(x) = 2{{\text{e}}^{ - 3x}}$ . The graph of $f$ passes through $\left( {\frac{1}{3}{\text{,}}\,\,5} \right)$ .

Find $f(x)$ .

Markscheme / solution

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

recognizing to integrate (M1)

eg $\int {f'}$ , $\int {2{{\text{e}}^{ - 3x}}{\text{d}}x}$ , ${\text{d}}u = - 3$

correct integral (do not penalize for missing + $C$ ) (A2)

eg $- \frac{2}{3}{{\text{e}}^{ - 3x}} + C$

substituting $\left( {\frac{1}{3}{\text{,}}\,\,5} \right)$ (in any order) into their integrated expression (must have + $C$ ) M1

eg $- \frac{2}{3}{{\text{e}}^{ - 3\left( {1/3} \right)}} + C = 5$

Note: Award M0 if they substitute into original or differentiated function.

$f(x) = - \frac{2}{3}{{\text{e}}^{ - 3x}} + 5 + \frac{2}{3}{{\text{e}}^{ - 1}}$ (or any equivalent form, eg $- \frac{2}{3}{{\text{e}}^{ - 3x}} + 5 - \frac{2}{{ - 3{\text{e}}}}$ ) A1 N4

[5 marks]

Examiners’ report

[N/A]